Þ ¥bonds€¬cell_resultsÞ TÙ$1a52ce21-ada0-4dea-b484-e6c4393328d4Š¦queued¤logs‘ˆ¤lineÿ£msg’Ù( 6.245 ms (556 allocations: 71.03 MiB)
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⎣⣿⣿⣿⣿⣿⣿⣿⣿⣿⎦ªtext/plain¤type¥Tuple¨objectid°82685dc88950114a¤mimeÙ!application/vnd.pluto.tree+object¬rootassigneeÀ²last_run_timestampËAÚ»»¹°persist_js_state·has_pluto_hook_features§cell_idÙ$1a52ce21-ada0-4dea-b484-e6c4393328d4¹depends_on_disabled_cells§runtimeÏ à’|صpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$acf9f7a7-300f-423d-8220-6014327ec39bŠ¦queued¤logs§running¦output†¤bodyÙ)get_ð“” (generic function with 1 method)¤mimeªtext/plain¬rootassigneeÀ²last_run_timestampËAÚ»® Zî°persist_js_state·has_pluto_hook_features§cell_idÙ$acf9f7a7-300f-423d-8220-6014327ec39b¹depends_on_disabled_cells§runtimeÎ ›µpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$42e36f72-e02c-435f-8fa6-30309b3ee3e6Š¦queued¤logs§running¦output†¤bodyÙ1get_steady_state (generic function with 1 method)¤mimeªtext/plain¬rootassigneeÀ²last_run_timestampËAÚ»ªOÿ.°persist_js_state·has_pluto_hook_features§cell_idÙ$42e36f72-e02c-435f-8fa6-30309b3ee3e6¹depends_on_disabled_cells§runtimeÎ Œ‹pµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$3107d9de-b4ec-4954-8f09-4e1896c83f62Š¦queued¤logs§running¦output†¤bodyÙgFor example, we can now also plot the response of the nominal rate:
The only slightly tricky thing is that for borrowing-constrained households, we can't find consumption through an inverted Euler equation. For these, we need to solve the above optimality condition for labor supply numerically. The function below (used in EGM_update above) does that using fixed-point iteration:
The model
The description (mostly) follows the ABRS paper.
Households
Skill $s$ follows a time-invariant discrete Markov Chain with $n_s$ states $\mathcal{S}:=\lbrace s_1,s_2,...,s_{n_s}\rbrace$ and exogenous transition probabilities $P(s',s)$ . This introduces the income risk. $\pi_s$ denotes the stationary distribution of $P$ . The average labor productivity $\int^1_0 s_{it} di$ is invariant and will be normalized to 1.
Skill $s$ does not only determine an households income per hour worked, but also the amount of lump-sum taxes she has to pay and the amount of dividends she receives, both of which are distributed proportionally to $s$ .
$$V_t(s,a) = \max_{a',n} \frac{c^{1-\sigma}-1}{1-\sigma} - \varphi\frac{n^{1+\nu}}{1+\nu} +\beta \sum_{s'\in \mathcal{S}}P(s',s)V_{t+1}(s',a') $$
$$\text{subject to}~~c+a' = (1+r_t)a + w_t n s - \underbrace{(d_t - \tau_t)}_{=\text{net transfer}}s ~~\text{and}~~a'\geq \bar{a}$$
Firms
A competitive final goods firm assembles its output using a CES production function $Y_t = (\int^1_0 y_{jt}^{\frac{1}{\mu}} dj)^{\mu}$ , giving rise to a standard CES-demand system for the continuum of intermediate goods. These intermediates, in turn, are supplied by monopolists who produce using only labor s.t. $y_{jt}=Z_t l_{jt}$ and are subject to quadratic price adjustment costs a la Rotemberg (1982). $Z_t$ denotes aggregate productivity that may be time-varying. In a symmetric equilibrium, gross inflation $1+\pi_t = P_t/P_{t-1}$ can then be shown to evolve according to a New Keynesian Phillips curve of the form
$$\log(1+\pi_t) = \kappa\left(\frac{w_t}{Z_t}- \frac{1}{\mu} \right) + \mathbb{E}_t\frac{1}{1+r_{t+1}}\frac{Y_{t+1}}{Y_{t}}\log(1+\pi_{t+1}).$$
Here $\kappa$ is the parameter governing firms' price adjustment costs, $\mu$ the steady state markup and $Y_t$ aggregate output, which is equal to the monopolists' individual output by symmetry. The firms' profits are distributed lump-sum to households, thus $d_t = Y_t - w_tL_t$ .
Policy
The model features a government consisting of two branches, a monetary and a fiscal authority. The former sets the nominal interest rate $i_t$ according to a Taylor rule of the form
$$i_t = r^* + \theta_\pi \pi_t + \epsilon^i_t$$
where $r^*$ is the economy's long run "natural rate" and $\epsilon^i_t$ a monetary policy shock. The real interest rate in period $t$ is determined by the previously set nominal rate and inflation so that $1+r_t = \frac{1+i_{t-1}}{1+\pi_t}$ .
The fiscal authority, in turn, issues a constant amount of government bonds $B$ each period and collects the lump sum taxes $\tau_t$ already mentioned above. Since there are no other spending items, it chooses the tax rate to cover the its interest rate payments every period, i.e.
$$\tau_t = r_t B~~.$$
Market clearing
In an equilibrium, the following market clearing conditions will have to hold:
$$B = \int^1_0 a_{it} di $$
$$\frac{Y_t}{Z_t} = L_t = \int^1_0 s_{it} n_{it} di $$
$$Y_t = \underbrace{\sum_{s\in \mathcal{S}}\int c(s,a) D(s,a)da}_{=\text{aggregate consumption}} - \underbrace{\frac{\mu}{\mu-1}\frac{1}{2\kappa}\left(\log(\pi_t)\right)^2Y_t}_{=\text{price adjustment costs}}$$
In this model, there are three markets that need to clear: Firstly, an asset market on which the sum of households' net savings need to equal the aggregate amount of bonds $B$ supplied by the government. Secondly a labor market on which the effective amount of labor supplied by the households' at the piece-rate real wage $w_t$ needs to equal the labor demanded by the firm. Finally, the goods market will have to clear, requiring aggregate consumption and real price adjustment costs to equal aggregate output. As is well known, price adjustment costs don't matter in linearized models and thus the latter will not be of concern here.
Calibration:
The calibration follows the example model in the ABRS paper and can be seen in the code block below. The skill process follows an AR(1)-process in logs that will be discretized to 7 points using the Rouwenhorst method (the function implementing the Rouwenhorst method is off-the-shelf from QuantEcon.jl
A "Dynare-like" method
Finally, reading the recent paper by Bhandari et al. (2023) features some analytical results suggesting yet another way to get the Aggregate Jacobians of $F$ . I will use a somewhat different formulation to aid exposition, but that is where I got the inspiration from.
Instead of considering a big system $F$ , they represent the model as a set of equations that need to hold in every period, as state space approaches implemented by DSGE modelling packages like Dynare $G$ that needs to hold every period, i.e. a perfect foresight equilibrium is defined as sequences $X$ so that
$$G(\tilde{X}_t,\tilde{Z}_t,x_t)=0~~\text{with}~~x_t = H(\lbrace X_t\rbrace_{t=0}^\infty,\lbrace Z_t\rbrace_{t=0}^\infty)~~\forall t$$
for exogenously given $Z$ , where $\tilde{X_t}=[X_{t-1},X_t,X_{t+1}]$ (and the same for $Z$ ) and $x_t$ denotes a set of variables that are the outcomes of the model's HA block, which I denote by $H$ . In the model studied here, $x_t$ would be asset demand and labor supply.
Differentiating the LHS w.r.t $\tilde{X_t}$ , we have
$$G_{\tilde{X}} + G_x\mathcal{J}_{t,\tilde{X}}~~\forall~t$$
and a similar expression if differentiating w.r.t. $Z$ . I am obviously abusing notation here.
Anyway, the point is that we can construct the Aggregate Jacobians $F_X$ and $F_Z$ by writing down the aggregate model part $G$ in a Dynare-like fashion. The below code Julia function displays how the function $G$ can look like if we set it up for all model variables (not just $Ï€$ , $w$ and $Y$ ):
GE Jacobians: The DAG approach
Having obtained the SSJ's of the HA block, we now want to obtain the aggregate Jacobians $F_X$ and $F_Z$ . For the former, it is often desirable to state it in terms of the minimum number of variables necessary: $F_X$ will have dimension $n_x \times T$ , where $T$ is the truncation horizon and $n_x$ the number of variables in $X$ . If $n_x$ becomes high, $F_X$ can become quite large and in turn computationally costly to invert.
To get $F_X$ and $F_Z$ , can we follow the approach outlined in ABRS, which is to split the model into different "blocks" and then chaining their derivatives along a Directed Acyclical Graph (DAG). For the model used here, the ABRS paper proposes a DAG as displayed below:
¤mimeimage/svg+xml¬rootassigneeÀ²last_run_timestampËAڻǸI•°persist_js_state·has_pluto_hook_features§cell_idÙ$57b4a641-8cf0-49ad-bc4c-c85e7071e355¹depends_on_disabled_cells§runtimeÎœ²Vµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$0303d80f-a072-4a51-bfaf-d37f721fe82fŠ¦queued¤logs§running¦output†¤bodyÙ+build_ð’¥ (generic function with 1 method)¤mimeªtext/plain¬rootassigneeÀ²last_run_timestampËAÚ»® Â;°persist_js_state·has_pluto_hook_features§cell_idÙ$0303d80f-a072-4a51-bfaf-d37f721fe82f¹depends_on_disabled_cells§runtimeÎ [& µpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$60fa7af4-1bd9-4acb-bfe0-61801317b177Š¦queued¤logs§running¦output†¤bodyÙ6solve_c_n_constrained (generic function with 1 method)¤mimeªtext/plain¬rootassigneeÀ²last_run_timestampËAÚ»¨#Ït°persist_js_state·has_pluto_hook_features§cell_idÙ$60fa7af4-1bd9-4acb-bfe0-61801317b177¹depends_on_disabled_cells§runtimeÎ.Ìeµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$fb7958ac-014b-40cb-9020-7d6f645a7f87Š¦queued¤logs‘ˆ¤lineÿ£msg’Ù) 13.750 ms (364 allocations: 80.30 MiB)
ªtext/plain§cell_idÙ$fb7958ac-014b-40cb-9020-7d6f645a7f87¦kwargs¢id´PlutoRunner_d1acb81e¤fileÙP/home/runner/.julia/packages/Pluto/F1RAN/src/runner/PlutoRunner/src/io/stdout.jl¥group¦stdout¥level®LogLevel(-555)§running¦output†¤body ¤mimeªtext/plain¬rootassigneeÀ²last_run_timestampËAÚ»¿ñ°persist_js_state·has_pluto_hook_features§cell_idÙ$fb7958ac-014b-40cb-9020-7d6f645a7f87¹depends_on_disabled_cells§runtimeÏ ÓùMµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$3e960018-54bf-4331-9ac0-e130e2393778Š¦queued¤logs§running¦output†¤bodyÚsArmed with the above functions, we can obtain steady state household policies by iterating over the EGM_update function for given $w$ , $r$ and net transfers.
For convenience below, I also define a version of the function that starts with a hard-coded initial guess.
The aggregate representation is terms of just three aggregate variables, output $Y$ , the real wage $w$ and inflation $Ï€$ and a corresponding number of GE conditions, which consist of the NK Phillips curve as well as labor- and asset market clearing conditions as stated above. However, these variables imply others which enter the different blocks that in turn provide additional inputs for more blocks and so on.
Either the way, the below function implements obtains the Aggregate Jacobians by chaining along the DAG "manually". Since all the blocks except the HA one have straightforward analytical derivatives, I just directly enter the analytical ones but one could equivalently obtain them by using some differentiation packages.
For simplicity, I will assume all exogenous variables except the monetary policy shock (represented by $r^*$ in the ABRS DAG) fixed here. The implementation of further shocks is left as exercise to the reader (I always wanted to write that phrase somewhere ).
Furthermore, I disregard terms w.r.t. the price adjustment costs since these turn out to be 0 anyway (feel free to check for yourself).
So, can we implement the above method for a reduced number of aggregate variables, say, again $Ï€$ , $w$ and $Y$ ? Indeed, I will demonstrate that below.
Solving the Household problem
For solving the household problem, we will again use the Endogenous Grid Method (EGM), which has to be modified to account for household's endogenous labor supply:
For given next period marginal value functions/consumption policy functions, we can still invert the Euler equation, but due to the endogenous labor supply choice, we don't immediately know how much wealth an household must have had to chose that consumption policy. However, we can link household's consumption and labor supply through the respective optimality condition
$$\varphi n_{it}^\nu = w_t s_{it} c_{it}^{-\sigma}~~,
$$
which requires the marginal disutility of working more to equal the marginal utility of the income the household earns from doing so. We can then implement the EGM backward step with the function below:
Given that it gives us what we want and seems easy, why not always use this method?
The issue is that the computational cost of getting the Jacobians of $F$ is much larger with the direct method, even if we use the HA Jacobian: In terms of computation time, for my implementations the direct method takes more than 50-100 times as long as the DAG method:
(Note that computation times displayed in the uploaded notebook differ from those I obtained on my laptop, as the code is run on a github server.)
Sequence Space Jacobians: Heterogeneous Agents (HA) Block
Having obtained the Steady State of our economy, we now wish to analyze the effects of a monetary policy shock to the central bank's nominal interest rate $i$ .
To do that, let's briefly recap the SSJ method: It is based on the idea that an equilibrium in the sequence space can be expressed as a solution to a system of the form
$$F(X,Z) = 0$$
where $X$ denotes the time-paths of a vector of endogenous variables and $Z$ the time-paths of a vector of exogenous shocks. To get linearized Impulse Response Functions (IRFs), we can truncate the system $F$ , compute its Jacobians with respect to $X$ and $Z$ and then obtain the IRFs as $dX = -F_X^{-1}F_Z dZ$ by the Implicit Function Theorem.
Now, the problem is that in an heterogenous agent model, $F$ is potentially quite costly to evaluate, as within the system we need to keep track of household's policy function along the grids and the joint income- and wealth distribution. To overcome the issue, ABRS proposed the efficient "fake news"-algorithm to separately linearize a model "block" containing these complicated objects.
Afterwards, one can then assemble $F_X$ and $F_Z$ around that "HA block". As said, this notebook deals with different ways to do so, but first, let us get the HA Jacobians using the ABRS "fake news" algorithm. Explaining in detail how it works would require quite a bit of text, so read the paper for that. As in the previous notebook, I'll just show you the implementation.
First define a function to get the object denoted as $\mathcal{E}$ in their paper:
With that, we can proceed as in the previous section. Instead of just the selection matrix P there are a few more matrices featuring derivatives of G_hetinput that need to be multiplied with the HA Block Jacobian, which now also appears in the expression for $F_Z$ . Overall, though, everything works according to the same principle.
¤mimeimage/svg+xml¬rootassigneeÀ²last_run_timestampËAÚ»µ\x‰°persist_js_state·has_pluto_hook_features§cell_idÙ$ffc02cf0-d10f-4dcb-a33a-9705ac6ccb12¹depends_on_disabled_cells§runtimeΨٵpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$0610663b-a8a9-4182-86eb-98093e77d35bŠ¦queued¤logs§running¦output†¤bodyÚûThe function takes as inputs current, lagged and (expected) next period values of the endogenous variables and shocks as well as the outputs of the HA block, denoted as het_out in the code.
As for State Space approaches, we need to ensure that the timing of the model is correct and the set of equation sufficient to fully characterize the equilibrium. Otherwise you'll get an error or weird results.
Now, to get the aggregate Jacobians of the model, we just need to differentiate G_fun with respect to the relevant inputs, set up some $(n_x\cdot T)\times (n_x\cdot T)$ matrices that have these derivatives at the right places, multiply or add them to the HA Block Jacobian $\mathcal{J}$ and we are done. It might require a bit of thinking to really understand how to do that correctly, in the function below I tried to do it in the simplest way possible.
"Dynare"-like method: Alternative Formulation
Compared to the case above, an issue is that the minimum set of aggregate variables necessary to characterize the dynamics of the economy may not contain all the inputs of the HA block.
For example, the households' wages and assset returns may both be a function of the aggregate capital stock. Or, for this model, if we again choose $Ï€$ , $w$ and $Y$ as aggregate variables, the household does not care about inflation per se but rather the real return it induces in conjunction with the nominal rate set by the central bank.
If that is the case, compared to above, we need to write an additional function that takes the aggregate $X$ and shocks $Z$ as inputs and returns the variables the housheolds actually care about (i.e. the inputs of the HA Block). The function below does just that:
With this, we can now get our HA Jacobians. We have 3 blocks inputs ($r$ , $w$ and the net transfers $d-Ï„$ ) and two outputs (aggregate labor supply $\mathcal{N}=\int^1_0 s_{i}n_{i}di$ and net savings $\mathcal{A}=\int^1_0 a_i di$ ), so need to construct 6 Sequence Space Jacobians in total. The code below does that using the functions defined above.
Everything is the same, as one would hope. And getting the aggregate Jacobians is now essentially as fast as for the DAG method:
We see that if the accommodative nominal interest rate shock is really persistent, inflation may react so much that the Central Bank is induced to actually raise the nominal rate upon impact of the shock. The real rate still decreases on impact though:
¤mimeimage/svg+xml¬rootassigneeÀ²last_run_timestampËAÚ»³¿²F°persist_js_state·has_pluto_hook_features§cell_idÙ$f7537f35-7549-4cdc-994d-616394084d3c¹depends_on_disabled_cells§runtimeΑuüVµpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$c23a83c1-4009-4378-b229-27b56cf4f640Š¦queued¤logs§running¦output†¤bodyÚ]Reassuringly, the responses look as in the ABRS notebook .
GE Jacobians: Direct Method with HA Jacobians
While this is certainly a matter of taste, I dislike implementing the DAG method manually for models that are a bit more complicated, mainly because setting up the DAG structure, differentiating all the different blocks and chaining them starts feeling tedious at some point.
Thus, I have been exploring different approaches to getting the aggregate Jacobians, which I'd like to share with the hope that others may find them useful as well.
I'll start with an alternative Method, which I'd call "Direct Methodd with HA Jacobian" and which is actually briefly outlined in the ABRS paper at the end of Section 4.1. While ABRS state it to be "less accurate", we will see that this concern is eliminated by the use of an AutoDiff package such as ForwardDiff.jl.
The idea is to set up the aggregate system of equation $F$ non-linearily but replace the non-linear HA-block with an linearized approaximation based on the HA SSJs we obtained using the "fake news" method. For the model analyzed here, this would look as follows:
As you may expect, I will again compare the resulting $F_X$ , $F_Z$ and inflation response to the DAG method.
Again, we will construct the GE Jacobian in terms of same three aggregate variables that fully characterize the aggregate dynamics of our model economy: The above function takes as inputs time-paths of these variables as well as the monetary policy shock and returns the residuals of the "target" equations along the time path.
What I like about this approach is that you don't need to think that much about how to set up the DAG and you don't have to differentiate and chain a lot of small blocks separately. And it gives us the same results:
That time difference shouldn't matter much if you are just interested in solving a model a few times for a given parameterization: After all, approx. 1 second for the direct method is not a long time. However, it will start to matter if you need to compute the Jacobians many times for estimating the model.
Thus, I think this "direct method" may be useful because it easy to implement and a) often models are just calibrated and not estimated anyway and b) even if that is the case, having a simple-to-implement additional method may be useful to check that a manually implemented DAG method gives the right results.
Conclusion
I hope you found this notebook useful. If yes, please consider starring its Github repo and share it with colleagues who might also find it useful. Feedback and suggestions are also very welcome.
Finally, since you seem interested in Heterogeneous Agent Macro, have a look at my Personal Website to learn about my research in this area and/or to get in contact with me.
¤mimeimage/svg+xml¬rootassigneeÀ²last_run_timestampËAÚ»¼çþ °persist_js_state·has_pluto_hook_features§cell_idÙ$48e1498e-2d8b-4051-b90c-799ab37c3338¹depends_on_disabled_cells§runtimeÎÀÿ'§µpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$a739d105-e648-420b-97b5-435532875218Š¦queued¤logs§running¦output†¤bodyÙŽIn addition to that, we just need a now smaller set of aggregate equations characterizing the equilibrium:
Overall, I feel this method provides a good compromise between the speed of the DAG method and the convenience of the Dynare-like method.
A downside compared to the former is that it is perhaps not as straightforward to use what ABRS call "Solved Blocks", although it can be adapted to some extent: If it were desirable to have one such "block" in a richer version of this model, one could e.g. add an additional set of inputs named solved_out to G2_fun and then proceed similar to the HA block.
¤mimeimage/svg+xml¬rootassigneeÀ²last_run_timestampËAÚ»Ë(6]°persist_js_state·has_pluto_hook_features§cell_idÙ$0933d143-a15b-49d9-8ced-9ddc42526a48¹depends_on_disabled_cells§runtimeÎ °¥µpublished_object_keys¸depends_on_skipped_cells§erroredÂÙ$e7a61ca2-b9a3-4dae-97d6-e7d1046acaa2Š¦queued¤logs‘ˆ¤lineÿ£msg’Ù( 6.986 ms (367 allocations: 44.20 MiB)
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The discretization of the model will work as in my previous notebook
While implementing the DAG method "manually" as above is perfectly doable for this not-that-complicated model, the above function looks somewhat tedious and I found it to be quite easy to make mistakes (try implementing if from scratch yourself). Naturally, a richer model would exacerbate this.
Nevertheless, we can now get the aggregate dynamics in response to a monetary shock. The code below obtains the model IRFs of inflation to a 25 basis point nominal rate disturbance that follows an AR(1)-process with different levels of persistence, ranging from 0.2 to 0.9.
Finally, you may ask yourself that even though we now got the GE Jacobians for $Y$ , $w$ and $\pi$ , what about the other variables? Do we still need to do DAG-ing for that?
Of course not, we can again just write a function that gives us the remaining variables as functions of $Y$ , $w$ and $\pi$ and the shocks, differentiate it and combine it with the GE Jacobian we already have. This is done by the two functions below:
Finally, a function that assembles the "fake news" matrix $\mathcal{F}$ and the HA block Jacobian $\mathcal{J}$ :
Let's briefly test this for some values:
Next, the below gets us the objects $\mathcal{D}$ and $\mathcal{Y}$ for which we need to conduct the backwards iteration. This works as in the previous notebook, except that we have more inputs and outputs of the HA block.
SSJ Notebook II - General Equilibrium with and without DAGs
This notebook serves as a sequel to my previous notebook Auclert et al. (2021 - henceforth ABRS)
While the HANC model as in my original notebook is attractive for learning purposes due to its simplicity, it essentially abstracted from another challenge of the ABRS method, getting the General Equilibrium (GE) Jacobians. ABRS propose a specific method for this, which is based on chaining partial equilibrium Jacobians along a Directed Acyclical Graph (DAG).
However, I personally found that just from reading the paper, it is not that easy to grap how that needs to be done in practice and while ABRS's comprehensive Python package
Furthermore, once you actually understood it, it turns out that implementing the DAG method without a package automating everything can be somewhat tedious and error-prone (at least in my opinion). Thus, in this notebook I will not only implement the DAG method manually for a simple HANK model but also demonstrate two alternative ways to obtain the GE Jacobians:
A Direct Method employing the HA Jacobians: This idea is briefly mentioned in the ABRS paper and while inefficient, I find it conceptionally much simpler than the DAG method.
A "Dynare-like" method: It is also possible to get Aggregate Jacobians by writing the aggregate parts of the model down in a way similar as one would e.g. in Dynare . This idea was inspired by insights in Bhandari et al. (BBEG) , who develop an interesting alterntive to the SSSJ method.
All methods are implemented relying solely on automatic differentiation through the ForwardDiff.jl package.
The example model for this notebook is the One-Asset HANK model also used in the original ABRS paper, which is briefly presented below. While still relatively simple, it has a more involved "macro" block due the presence of a Forward-Looking Phillips curve, endogenous labor supply and so on. Additionally, ABRS have a Python notebook also implementing the model so we can check the results here against the "original".
The notebook assumes some familiarity with Julia and features the following parts:
Description of the model
Solving/Calibrating the steady state
Obtaining the Heterogeneous Agent Jacobian
Different ways to obtain the GE Jacobians
As previously, I tried to focus more on a clear and simple implementation instead one optimized for speed.
Nevertheless, while the above worked and conveniently returns the Jacobians and time-paths for all model variables at the same time, that is not always desirable: Effectively, that gave us a bigger $F_X$ -Matrix and inverting big matrices is costly. Here the difference is not so big, but if you have, say, 20 aggregate variables and a longer truncation horizon (say $T=500$ ), it will start to matter.
Since $F_X$ and $F_Z$ are different here as they relate to bigger number of variables, let us instead test whether we get the same results as before:
Indeed, we see that the Jacobians obtained using the DAG method and the Direct Method with HA Jacobian are the same (up to some small numerical error). Let's also plot the inflation response as visual demonstration.
As the plot above confirms, the model responses to the nominal rate shocks are the same as for the DAG method (up to some Floating Point Error). And in contrast to the Direct method, getting $F_X$ and $F_Z$ is rather fast:
That worked, nice!
For getting the steady state distribution, I again use the Young (2010) histogram method. Since this was already discussed in the previous notebook $\Lambda$ and solve for the stationary distribution $D$ are relegated to the hidden code fields below (they are called build_Λ and inv_dist, respectively).
We can check our calibrated parameter values for $\beta$ as well as $\varphi$ and see that they correspond to those obtained by ABRS:
Solving and Calibrating the Steady State
We are now ready to calibrate/solve the model's steady state. As mentioned above, I will choose $\beta$ so as to induce an SS interest rate of $r=0.005$ and $\varphi$ to achieve $L=Y=1$ . The below function implements this:
(Yes, these responses look somewhat ugly. Feel free to convince yourself that the ones in the ABRS notebook look the same.)
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No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
[90m[05823500] [39m[93m↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0[39m
[0m[1mInstantiating...[22m
[90m===[39m
[0m[1mPrecompiling...[22m
[90m===[39m
[32m[1m Activating[22m[39m project at `/tmp/jl_grBpna`
[92m[1mPrecompiling[22m[39m project...
181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.«ForwardDiffÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
Installed Unitful ─────────────────── v1.22.1 [32m[1m[22m[39m
Installed DSP ─────────────────────── v0.7.10 [32m[1m[22m[39m
Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
Installed Xorg_libxkbfile_jll ─────── v1.1.3+0 [32m[1m[22m[39m
Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
Installed Polynomials ─────────────── v4.0.19 [32m[1m[22m[39m
Installed PlutoUI ─────────────────── v0.7.62 [32m[1m[22m[39m
Installed Pango_jll ───────────────── v1.56.3+0 [32m[1m[22m[39m
Installed xkbcommon_jll ───────────── v1.8.1+0 [32m[1m[22m[39m
Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
Installed BenchmarkTools ──────────── v1.6.0 [32m[1m[22m[39m
Installed HarfBuzz_jll ────────────── v8.5.1+0 [32m[1m[22m[39m
Installed fzf_jll ─────────────────── v0.61.1+0 [32m[1m[22m[39m
Installed NLSolversBase ───────────── v7.9.1 [32m[1m[22m[39m
Installed DiffEqBase ──────────────── v6.174.0 [32m[1m[22m[39m
Installed ThreadingUtilities ──────── v0.5.4 [32m[1m[22m[39m
Installed SciMLBase ───────────────── v2.87.0 [32m[1m[22m[39m
Installed Graphs ──────────────────── v1.12.1 [32m[1m[22m[39m
Installed Rmath_jll ───────────────── v0.5.1+0 [32m[1m[22m[39m
Installed libpng_jll ──────────────── v1.6.48+0 [32m[1m[22m[39m
Installed StatsAPI ────────────────── v1.7.1 [32m[1m[22m[39m
Installed JLFzf ───────────────────── v0.1.11 [32m[1m[22m[39m
Installed ColorTypes ──────────────── v0.11.5 [32m[1m[22m[39m
Installed gperf_jll ───────────────── v3.3.0+0 [32m[1m[22m[39m
Installed StatsBase ───────────────── v0.34.5 [32m[1m[22m[39m
Installed Colors ──────────────────── v0.13.1 [32m[1m[22m[39m
Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
Installed EnzymeCore ──────────────── v0.8.11 [32m[1m[22m[39m
Installed Inflate ─────────────────── v0.1.5 [32m[1m[22m[39m
Installed AbstractFFTs ────────────── v1.5.0 [32m[1m[22m[39m
Installed ColorVectorSpace ────────── v0.10.0 [32m[1m[22m[39m
Installed GR ──────────────────────── v0.73.16 [32m[1m[22m[39m
Installed Plots ───────────────────── v1.40.13 [32m[1m[22m[39m
Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
[90m[05823500] [39m[93m↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0[39m
[0m[1mInstantiating...[22m
[90m===[39m
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[90m===[39m
[32m[1m Activating[22m[39m project at `/tmp/jl_grBpna`
[92m[1mPrecompiling[22m[39m project...
181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.²BasicInterpolatorsÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
Installed Unitful ─────────────────── v1.22.1 [32m[1m[22m[39m
Installed DSP ─────────────────────── v0.7.10 [32m[1m[22m[39m
Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
Installed Xorg_libxkbfile_jll ─────── v1.1.3+0 [32m[1m[22m[39m
Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
Installed Polynomials ─────────────── v4.0.19 [32m[1m[22m[39m
Installed PlutoUI ─────────────────── v0.7.62 [32m[1m[22m[39m
Installed Pango_jll ───────────────── v1.56.3+0 [32m[1m[22m[39m
Installed xkbcommon_jll ───────────── v1.8.1+0 [32m[1m[22m[39m
Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
Installed BenchmarkTools ──────────── v1.6.0 [32m[1m[22m[39m
Installed HarfBuzz_jll ────────────── v8.5.1+0 [32m[1m[22m[39m
Installed fzf_jll ─────────────────── v0.61.1+0 [32m[1m[22m[39m
Installed NLSolversBase ───────────── v7.9.1 [32m[1m[22m[39m
Installed DiffEqBase ──────────────── v6.174.0 [32m[1m[22m[39m
Installed ThreadingUtilities ──────── v0.5.4 [32m[1m[22m[39m
Installed SciMLBase ───────────────── v2.87.0 [32m[1m[22m[39m
Installed Graphs ──────────────────── v1.12.1 [32m[1m[22m[39m
Installed Rmath_jll ───────────────── v0.5.1+0 [32m[1m[22m[39m
Installed libpng_jll ──────────────── v1.6.48+0 [32m[1m[22m[39m
Installed StatsAPI ────────────────── v1.7.1 [32m[1m[22m[39m
Installed JLFzf ───────────────────── v0.1.11 [32m[1m[22m[39m
Installed ColorTypes ──────────────── v0.11.5 [32m[1m[22m[39m
Installed gperf_jll ───────────────── v3.3.0+0 [32m[1m[22m[39m
Installed StatsBase ───────────────── v0.34.5 [32m[1m[22m[39m
Installed Colors ──────────────────── v0.13.1 [32m[1m[22m[39m
Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
Installed EnzymeCore ──────────────── v0.8.11 [32m[1m[22m[39m
Installed Inflate ─────────────────── v0.1.5 [32m[1m[22m[39m
Installed AbstractFFTs ────────────── v1.5.0 [32m[1m[22m[39m
Installed ColorVectorSpace ────────── v0.10.0 [32m[1m[22m[39m
Installed GR ──────────────────────── v0.73.16 [32m[1m[22m[39m
Installed Plots ───────────────────── v1.40.13 [32m[1m[22m[39m
Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
[90m[05823500] [39m[93m↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0[39m
[0m[1mInstantiating...[22m
[90m===[39m
[0m[1mPrecompiling...[22m
[90m===[39m
[32m[1m Activating[22m[39m project at `/tmp/jl_grBpna`
[92m[1mPrecompiling[22m[39m project...
181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.§PlutoUIÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
Installed Unitful ─────────────────── v1.22.1 [32m[1m[22m[39m
Installed DSP ─────────────────────── v0.7.10 [32m[1m[22m[39m
Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
Installed Xorg_libxkbfile_jll ─────── v1.1.3+0 [32m[1m[22m[39m
Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
Installed Polynomials ─────────────── v4.0.19 [32m[1m[22m[39m
Installed PlutoUI ─────────────────── v0.7.62 [32m[1m[22m[39m
Installed Pango_jll ───────────────── v1.56.3+0 [32m[1m[22m[39m
Installed xkbcommon_jll ───────────── v1.8.1+0 [32m[1m[22m[39m
Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
Installed BenchmarkTools ──────────── v1.6.0 [32m[1m[22m[39m
Installed HarfBuzz_jll ────────────── v8.5.1+0 [32m[1m[22m[39m
Installed fzf_jll ─────────────────── v0.61.1+0 [32m[1m[22m[39m
Installed NLSolversBase ───────────── v7.9.1 [32m[1m[22m[39m
Installed DiffEqBase ──────────────── v6.174.0 [32m[1m[22m[39m
Installed ThreadingUtilities ──────── v0.5.4 [32m[1m[22m[39m
Installed SciMLBase ───────────────── v2.87.0 [32m[1m[22m[39m
Installed Graphs ──────────────────── v1.12.1 [32m[1m[22m[39m
Installed Rmath_jll ───────────────── v0.5.1+0 [32m[1m[22m[39m
Installed libpng_jll ──────────────── v1.6.48+0 [32m[1m[22m[39m
Installed StatsAPI ────────────────── v1.7.1 [32m[1m[22m[39m
Installed JLFzf ───────────────────── v0.1.11 [32m[1m[22m[39m
Installed ColorTypes ──────────────── v0.11.5 [32m[1m[22m[39m
Installed gperf_jll ───────────────── v3.3.0+0 [32m[1m[22m[39m
Installed StatsBase ───────────────── v0.34.5 [32m[1m[22m[39m
Installed Colors ──────────────────── v0.13.1 [32m[1m[22m[39m
Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
Installed EnzymeCore ──────────────── v0.8.11 [32m[1m[22m[39m
Installed Inflate ─────────────────── v0.1.5 [32m[1m[22m[39m
Installed AbstractFFTs ────────────── v1.5.0 [32m[1m[22m[39m
Installed ColorVectorSpace ────────── v0.10.0 [32m[1m[22m[39m
Installed GR ──────────────────────── v0.73.16 [32m[1m[22m[39m
Installed Plots ───────────────────── v1.40.13 [32m[1m[22m[39m
Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
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181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.®BenchmarkToolsÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
Installed Unitful ─────────────────── v1.22.1 [32m[1m[22m[39m
Installed DSP ─────────────────────── v0.7.10 [32m[1m[22m[39m
Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
Installed Xorg_libxkbfile_jll ─────── v1.1.3+0 [32m[1m[22m[39m
Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
Installed Polynomials ─────────────── v4.0.19 [32m[1m[22m[39m
Installed PlutoUI ─────────────────── v0.7.62 [32m[1m[22m[39m
Installed Pango_jll ───────────────── v1.56.3+0 [32m[1m[22m[39m
Installed xkbcommon_jll ───────────── v1.8.1+0 [32m[1m[22m[39m
Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
Installed BenchmarkTools ──────────── v1.6.0 [32m[1m[22m[39m
Installed HarfBuzz_jll ────────────── v8.5.1+0 [32m[1m[22m[39m
Installed fzf_jll ─────────────────── v0.61.1+0 [32m[1m[22m[39m
Installed NLSolversBase ───────────── v7.9.1 [32m[1m[22m[39m
Installed DiffEqBase ──────────────── v6.174.0 [32m[1m[22m[39m
Installed ThreadingUtilities ──────── v0.5.4 [32m[1m[22m[39m
Installed SciMLBase ───────────────── v2.87.0 [32m[1m[22m[39m
Installed Graphs ──────────────────── v1.12.1 [32m[1m[22m[39m
Installed Rmath_jll ───────────────── v0.5.1+0 [32m[1m[22m[39m
Installed libpng_jll ──────────────── v1.6.48+0 [32m[1m[22m[39m
Installed StatsAPI ────────────────── v1.7.1 [32m[1m[22m[39m
Installed JLFzf ───────────────────── v0.1.11 [32m[1m[22m[39m
Installed ColorTypes ──────────────── v0.11.5 [32m[1m[22m[39m
Installed gperf_jll ───────────────── v3.3.0+0 [32m[1m[22m[39m
Installed StatsBase ───────────────── v0.34.5 [32m[1m[22m[39m
Installed Colors ──────────────────── v0.13.1 [32m[1m[22m[39m
Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
Installed EnzymeCore ──────────────── v0.8.11 [32m[1m[22m[39m
Installed Inflate ─────────────────── v0.1.5 [32m[1m[22m[39m
Installed AbstractFFTs ────────────── v1.5.0 [32m[1m[22m[39m
Installed ColorVectorSpace ────────── v0.10.0 [32m[1m[22m[39m
Installed GR ──────────────────────── v0.73.16 [32m[1m[22m[39m
Installed Plots ───────────────────── v1.40.13 [32m[1m[22m[39m
Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
[90m[05823500] [39m[93m↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0[39m
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181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.LinearAlgebraÚ(ì
[0m[1mResolving...[22m
[90m===[39m
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Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
Installed Unitful ─────────────────── v1.22.1 [32m[1m[22m[39m
Installed DSP ─────────────────────── v0.7.10 [32m[1m[22m[39m
Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
Installed Xorg_libxkbfile_jll ─────── v1.1.3+0 [32m[1m[22m[39m
Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
Installed Polynomials ─────────────── v4.0.19 [32m[1m[22m[39m
Installed PlutoUI ─────────────────── v0.7.62 [32m[1m[22m[39m
Installed Pango_jll ───────────────── v1.56.3+0 [32m[1m[22m[39m
Installed xkbcommon_jll ───────────── v1.8.1+0 [32m[1m[22m[39m
Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
Installed BenchmarkTools ──────────── v1.6.0 [32m[1m[22m[39m
Installed HarfBuzz_jll ────────────── v8.5.1+0 [32m[1m[22m[39m
Installed fzf_jll ─────────────────── v0.61.1+0 [32m[1m[22m[39m
Installed NLSolversBase ───────────── v7.9.1 [32m[1m[22m[39m
Installed DiffEqBase ──────────────── v6.174.0 [32m[1m[22m[39m
Installed ThreadingUtilities ──────── v0.5.4 [32m[1m[22m[39m
Installed SciMLBase ───────────────── v2.87.0 [32m[1m[22m[39m
Installed Graphs ──────────────────── v1.12.1 [32m[1m[22m[39m
Installed Rmath_jll ───────────────── v0.5.1+0 [32m[1m[22m[39m
Installed libpng_jll ──────────────── v1.6.48+0 [32m[1m[22m[39m
Installed StatsAPI ────────────────── v1.7.1 [32m[1m[22m[39m
Installed JLFzf ───────────────────── v0.1.11 [32m[1m[22m[39m
Installed ColorTypes ──────────────── v0.11.5 [32m[1m[22m[39m
Installed gperf_jll ───────────────── v3.3.0+0 [32m[1m[22m[39m
Installed StatsBase ───────────────── v0.34.5 [32m[1m[22m[39m
Installed Colors ──────────────────── v0.13.1 [32m[1m[22m[39m
Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
Installed EnzymeCore ──────────────── v0.8.11 [32m[1m[22m[39m
Installed Inflate ─────────────────── v0.1.5 [32m[1m[22m[39m
Installed AbstractFFTs ────────────── v1.5.0 [32m[1m[22m[39m
Installed ColorVectorSpace ────────── v0.10.0 [32m[1m[22m[39m
Installed GR ──────────────────────── v0.73.16 [32m[1m[22m[39m
Installed Plots ───────────────────── v1.40.13 [32m[1m[22m[39m
Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
[90m[05823500] [39m[93m↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0[39m
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[92m[1mPrecompiling[22m[39m project...
181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.¬SparseArraysÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
Installed Unitful ─────────────────── v1.22.1 [32m[1m[22m[39m
Installed DSP ─────────────────────── v0.7.10 [32m[1m[22m[39m
Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
Installed Xorg_libxkbfile_jll ─────── v1.1.3+0 [32m[1m[22m[39m
Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
Installed Polynomials ─────────────── v4.0.19 [32m[1m[22m[39m
Installed PlutoUI ─────────────────── v0.7.62 [32m[1m[22m[39m
Installed Pango_jll ───────────────── v1.56.3+0 [32m[1m[22m[39m
Installed xkbcommon_jll ───────────── v1.8.1+0 [32m[1m[22m[39m
Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
Installed BenchmarkTools ──────────── v1.6.0 [32m[1m[22m[39m
Installed HarfBuzz_jll ────────────── v8.5.1+0 [32m[1m[22m[39m
Installed fzf_jll ─────────────────── v0.61.1+0 [32m[1m[22m[39m
Installed NLSolversBase ───────────── v7.9.1 [32m[1m[22m[39m
Installed DiffEqBase ──────────────── v6.174.0 [32m[1m[22m[39m
Installed ThreadingUtilities ──────── v0.5.4 [32m[1m[22m[39m
Installed SciMLBase ───────────────── v2.87.0 [32m[1m[22m[39m
Installed Graphs ──────────────────── v1.12.1 [32m[1m[22m[39m
Installed Rmath_jll ───────────────── v0.5.1+0 [32m[1m[22m[39m
Installed libpng_jll ──────────────── v1.6.48+0 [32m[1m[22m[39m
Installed StatsAPI ────────────────── v1.7.1 [32m[1m[22m[39m
Installed JLFzf ───────────────────── v0.1.11 [32m[1m[22m[39m
Installed ColorTypes ──────────────── v0.11.5 [32m[1m[22m[39m
Installed gperf_jll ───────────────── v3.3.0+0 [32m[1m[22m[39m
Installed StatsBase ───────────────── v0.34.5 [32m[1m[22m[39m
Installed Colors ──────────────────── v0.13.1 [32m[1m[22m[39m
Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
Installed EnzymeCore ──────────────── v0.8.11 [32m[1m[22m[39m
Installed Inflate ─────────────────── v0.1.5 [32m[1m[22m[39m
Installed AbstractFFTs ────────────── v1.5.0 [32m[1m[22m[39m
Installed ColorVectorSpace ────────── v0.10.0 [32m[1m[22m[39m
Installed GR ──────────────────────── v0.73.16 [32m[1m[22m[39m
Installed Plots ───────────────────── v1.40.13 [32m[1m[22m[39m
Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
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181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.¥PlotsÚ(ì
[0m[1mResolving...[22m
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[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
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Installed PDMats ──────────────────── v0.11.35 [32m[1m[22m[39m
Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
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Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
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Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
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Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
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Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
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Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
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181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.ªnbpkg_syncÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
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Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
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Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
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Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
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Installed NLopt ───────────────────── v1.1.4 [32m[1m[22m[39m
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Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
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Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
No Changes to `/tmp/jl_grBpna/Project.toml` [32m[1m[22m[39m
[32m[1m Updating[22m[39m `/tmp/jl_grBpna/Manifest.toml`
[90m[05823500] [39m[93m↑ OpenLibm_jll v0.8.1+2 ⇒ v0.8.5+0[39m
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[32m[1m Activating[22m[39m project at `/tmp/jl_grBpna`
[92m[1mPrecompiling[22m[39m project...
181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.ªParametersÚ(ì
[0m[1mResolving...[22m
[90m===[39m
[32m[1m Installed[22m[39m Xorg_xkbcomp_jll ────────── v1.4.7+0
Installed StatsFuns ───────────────── v1.5.0 [32m[1m[22m[39m
Installed HypergeometricFunctions ─── v0.3.28 [32m[1m[22m[39m
Installed GR_jll ──────────────────── v0.73.16+0 [32m[1m[22m[39m
Installed FFTW ────────────────────── v1.8.1 [32m[1m[22m[39m
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Installed Hyperscript ─────────────── v0.0.5 [32m[1m[22m[39m
Installed TimerOutputs ────────────── v0.5.29 [32m[1m[22m[39m
Installed QuantEcon ───────────────── v0.16.6 [32m[1m[22m[39m
Installed Polyester ───────────────── v0.7.18 [32m[1m[22m[39m
Installed CEnum ───────────────────── v0.5.0 [32m[1m[22m[39m
Installed StaticArrays ────────────── v1.9.13 [32m[1m[22m[39m
Installed Xorg_libSM_jll ──────────── v1.2.6+0 [32m[1m[22m[39m
Installed Cairo_jll ───────────────── v1.18.5+0 [32m[1m[22m[39m
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Installed Xorg_libXinerama_jll ────── v1.1.6+0 [32m[1m[22m[39m
Installed ArnoldiMethod ───────────── v0.4.0 [32m[1m[22m[39m
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Installed Dbus_jll ────────────────── v1.16.2+0 [32m[1m[22m[39m
Installed PositiveFactorizations ──── v0.2.4 [32m[1m[22m[39m
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Installed Libffi_jll ──────────────── v3.4.7+0 [32m[1m[22m[39m
Installed Xorg_libXcursor_jll ─────── v1.2.4+0 [32m[1m[22m[39m
Installed Qt6Declarative_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed FFTW_jll ────────────────── v3.3.11+0 [32m[1m[22m[39m
Installed Xorg_libXi_jll ──────────── v1.8.3+0 [32m[1m[22m[39m
Installed Qt6ShaderTools_jll ──────── v6.8.2+1 [32m[1m[22m[39m
Installed ArrayInterface ──────────── v7.19.0 [32m[1m[22m[39m
Installed LineSearches ────────────── v7.3.0 [32m[1m[22m[39m
Installed NLopt_jll ───────────────── v2.10.0+0 [32m[1m[22m[39m
Installed Wayland_jll ─────────────── v1.23.1+0 [32m[1m[22m[39m
Installed Rmath ───────────────────── v0.8.0 [32m[1m[22m[39m
Installed StableRNGs ──────────────── v1.0.3 [32m[1m[22m[39m
Installed IntegerMathUtils ────────── v0.1.2 [32m[1m[22m[39m
Installed SimpleTraits ────────────── v0.9.4 [32m[1m[22m[39m
Installed Latexify ────────────────── v0.16.8 [32m[1m[22m[39m
Installed DifferentiationInterface ── v0.6.54 [32m[1m[22m[39m
Installed Xorg_xkeyboard_config_jll ─ v2.44.0+0 [32m[1m[22m[39m
Installed Xorg_libXfixes_jll ──────── v6.0.1+0 [32m[1m[22m[39m
Installed Qt6Wayland_jll ──────────── v6.8.2+0 [32m[1m[22m[39m
Installed Xorg_libXrandr_jll ──────── v1.5.5+0 [32m[1m[22m[39m
Installed Primes ──────────────────── v0.5.7 [32m[1m[22m[39m
Installed RuntimeGeneratedFunctions ─ v0.5.15 [32m[1m[22m[39m
Installed IterTools ───────────────── v1.10.0 [32m[1m[22m[39m
Installed Optim ───────────────────── v1.12.0 [32m[1m[22m[39m
Installed Glib_jll ────────────────── v2.84.0+0 [32m[1m[22m[39m
Installed Qt6Base_jll ─────────────── v6.8.2+1 [32m[1m[22m[39m
Installed QuadGK ──────────────────── v2.11.2 [32m[1m[22m[39m
Installed UnitfulLatexify ─────────── v1.7.0 [32m[1m[22m[39m
Installed Distributions ───────────── v0.25.120 [32m[1m[22m[39m
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181 dependencies successfully precompiled in 219 seconds. 209 already precompiled.§enabled÷restart_recommended_msgÀ´restart_required_msgÀbusy_packages¶waiting_for_permissionÂÙ,waiting_for_permission_but_probably_disabled«cell_inputsÞ TÙ$1a52ce21-ada0-4dea-b484-e6c4393328d4„§cell_idÙ$1a52ce21-ada0-4dea-b484-e6c4393328d4¤codeÙ%@btime get_GE_jacob_DAG($Js,$SS_objs)¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$acf9f7a7-300f-423d-8220-6014327ec39b„§cell_idÙ$acf9f7a7-300f-423d-8220-6014327ec39b¤codeÙÀfunction get_ð“”(Λ,y_ss,np::NumericalParameters)
ð“” = zeros(length(y_ss),np.T+1)
ð“”[:,1] = y_ss[:]
for tt = 2:(np.T+1)
@views ð“”[:,tt] = Λ*ð“”[:,tt-1]
end
return ð“”
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$42e36f72-e02c-435f-8fa6-30309b3ee3e6„§cell_idÙ$42e36f72-e02c-435f-8fa6-30309b3ee3e6¤codeÚ #Note: Since NumericalParameters is an immutable struct, I use @set!
#rom the Setfield package to change its elements.
function get_steady_state(np::NumericalParameters)
#normalize avg. labor efficiency to 1
s_dist = inv_dist(np.Ps)
@set! np.s_grid = np.s_grid./sum(s_dist.*np.s_grid)
#unpack objects
@unpack mp, s_grid, na, ns = np
@unpack μ, B, Y_target, r_target, Z_ss = mp
r_ss = r_target
Y_ss = Y_target ; L_ss = Y_ss/Z_ss
#get steady state objects implied by parameters:
w_ss = 1/μ #wage
div_ss = Y_ss - w_ss*L_ss #dividend
Ï„_level_ss = r_target*B #tax rate
T_level_ss = div_ss - Ï„_level_ss #net lump sum transfer
#Solve for β and φ consistent with targets using Broyden's Method
prob = NonlinearProblem((x,p) ->
β_φ_objective(x[1],x[2],p,np),
[np.mp.β,np.mp.φ],[w_ss,r_target,T_level_ss])
sol = solve(prob,Broyden(alpha = 100.0))
#Note: alpha = 100.0 ensures that the solvers does not take too big steps
#at the beginning
if sol.retcode != :Success
@warn "Calibration didn't solve"
end
@set! np.mp.β = sol.u[1]
@set! np.mp.φ = sol.u[2]
#get remaining steady state objectives for correct β and φ
c_ss, a_ss, n_ss = solve_EGM_SS(w_ss,r_target,T_level_ss,np)
#get distribution as (na × ns matrix)
Λ_ss = build_Λ(a_ss, np)
D_ss = inv_dist(Λ_ss)
D_ss = reshape(D_ss,(np.na,np.ns))
#aggregate capital stock implied by HH savings
A_ss = sum(D_ss.*np.a_grid)
L_ss = sum(D_ss.*(n_ss.*s_grid'))
N_ss = sum(D_ss.*n_ss) #aggregate hours
C_ss = L_ss*Z_ss #aggregate consumption
#check that Walras' Law holds
println("Goods market clearing residual: ",abs(sum(D_ss.*c_ss) - C_ss))
return (; L_ss, w_ss, Y_ss, r_ss, τ_level_ss, div_ss, T_level_ss, c_ss, a_ss, n_ss, Λ_ss, D_ss, A_ss, N_ss, C_ss ,np)
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$3107d9de-b4ec-4954-8f09-4e1896c83f62„§cell_idÙ$3107d9de-b4ec-4954-8f09-4e1896c83f62¤codeÙMmd"""
For example, we can now also plot the response of the nominal rate:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$575be7fa-ae2f-4fac-b7b7-c6ae6a1fec93„§cell_idÙ$575be7fa-ae2f-4fac-b7b7-c6ae6a1fec93¤codeÚÓ#function that conducts the backwards iterations, taking as inputs the SS and perturbations of w and r and the level of net transfers (dividends minus lump sum taxes)
function backward_objective(dx_w,dx_r,dx_T,c_ss,w_ss,r_ss,T_level_ss,
D_ss,np::NumericalParameters)
#Note: dx_w and dx_r are the perturbations to the wage and interest rate
#T periods in advance
@unpack T, s_grid = np
type_ind = dx_w+dx_r+dx_T #get type of perturbance term to initialize containers
#initialize containers
A_terms = zeros(eltype(type_ind),T)
L_terms = zeros(eltype(type_ind),T)
D_terms = zeros(eltype(type_ind),length(D_ss),T)
#get values for re-centering
#(cf. Appendix C of Auclert et al. (2021))
c_nc = EGM_update(c_ss,w_ss,T_level_ss,r_ss,r_ss,np)[1]
c_rc = c_ss
#backward iteration
for tt in Iterators.reverse(1:T)
if tt==T
c_t, aPrime_t, n_t = EGM_update(c_rc,w_ss + dx_w, T_level_ss + dx_T,
r_ss + dx_r,r_ss,np)
elseif tt == (T-1)
c_t, aPrime_t, n_t = EGM_update(c_rc,w_ss,T_level_ss,r_ss,r_ss + dx_r,np)
else
c_t, aPrime_t, n_t = EGM_update(c_rc,w_ss,T_level_ss,r_ss,r_ss,np)
end
Λ = build_Λ(aPrime_t,np)
@views begin #@views reduces the number of allocations
A_terms[tt] = (aPrime_t[:])'*D_ss[:]
L_terms[tt] = ((n_t.*s_grid')[:])'*D_ss[:]
D_terms[:,tt] = Λ'*D_ss[:]
end
#do re-centering
c_rc = c_ss .+ (c_t .- c_nc)
end
return vcat(A_terms,L_terms,D_terms[:])
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$78f02c49-becd-4060-9f2f-ac250f468110„§cell_idÙ$78f02c49-becd-4060-9f2f-ac250f468110¤codeÚEmd"""
The only slightly tricky thing is that for borrowing-constrained households, we can't find consumption through an inverted Euler equation. For these, we need to solve the above optimality condition for labor supply numerically. The function below (used in `EGM_update` above) does that using fixed-point iteration:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$85614fee-9eea-4b9a-ac00-91847d8d7cce„§cell_idÙ$85614fee-9eea-4b9a-ac00-91847d8d7cce¤codeÚ„md"""
## The model
The description (mostly) follows the ABRS paper.
### Households
* There is a unit mass of ex-ante identical households who differ ex-post by their labor productivity $s$ (''skill'') and asset holdings $a$.
* Skill $s$ follows a time-invariant discrete Markov Chain with $n_s$ states $\mathcal{S}:=\lbrace s_1,s_2,...,s_{n_s}\rbrace$ and exogenous transition probabilities $P(s',s)$. This introduces the income risk. $\pi_s$ denotes the stationary distribution of $P$. The average labor productivity $\int^1_0 s_{it} di$ is invariant and will be normalized to 1.
* Skill $s$ does not only determine an households income per hour worked, but also the amount of lump-sum taxes she has to pay and the amount of dividends she receives, both of which are distributed proportionally to $s$.
* A household can freely choose the amount of hours $n$ she works subject to an additively separable utility cost of working
* Savings- and labor choices $a$ and $n$ are the solution to the household's utility maximization problem, which is characterized by the following Bellman equation
```math
V_t(s,a) = \max_{a',n} \frac{c^{1-\sigma}-1}{1-\sigma} - \varphi\frac{n^{1+\nu}}{1+\nu} +\beta \sum_{s'\in \mathcal{S}}P(s',s)V_{t+1}(s',a')
```
```math
\text{subject to}~~c+a' = (1+r_t)a + w_t n s - \underbrace{(d_t - \tau_t)}_{=\text{net transfer}}s ~~\text{and}~~a'\geq \bar{a}
```
* Above, $d_t$ and $\tau_t$ denote the aggregate level of dividends and taxes.
* We denote by $D$ the distribution of households, i.e. $D_t(s,a)$ is the mass of households who start the period with skill $s$ and assets $a$.
### Firms
A competitive final goods firm assembles its output using a CES production function $Y_t = (\int^1_0 y_{jt}^{\frac{1}{\mu}} dj)^{\mu}$, giving rise to a standard CES-demand system for the continuum of intermediate goods. These intermediates, in turn, are supplied by monopolists who produce using only labor s.t. $y_{jt}=Z_t l_{jt}$ and are subject to quadratic price adjustment costs a la Rotemberg (1982). $Z_t$ denotes aggregate productivity that may be time-varying. In a symmetric equilibrium, gross inflation $1+\pi_t = P_t/P_{t-1}$ can then be shown to evolve according to a New Keynesian Phillips curve of the form
```math
\log(1+\pi_t) = \kappa\left(\frac{w_t}{Z_t}- \frac{1}{\mu} \right) + \mathbb{E}_t\frac{1}{1+r_{t+1}}\frac{Y_{t+1}}{Y_{t}}\log(1+\pi_{t+1}).
```
Here $\kappa$ is the parameter governing firms' price adjustment costs, $\mu$ the steady state markup and $Y_t$ aggregate output, which is equal to the monopolists' individual output by symmetry. The firms' profits are distributed lump-sum to households, thus $d_t = Y_t - w_tL_t$.
### Policy
The model features a government consisting of two branches, a **monetary** and a **fiscal** authority. The former sets the nominal interest rate $i_t$ according to a Taylor rule of the form
```math
i_t = r^* + \theta_\pi \pi_t + \epsilon^i_t
```
where $r^*$ is the economy's long run "natural rate" and $\epsilon^i_t$ a monetary policy shock. The real interest rate in period $t$ is determined by the previously set nominal rate and inflation so that $1+r_t = \frac{1+i_{t-1}}{1+\pi_t}$.
The fiscal authority, in turn, issues a constant amount of government bonds $B$ each period and collects the lump sum taxes $\tau_t$ already mentioned above. Since there are no other spending items, it chooses the tax rate to cover the its interest rate payments every period, i.e.
```math
\tau_t = r_t B~~.
```
### Market clearing
In an equilibrium, the following market clearing conditions will have to hold:
* Asset market:
```math
B = \int^1_0 a_{it} di
```
* Labor market
```math
\frac{Y_t}{Z_t} = L_t = \int^1_0 s_{it} n_{it} di
```
* Good market
```math
Y_t = \underbrace{\sum_{s\in \mathcal{S}}\int c(s,a) D(s,a)da}_{=\text{aggregate consumption}} - \underbrace{\frac{\mu}{\mu-1}\frac{1}{2\kappa}\left(\log(\pi_t)\right)^2Y_t}_{=\text{price adjustment costs}}
```
In this model, there are three markets that need to clear: Firstly, an asset market on which the sum of households' net savings need to equal the aggregate amount of bonds $B$ supplied by the government. Secondly a labor market on which the effective amount of labor supplied by the households' at the piece-rate real wage $w_t$ needs to equal the labor demanded by the firm. Finally, the goods market will have to clear, requiring aggregate consumption and real price adjustment costs to equal aggregate output.
As is well known, price adjustment costs don't matter in linearized models and thus the latter will not be of concern here.
### Calibration:
The calibration follows the example model in the ABRS paper and can be seen in the code block below.
The skill process follows an AR(1)-process in logs that will be discretized to 7 points using the Rouwenhorst method
(the function implementing the Rouwenhorst method is off-the-shelf from *[QuantEcon.jl](https://github.com/QuantEcon/QuantEcon.jl)*).
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$fed0b8e1-fe6c-4c17-8c03-bc819ac7c45d„§cell_idÙ$fed0b8e1-fe6c-4c17-8c03-bc819ac7c45d¤codeÚ/#function that returns HA SSJ for given parameters and SS inputs
function get_ð’´_ð’Ÿ(c_ss,w_ss,r_ss,T_level_ss,D_ss,np::NumericalParameters)
@unpack T, na, ns = np
#objective function to be differentiated
obj_fun(x) = backward_objective(x[1],x[2],x[3],c_ss,w_ss,r_ss,T_level_ss,D_ss,
np::NumericalParameters)
#automatic differentiation using ForwardDiff
derivs = ForwardDiff.jacobian(x -> obj_fun(x),zeros(3))
#note given way the outputs of obj_fun are saved, need to flip arrays
#(done by reverse())
ð’´ = Matrix{Vector{Float64}}(undef,3,2)
ð’Ÿ = Vector{Matrix{Float64}}(undef,3)
for ii = 1:3
ð’´[ii,1] = reverse(derivs[1:T,ii])
ð’´[ii,2] = reverse(derivs[T+1:2*T,ii])
ð’Ÿ[ii] = reverse(reshape(derivs[(2*T+1):end,ii],(na*ns,T)),dims=2)
end
return ð’´, ð’Ÿ
end
¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$62e7a938-89b2-4123-b431-1f4cd98b3da8„§cell_idÙ$62e7a938-89b2-4123-b431-1f4cd98b3da8¤codeÚ–md"""
## A "Dynare-like" method
Finally, reading the recent paper by [Bhandari et al. (2023)](https://www.nber.org/papers/w31744) features some analytical results suggesting yet another way to get the Aggregate Jacobians of $F$. I will use a somewhat different formulation to aid exposition, but that is where I got the inspiration from.
Instead of considering a big system $F$, they represent the model as a set of equations that need to hold in every period, as state space approaches implemented by DSGE modelling packages like *[Dynare](dynare.org)* typically do. In my own notation, we formulate the model as a system of equations $G$ that needs to hold every period, i.e. a perfect foresight equilibrium is defined as sequences $X$ so that
```math
G(\tilde{X}_t,\tilde{Z}_t,x_t)=0~~\text{with}~~x_t = H(\lbrace X_t\rbrace_{t=0}^\infty,\lbrace Z_t\rbrace_{t=0}^\infty)~~\forall t
```
for exogenously given $Z$, where $\tilde{X_t}=[X_{t-1},X_t,X_{t+1}]$ (and the same for $Z$) and $x_t$ denotes a set of variables that are the outcomes of the model's HA block, which I denote by $H$. In the model studied here, $x_t$ would be asset demand and labor supply.
Differentiating the LHS w.r.t $\tilde{X_t}$ , we have
```math
G_{\tilde{X}} + G_x\mathcal{J}_{t,\tilde{X}}~~\forall~t
```
and a similar expression if differentiating w.r.t. $Z$. I am obviously abusing notation here.
Anyway, the point is that we can construct the Aggregate Jacobians $F_X$ and $F_Z$ by writing down the aggregate model part $G$ in a Dynare-like fashion. The below code Julia function displays how the function $G$ can look like if we set it up for *all* model variables (not just $Ï€$, $w$ and $Y$):
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$6da15895-81a6-434e-9002-522a3b3348a1„§cell_idÙ$6da15895-81a6-434e-9002-522a3b3348a1¤codeÚMmd"""
## GE Jacobians: The DAG approach
Having obtained the SSJ's of the HA block, we now want to obtain the aggregate Jacobians $F_X$ and $F_Z$. For the former, it is often desirable to state it in terms of the minimum number of variables necessary: $F_X$ will have dimension $n_x \times T$, where $T$ is the truncation horizon and $n_x$ the number of variables in $X$. If $n_x$ becomes high, $F_X$ can become quite large and in turn computationally costly to invert.
To get $F_X$ and $F_Z$, can we follow the approach outlined in ABRS, which is to split the model into different "blocks" and then chaining their derivatives along a Directed Acyclical Graph (DAG). For the model used here, the ABRS paper proposes a DAG as displayed below:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$57b4a641-8cf0-49ad-bc4c-c85e7071e355„§cell_idÙ$57b4a641-8cf0-49ad-bc4c-c85e7071e355¤codeÚhbegin
F_x4, F_RShock4= get_jacobs_G2(Js,SS_objs)
println("Maximum difference for F_x: ",maximum(abs.(F_x1 .- F_x4)))
println("Maximum difference for F_RShock: ",maximum(abs.(F_RShock1 .- F_RShock4)))
#get aggregate variables
GEJ = -F_x4\Matrix(F_RShock4)
AggVars_dyn2 = GEJ*dRShock
#extract inlfation response
Ï€_dyn2 = AggVars_dyn2[1:300,:]
println("Max. abs. diff. π compared to DAG: ", maximum(abs.(π_dyn2 .- π_dag)))
#plot inflation response
plot(1:20,10_000*Ï€_dyn2[1:20,:], labels = labels, linewidth = 3.0, ylabel = "Basis Points", legend = :topright,title = "Inflation Response (G method 2)")
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$0303d80f-a072-4a51-bfaf-d37f721fe82f„§cell_idÙ$0303d80f-a072-4a51-bfaf-d37f721fe82f¤codeÚªfunction build_ð’¥(ð’´,ð““,ð“”,np::NumericalParameters)
#construct fake news matrix ð“•
ð“• = zeros(np.T,np.T)
ð“•[1,:] = ð’´
ð“•[2:end,:] .= ð“”[:,1:(np.T-1)]'*ð““
#assemble the Sequence Space Sacobian ð’¥
ð’¥ = zeros(np.T,np.T)
ð’¥[:,1] .= ð“•[:,1]
ð’¥[1,2:end] .= ð“•[1,2:end]
for tt = 2:np.T
@views ð’¥[2:end,tt] .= ð“•[2:end,tt] .+ ð’¥[1:(end-1),tt-1]
end
return ð’¥, ð“•
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$60fa7af4-1bd9-4acb-bfe0-61801317b177„§cell_idÙ$60fa7af4-1bd9-4acb-bfe0-61801317b177¤codeÚÉ#solve consumption of constrained HHs via fixed point iteration
function solve_c_n_constrained(a_grid_c,c_guess,ws,r,T,np::NumericalParameters;
max_iter = 100, tol = 1e-11)
@unpack mp, amin = np
@unpack σ,φ,ν = mp
iter = 0 ; dist = 100.0
while (iter < max_iter) & (dist > tol)
#labor supply implied by consumption guess
n_c = ((ws*(c_guess).^(-σ))./φ).^(1/(ν))
#consumption implied by n_c (assuming constraint binding)
c_implied = ws*n_c .+ T .+ (1+r)*a_grid_c .- amin
dist = maximum(abs.(c_implied .- c_guess))
#update
c_guess = 0.25*c_implied .+ 0.75*c_guess
iter += 1
end
if iter == max_iter
@warn "Constrained labor supply didn't solve"
end
return c_guess
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$fb7958ac-014b-40cb-9020-7d6f645a7f87„§cell_idÙ$fb7958ac-014b-40cb-9020-7d6f645a7f87¤codeÙ"@btime get_jacobs_G($Js,$SS_objs);¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$3e960018-54bf-4331-9ac0-e130e2393778„§cell_idÙ$3e960018-54bf-4331-9ac0-e130e2393778¤codeÚmd"""
Armed with the above functions, we can obtain steady state household policies by iterating over the `EGM_update` function for given $w$, $r$ and net transfers.
For convenience below, I also define a version of the function that starts with a hard-coded initial guess.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$7a2bba2b-7084-4979-82c8-6cf3e5daec3d„§cell_idÙ$7a2bba2b-7084-4979-82c8-6cf3e5daec3d¤codeÚ{md"""
The aggregate representation is terms of just three aggregate variables, output $Y$, the real wage $w$ and inflation $Ï€$ and a corresponding number of GE conditions, which consist of the NK Phillips curve as well as labor- and asset market clearing conditions as stated above. However, these variables imply others which enter the different blocks that in turn provide additional inputs for more blocks and so on.
Either the way, the below function implements obtains the Aggregate Jacobians by chaining along the DAG "manually". Since all the blocks except the HA one have straightforward analytical derivatives, I just directly enter the analytical ones but one could equivalently obtain them by using some differentiation packages.
For simplicity, I will assume all exogenous variables except the monetary policy shock (represented by $r^*$ in the ABRS DAG) fixed here. The implementation of further shocks is left as exercise to the reader (*I always wanted to write that phrase somewhere*).
Furthermore, I disregard terms w.r.t. the price adjustment costs since these turn out to be 0 anyway (feel free to check for yourself).
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$9790e1b7-cb54-4086-ba92-4006667359b0„§cell_idÙ$9790e1b7-cb54-4086-ba92-4006667359b0¤codeÙ¡md"""
So, can we implement the above method for a reduced number of aggregate variables, say, again $Ï€$, $w$ and $Y$? Indeed, I will demonstrate that below.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$b1d47ec8-4958-449a-b9df-9f4cca062ed9„§cell_idÙ$b1d47ec8-4958-449a-b9df-9f4cca062ed9¤codeÚ`function build_Λ(a_choice::Matrix, np::NumericalParameters)
@unpack na, ns, a_grid, Ps = np
#pre-allocate arrays
weights_R = zeros(eltype(a_choice),na*ns,ns)
weights_L = zeros(eltype(a_choice),na*ns,ns)
IDX_col_R = zeros(Int64,na*ns,ns) #where agents "go"
@views begin
for ss = 1:ns
for aa = 1:na
#find closest smaller capital value on grid
al_idx = searchsortedlast(a_grid,a_choice[aa,ss])
ss_shifter = (ss-1)*na #inner helper variable
#if clause checks for boundary cases
if al_idx == na
#if higher than highest grid point, I assign entire mass to maximum grid point
for sss = 1:ns
weights_R[ss_shifter + aa,sss] += Ps[ss,sss]
IDX_col_R[ss_shifter + aa,sss] = al_idx + (sss-1)*na
end
elseif al_idx == 0
#if lower than smallest grid point, I assign entire mass to lowest grid point
for sss = 1:ns
weights_L[ss_shifter + aa,sss] += Ps[ss,sss]
IDX_col_R[ss_shifter + aa,sss] = (sss-1)*na + 2
end
else #regular case - using weights formula
wr = ((a_choice[aa,ss] - a_grid[al_idx]) / (a_grid[al_idx+1] - a_grid[al_idx]))
lr = 1.0 - wr
for sss = 1:ns
weights_R[ss_shifter + aa,sss] += Ps[ss,sss]*wr
weights_L[ss_shifter + aa,sss] += Ps[ss,sss]*lr
IDX_col_R[ss_shifter + aa,sss] = (sss-1)*na + al_idx + 1
end
end
end
end
IDX_from = repeat(1:(na*ns),outer=2*ns)
weights = vcat(weights_R[:], weights_L[:])
IDX_to = vcat(IDX_col_R[:],IDX_col_R[:] .- 1)
end
#return sparse transition matrix
return sparse(IDX_from,IDX_to,weights,na*ns,na*ns)
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$12e86c4c-dbae-4095-9c79-293ba38df5dc„§cell_idÙ$12e86c4c-dbae-4095-9c79-293ba38df5dc¤codeÚ5#function that returns the Jacobians of F using the direct method
function get_jacobs_direct(Js,SS_objs)
@unpack np,r_ss,w_ss,Y_ss = SS_objs
@unpack T = np
#check that system of equationsfulilled
check = direct_HAjacob_objective(zeros(T),ones(T),
w_ss*ones(T),zeros(T),Js,SS_objs)
#check that equations hold in SS
if maximum(abs.(check)) > 1e-8
@warn "large F residuals"
end
#differentiate system w.r.t endogenous variables
F_x = ForwardDiff.jacobian( x ->
direct_HAjacob_objective(x[1:T],x[T+1:2*np.T],
x[2*T+1:3*T],zeros(T),Js,SS_objs),
vcat(zeros(T),ones(T),w_ss*ones(T)))
#differentiate system w.r.t shocks
F_RShock = ForwardDiff.jacobian( x ->
direct_HAjacob_objective(zeros(T),ones(T),
w_ss*ones(T),x,Js,SS_objs),zeros(T))
return F_x, F_RShock
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$9bae8cdb-18bb-4c5b-8984-c27fd9417874„§cell_idÙ$9bae8cdb-18bb-4c5b-8984-c27fd9417874¤codeÚ begin
#Lets start with loading some Julia packages that will be useful below
using Parameters, LinearAlgebra, SparseArrays
using Setfield,ForwardDiff, BasicInterpolators
using NonlinearSolve,Plots
using QuantEcon: rouwenhorst
using PlutoUI: TableOfContents
using BenchmarkTools
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$5f99a5dd-abab-4fd4-bb67-0ba7dd201977„§cell_idÙ$5f99a5dd-abab-4fd4-bb67-0ba7dd201977¤codeÚTmd"""
## Solving the Household problem
For solving the household problem, we will again use the Endogenous Grid Method (EGM), which has to be modified to account for household's endogenous labor supply:
For given *next period* marginal value functions/consumption policy functions, we can still invert the Euler equation, but due to the endogenous labor supply choice, we don't immediately know how much wealth an household must have had to chose that consumption policy. However, we can link household's consumption and labor supply through the respective optimality condition
```math
\varphi n_{it}^\nu = w_t s_{it} c_{it}^{-\sigma}~~,
```
which requires the marginal disutility of working more to equal the marginal utility of the income the household earns from doing so. We can then implement the EGM backward step with the function below:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$3ca6f293-b683-4c70-b9eb-93fff236b786„§cell_idÙ$3ca6f293-b683-4c70-b9eb-93fff236b786¤codeÚmd"""
Given that it gives us what we want and seems easy, why not always use this method?
The issue is that the computational cost of getting the Jacobians of $F$ is much larger with the direct method, even if we use the HA Jacobian: In terms of computation time, for my implementations the direct method takes more than 50-100 times as long as the DAG method:
(Note that computation times displayed in the uploaded notebook differ from those I obtained on my laptop, as the code is run on a github server.)
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$2e128a66-9a3b-460a-9641-2bb8cba9c6b3„§cell_idÙ$2e128a66-9a3b-460a-9641-2bb8cba9c6b3¤codeÚcfunction get_jacobs_G(Js,SS_objs)
np = SS_objs.np
#define vectors of steady state variables
Xss = [SS_objs.w_ss,SS_objs.r_ss,SS_objs.T_level_ss,SS_objs.div_ss,
SS_objs.r_ss,SS_objs.Y_ss,SS_objs.Ï„_level_ss,0.0]
het_out_ss = [SS_objs.L_ss,SS_objs.A_ss]
#check that equations hold in SS
check = G_fun(Xss,Xss,Xss,het_out_ss,0.0,np)
if maximum(abs.(check)) > 1e-8
@warn "large F residuals"
end
#compute derivatives derivatives
dG_X = ForwardDiff.jacobian(x ->
G_fun(Xss,x,Xss,het_out_ss,0.0,np),Xss)
dG_Xlag = ForwardDiff.jacobian(x ->
G_fun(x,Xss,Xss,het_out_ss,0.0,np),Xss)
dG_XPrime = ForwardDiff.jacobian(x ->
G_fun(Xss,Xss,x,het_out_ss,0.0,np),Xss)
dG_het = ForwardDiff.jacobian(x ->
G_fun(Xss,Xss,Xss,x,0.0,np),het_out_ss)
dG_RShock = ForwardDiff.derivative(x ->
G_fun(Xss,Xss,Xss,het_out_ss,x,np),0.0)
#define a selection matrix for HA block inputs
P = zeros(3,8) ; P[1,1] = 1.0 ; P[2,2] = 1.0 ; P[3,3] = 1.0
#Note the ordering: w is first HA block input and also first element of X etc.
#define some helper matrices
Im = sparse(I,np.T,np.T)
Im_ = spdiagm(-1=>ones(np.T-1))
Imp = spdiagm(1=>ones(np.T-1))
#get J into suitable shape (again, note ordering of variables)
J_reshaped = [hcat(Js[:,2]...) ; hcat(Js[:,1]...)]
#assemble Aggregate Jacobians
F_x = kron(dG_het,Im)*J_reshaped*kron(P,Im) .+
kron(dG_X,Im) .+ kron(dG_XPrime,Imp) .+ kron(dG_Xlag,Im_)
F_RShock = kron(dG_RShock,Im)
test_M1 = kron(dG_X,Im) .+ kron(dG_XPrime,Imp) .+ kron(dG_Xlag,Im_)
test_M2 = kron(dG_het,Im)
test_M3 = J_reshaped
test_M4 = kron(P,Im)
return F_x, F_RShock
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$d0385d4e-08ca-43b1-ae70-8da38175cb6e„§cell_idÙ$d0385d4e-08ca-43b1-ae70-8da38175cb6e¤codeÙ«function log10_grid(amin,amax, na)
pivot = 0.25
grid = Iterators.map(exp10, range(log10(amin + pivot),log10(amax + pivot), na)) .- pivot
grid[1] = amin
return grid
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$d9e401e9-b253-425f-9d98-340088757c2a„§cell_idÙ$d9e401e9-b253-425f-9d98-340088757c2a¤codeÚZmd"""
## Sequence Space Jacobians: Heterogeneous Agents (HA) Block
Having obtained the Steady State of our economy, we now wish to analyze the effects of a monetary policy shock to the central bank's nominal interest rate $i$.
To do that, let's briefly recap the SSJ method: It is based on the idea that an equilibrium in the sequence space can be expressed as a solution to a system of the form
```math
F(X,Z) = 0
```
where $X$ denotes the time-paths of a vector of endogenous variables and $Z$ the time-paths of a vector of exogenous shocks. To get linearized Impulse Response Functions (IRFs), we can truncate the system $F$, compute its Jacobians with respect to $X$ and $Z$ and then obtain the IRFs as $dX = -F_X^{-1}F_Z dZ$ by the Implicit Function Theorem.
Now, the problem is that in an heterogenous agent model, $F$ is potentially quite costly to evaluate, as within the system we need to keep track of household's policy function along the grids and the joint income- and wealth distribution. To overcome the issue, ABRS proposed the efficient "fake news"-algorithm to separately linearize a model "block" containing these complicated objects.
Afterwards, one can then assemble $F_X$ and $F_Z$ around that "HA block". As said, this notebook deals with different ways to do so, but first, let us get the HA Jacobians using the ABRS "fake news" algorithm. Explaining in detail how it works would require quite a bit of text, so read the paper for that. As in the previous notebook, I'll just show you the implementation.
First define a function to get the object denoted as $\mathcal{E}$ in their paper:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$84dec990-0264-4d5d-994f-87cdd7e97a42„§cell_idÙ$84dec990-0264-4d5d-994f-87cdd7e97a42¤codeÚ[md"""
With that, we can proceed as in the previous section. Instead of just the selection matrix `P` there are a few more matrices featuring derivatives of `G_hetinput` that need to be multiplied with the HA Block Jacobian, which now also appears in the expression for $F_Z$. Overall, though, everything works according to the same principle.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$ffc02cf0-d10f-4dcb-a33a-9705ac6ccb12„§cell_idÙ$ffc02cf0-d10f-4dcb-a33a-9705ac6ccb12¤codeÚêbegin
println("Maximum difference for F_x: ",maximum(abs.(F_x1 .- F_x2)))
println("Maximum difference for F_RShock: ",maximum(abs.(F_RShock1 .- F_RShock2)))
#get aggregate variables
AggVars_direct = (-(F_x2\Matrix(F_RShock2))*dRShock)
#extract inlfation response
Ï€_direct = AggVars_direct[1:300,:]
#plot inflation response
plot(1:20,10_000*Ï€_dag[1:20,:], labels = labels, linewidth = 3.0, ylabel = "Basis Points", legend = :topright,title = "Inflation Response (Direct method)")
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$0610663b-a8a9-4182-86eb-98093e77d35b„§cell_idÙ$0610663b-a8a9-4182-86eb-98093e77d35b¤codeÚgmd"""
The function takes as inputs current, lagged and (expected) next period values of the endogenous variables and shocks as well as the outputs of the HA block, denoted as `het_out` in the code.
As for State Space approaches, we need to ensure that the timing of the model is correct and the set of equation sufficient to fully characterize the equilibrium. Otherwise you'll get an error or weird results.
Now, to get the aggregate Jacobians of the model, we just need to differentiate `G_fun` with respect to the relevant inputs, set up some $(n_x\cdot T)\times (n_x\cdot T)$ matrices that have these derivatives at the right places, multiply or add them to the HA Block Jacobian $\mathcal{J}$ and we are done. It might require a bit of thinking to really understand how to do that correctly, in the function below I tried to do it in the simplest way possible.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$331dfd8e-c4ad-4c6d-89c9-691eb5a16f0c„§cell_idÙ$331dfd8e-c4ad-4c6d-89c9-691eb5a16f0c¤codeÚWmd"""
## "Dynare"-like method: Alternative Formulation
Compared to the case above, an issue is that the minimum set of aggregate variables necessary to characterize the dynamics of the economy may not contain all the inputs of the HA block.
For example, the households' wages and assset returns may both be a function of the aggregate capital stock. Or, for this model, if we again choose $Ï€$, $w$ and $Y$ as aggregate variables, the household does not care about inflation per se but rather the real return it induces in conjunction with the nominal rate set by the central bank.
If that is the case, compared to above, we need to write an additional function that takes the aggregate $X$ and shocks $Z$ as inputs and returns the variables the housheolds actually care about (i.e. the inputs of the HA Block). The function below does just that:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$e9772d25-e08b-4888-833c-1a3e79518b22„§cell_idÙ$e9772d25-e08b-4888-833c-1a3e79518b22¤codeÚgmd"""
With this, we can now get our HA Jacobians. We have 3 blocks inputs ($r$, $w$ and the net transfers $d-Ï„$) and two outputs (aggregate labor supply $\mathcal{N}=\int^1_0 s_{i}n_{i}di$ and net savings $\mathcal{A}=\int^1_0 a_i di$), so need to construct 6 Sequence Space Jacobians in total. The code below does that using the functions defined above.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$76149d26-923a-4cd0-b3f6-285d1e00cb85„§cell_idÙ$76149d26-923a-4cd0-b3f6-285d1e00cb85¤codeÙŠmd"""
Everything is the same, as one would hope. And getting the aggregate Jacobians is now essentially as fast as for the DAG method:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$dac31547-cebf-4c1b-be47-5253b4b86abc„§cell_idÙ$dac31547-cebf-4c1b-be47-5253b4b86abc¤codeÚbegin
#put SSJs together
@unpack c_ss,w_ss,r_ss,np,D_ss,a_ss,n_ss,T_level_ss,np = SS_objs
ð’´s, ð““s = get_ð’´_ð’Ÿ(c_ss,w_ss,r_ss,T_level_ss,D_ss,np)
#container for Jacobians
Js = Matrix{Matrix{Float64}}(undef,3,2)
for oo = 1:2 #loop over HA block outputs
if oo == 1
x_ss = a_ss
else
x_ss = n_ss.*np.s_grid'
end
ð“” = get_ð“”(SS_objs.Λ_ss,x_ss,SS_objs.np)
for ii = 1:3 #loop over HA block inputs
Js[ii,oo] = build_ð’¥(ð’´s[ii,oo],ð““s[ii],ð“”,np::NumericalParameters)[1]
end
end
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$d1788944-9c32-4e98-b60e-b8834dba9dde„§cell_idÙ$d1788944-9c32-4e98-b60e-b8834dba9dde¤codeÚ md"""
We see that if the accommodative nominal interest rate shock is really persistent, inflation may react so much that the Central Bank is induced to actually *raise* the nominal rate upon impact of the shock. The *real* rate still decreases on impact though:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f7537f35-7549-4cdc-994d-616394084d3c„§cell_idÙ$f7537f35-7549-4cdc-994d-616394084d3c¤codeÚ8begin
#get F_x and F_R
F_x1, F_RShock1 = get_GE_jacob_DAG(Js,SS_objs)
#get vectors of monetary policy shocks with different persistence
dRShock = -0.0025.*([0.2,0.4,0.6,0.8,0.9]'.^(0:299))
#get aggregate variables
AggVars_DAG = (-(F_x1\Matrix(F_RShock1))*dRShock)
#extract inlfation response
Ï€_dag = AggVars_DAG[1:300,:]
#plot inflation response
labels = ["Ï=0.2" "Ï=0.4" "Ï=0.6" "Ï=0.8" "Ï=0.9"]
plot(1:20,10_000*Ï€_dag[1:20,:], labels = labels, linewidth = 3.0, ylabel = "Basis Points", legend = :topright,title = "Inflation Response (DAG method)")
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$c23a83c1-4009-4378-b229-27b56cf4f640„§cell_idÙ$c23a83c1-4009-4378-b229-27b56cf4f640¤codeÚÏmd"""
Reassuringly, the responses look as in the [ABRS notebook](https://github.com/shade-econ/sequence-jacobian/blob/master/notebooks/hank.ipynb).
## GE Jacobians: Direct Method with HA Jacobians
While this is certainly a matter of taste, I dislike implementing the DAG method manually for models that are a bit more complicated, mainly because setting up the DAG structure, differentiating all the different blocks and chaining them starts feeling tedious at some point.
Thus, I have been exploring different approaches to getting the aggregate Jacobians, which I'd like to share with the hope that others may find them useful as well.
I'll start with an alternative Method, which I'd call "Direct Methodd with HA Jacobian" and which is actually briefly outlined in the ABRS paper at the end of Section 4.1. While ABRS state it to be "less accurate", we will see that this concern is eliminated by the use of an AutoDiff package such as `ForwardDiff.jl`.
The idea is to set up the aggregate system of equation $F$ non-linearily but replace the non-linear HA-block with an linearized approaximation based on the HA SSJs we obtained using the "fake news" method. For the model analyzed here, this would look as follows:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$d1f4ce92-1bd4-42ef-b4ea-e37cdb117579„§cell_idÙ$d1f4ce92-1bd4-42ef-b4ea-e37cdb117579¤codeÙvmd"""
As you may expect, I will again compare the resulting $F_X$, $F_Z$ and inflation response to the DAG method.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$d7530a99-2c7c-4cde-8263-5707b8cd1974„§cell_idÙ$d7530a99-2c7c-4cde-8263-5707b8cd1974¤codeÚifunction get_jacobs_G2(Js,SS_objs)
np = SS_objs.np
#define vectors of steady state variables
Xss = [0.0,SS_objs.Y_ss,SS_objs.w_ss]
het_out_ss = [SS_objs.L_ss,SS_objs.A_ss]
#check that equations hold in SS
check = G2_fun(Xss,Xss,Xss,het_out_ss,0.0,np)
if maximum(abs.(check)) > 1e-8
@warn "large F residuals"
end
#compute derivatives of G2
dG2_X = ForwardDiff.jacobian(x ->
G2_fun(Xss,x,Xss,het_out_ss,0.0,np),Xss)
dG2_Xlag = ForwardDiff.jacobian(x ->
G2_fun(x,Xss,Xss,het_out_ss,0.0,np),Xss)
dG2_XPrime = ForwardDiff.jacobian(x ->
G2_fun(Xss,Xss,x,het_out_ss,0.0,np),Xss)
dG2_het = ForwardDiff.jacobian(x ->
G2_fun(Xss,Xss,Xss,x,0.0,np),het_out_ss)
dG2_RShock = ForwardDiff.derivative(x ->
G2_fun(Xss,Xss,Xss,het_out_ss,x,np),0.0)
#Note: derivative w.r.t. XPrime not needed
#derivatives of HA block inputs
dG_hi = ForwardDiff.jacobian(x -> G_hetinput(Xss,x,Xss,0.0,np),Xss)
dG_hilag = ForwardDiff.jacobian(x -> G_hetinput(x,Xss,Xss,0.0,np),Xss)
dG_hi_Rshocks = ForwardDiff.derivative(x ->
G_hetinput(Xss,Xss,Xss,x,np),0.0)
#define some helper matrices
Im = sparse(I,np.T,np.T)
Im_ = spdiagm(-1=>ones(np.T-1))
Imp = spdiagm(1=>ones(np.T-1))
#get J into suitable shape (again, note ordering of variables)
J_reshaped = [hcat(Js[:,2]...) ; hcat(Js[:,1]...)]
#assemble Aggregate Jacobians
F_x = kron(dG2_het,Im)*J_reshaped*(kron(dG_hi,Im) .+ kron(dG_hilag,Im_)) .+
kron(dG2_X,Im) .+ kron(dG2_XPrime,Imp) .+ kron(dG2_Xlag,Im_)
F_RShock = kron(dG2_het,Im)*J_reshaped*kron(dG_hi_Rshocks,Im_) .+
kron(dG2_RShock,Im)
return F_x, F_RShock
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$06413e4c-9b9e-4360-9222-3449bb751418„§cell_idÙ$06413e4c-9b9e-4360-9222-3449bb751418¤codeÚ+md"""
Again, we will construct the GE Jacobian in terms of same three aggregate variables that fully characterize the aggregate dynamics of our model economy: The above function takes as inputs time-paths of these variables as well as the monetary policy shock and returns the residuals of the "target" equations along the time path.
What I like about this approach is that you don't need to think that much about how to set up the DAG and you don't have to differentiate and chain a lot of small blocks separately. And it gives us the same results:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$dfb79d2c-d29c-4e27-b677-959b3d7b57f1„§cell_idÙ$dfb79d2c-d29c-4e27-b677-959b3d7b57f1¤codeÚ©begin #begin codeblock
function solve_EGM_SS(cGuess::AbstractArray,wSS,
rSS,T_level,np::NumericalParameters;
max_iter::Int = 3000, print::Bool = false)
c0 = cGuess
local c1, a1, n1
#iterate until convergence
dist = 1.0; iter = 0
while (dist>np.tolHH) & (iter < max_iter)
c1, a1, n1 = EGM_update(c0,wSS,T_level,rSS,rSS,np)
#compute distance
dist = maximum(abs.(c1 .- c0))
if print & (rem(iter,100) == 0.0)
println("iteration: ",iter," current distance: ",dist)
end
#update
c0 = copy(c1)
iter += 1
end
#check convergence
println("Done after ",iter," iterations, final distance ",dist)
if iter == max_iter
@warn "Non-convergence of solve_EGM_SS"
end
return c1, a1, n1
end
#helper function providing initial guess
function get_cguess(w,r,np)
incs = w*np.s_grid
return 0.1 .+ 0.1*((1+r)*np.a_grid .+ incs')
end
#additional method definition that calls get_cguess
function solve_EGM_SS(wSS,rSS,T_level,np::NumericalParameters;
max_iter::Int = 3000, print::Bool = false)
c_guess = get_cguess(wSS,rSS,np)
return solve_EGM_SS(c_guess,wSS,rSS,T_level,np, max_iter = max_iter, print = print)
end
end #end codeblock¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$a79aebd9-44b4-4267-87bc-62b8c8723766„§cell_idÙ$a79aebd9-44b4-4267-87bc-62b8c8723766¤codeÙ/F_x2, F_RShock2 = get_jacobs_direct(Js,SS_objs)¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$6555e425-097c-45d3-9fc2-6c579ca099d5„§cell_idÙ$6555e425-097c-45d3-9fc2-6c579ca099d5¤codeÚtmd"""
That time difference shouldn't matter much if you are just interested in solving a model a few times for a given parameterization: After all, approx. 1 second for the direct method is not a long time. However, it will start to matter if you need to compute the Jacobians many times for estimating the model.
Thus, I think this "direct method" may be useful because it easy to implement and a) often models are just calibrated and not estimated anyway and b) even if that is the case, having a simple-to-implement additional method may be useful to check that a manually implemented DAG method gives the right results.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$03eb6ef0-6b44-4d2d-a972-fff171c22772„§cell_idÙ$03eb6ef0-6b44-4d2d-a972-fff171c22772¤codeÚ5#takes in small set of aggr variables and shocks and returns all aggregate variables
function AggVars(Xlag,X,RShocks,np)
@unpack mp = np
@unpack μ, κ, B, θ_π, Z_ss, r_target = mp
#unpack stuff
Ï€i, Y, w = X
Ï€i_, Y_, w_ = Xlag
RShock_, RShock = RShocks
#get aggregates
#nominal rates
i = (r_target + θ_π*πi + RShock)
i_ = (r_target + θ_π*πi_ + RShock_)
#real rate
r = (1+i_)/(1+Ï€i) - 1
#taxes
Ï„ = r*B
#labor
L = Y/Z_ss
#dividend
div = Y - w*L
#net transfer
T_level = div - Ï„
return [w,r,T_level,div,i,Y,Ï„,Ï€i]
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$dbf4d282-2c04-4d9d-b1f1-2ad8f6e9098b„§cell_idÙ$dbf4d282-2c04-4d9d-b1f1-2ad8f6e9098b¤codeÚv@with_kw struct NumericalParameters{F,I}
#include instance of model parameters
mp::ModelParameters{F} = ModelParameters()
# specification for grids
na::I = 500 #no. of points on asset grids
ns::I = 7 #no. of points on skill grid
amin::F = 0.0 #borrowing limit
amax::F = 150.0 #upper end grid
#set grids and transition matrix for s process
s_grid::Vector{F} = exp.(rouwenhorst(ns, mp.Ïs, mp.σs).state_values)
Ps::Matrix{F} = rouwenhorst(ns, mp.Ïs, mp.σs).p
a_grid::Vector{F} = log10_grid(amin,amax,na)
#tolerance value for HH problem
tolHH::F = 1e-9
#size of Sequence Space Jacobians
T::I = 300
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$7e4416f4-1211-45bc-9511-e17028af154b„§cell_idÙ$7e4416f4-1211-45bc-9511-e17028af154b¤codeÚZfunction EGM_update(cPrime::AbstractArray,w,T_level,r,rPrime,np::NumericalParameters)
@unpack mp,a_grid,s_grid,ns,na,amin = np
@unpack β,σ,φ,ν = mp
#get expected marginal utility
EVa = update_EVa(cPrime,rPrime,np)
#invert Euler equation
MU_endog = (β.*EVa)
#use labor supply optimality condition to back out n on endog grid
n_endog = ((w*MU_endog.*s_grid')./φ).^(1/(ν))
#consumption on endog. grid
c_endog = MU_endog.^(-1/σ)
#check consumption is positive
@assert all(x -> x>0, c_endog)
#vector of labor/transfer incomes for different types
inc_endog = w.*s_grid'.*n_endog
#calculate assets today consistent with savings choice
a_endog = (c_endog .- inc_endog .- T_level*s_grid' .+ a_grid)./(1.0 + r)
#below the points, borrowing constraints become binding
constr = a_endog[1,:]
#pre-allocate array
c = zeros(eltype(n_endog),na,ns)
#for interoperability with ForwardDiff:
ag = a_grid .+ zeros(eltype(n_endog),1)
#ensures that the grid has the same type as a_endog
@views for si = 1:ns #loop over ind. producitivity state
#interpolate consumption, ignoring constraint for now
itp_c = LinearInterpolator(a_endog[:,si],c_endog[:,si],NoBoundaries())
c[:,si] = itp_c.(ag)
#find constrained grid points
constr_ind = (a_grid .< constr[si])
#for constrained households: back out consumption using the labor supply optimality condition
if any(constr_ind)
c[constr_ind,si] = solve_c_n_constrained(a_grid[constr_ind],
c[constr_ind,si],
w*s_grid[si],r,T_level*s_grid[si],np)
end
end
#labor supply
n = ((w*((c).^(-σ)).*s_grid')./φ).^(1/(ν))
#savings policy
aPrime = (1+r)*a_grid .+ (w*n .+ T_level).*s_grid' .- c
#impose borrowing limit exactly
aPrime[aPrime .< constr'] .= amin
return c,aPrime, n
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$84f43682-fb05-4b0e-aa37-c12819b4f666„§cell_idÙ$84f43682-fb05-4b0e-aa37-c12819b4f666¤codeÚÝ
md"""
## Conclusion
I hope you found this notebook useful. If yes, please consider starring its [Github repo](https://github.com/mhaense1/SSJ_Julia_Notebook) and share it with colleagues who might also find it useful. Feedback and suggestions are also very welcome.
Finally, since you seem interested in Heterogeneous Agent Macro, have a look at my [Personal Website](https://mhaense1.github.io/) to learn about my research in this area and/or to get in contact with me.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$48e1498e-2d8b-4051-b90c-799ab37c3338„§cell_idÙ$48e1498e-2d8b-4051-b90c-799ab37c3338¤codeÚÇbegin
#get Aggregate Jacobians
F_x3, F_RShock3 = get_jacobs_G(Js,SS_objs)
#obtain impulse response
AggVars_dyn = -F_x3\Matrix(F_RShock3)*dRShock
#extract inflation response
Ï€_dyn = AggVars_dyn[2101:2400,:]
println("Max. abs. difference compared to DAG: ", maximum(abs.(π_dyn .- π_dag)))
plot(1:20,10_000*Ï€_dyn[1:20,:], labels = labels, linewidth = 3.0, ylabel = "Basis Points", legend = :topright,title = "Inflation Response (G method)")
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$a739d105-e648-420b-97b5-435532875218„§cell_idÙ$a739d105-e648-420b-97b5-435532875218¤codeÙtmd"""
In addition to that, we just need a now smaller set of aggregate equations characterizing the equilibrium:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$f0cdbfc2-6216-491b-a874-0313e20558b4„§cell_idÙ$f0cdbfc2-6216-491b-a874-0313e20558b4¤codeÚã#returns takes in aggregate variables and returns inputs to heterogeneous agents block
function G_hetinput(Xlag,X,XPrime,RShock_,np::NumericalParameters)
@unpack mp = np
@unpack μ, κ, B, θ_π, Z_ss, r_target = mp
#unpack variables
Ï€ip, Yp, wp = XPrime
Ï€i, Y, w = X
Ï€i_, Y_, w_ = Xlag
#get nominal and real rates
i_ = (r_target + θ_π*πi_ + RShock_)
r = (1+i_)/(1+Ï€i) - 1
#net transfer level T_level
T_level = Y*(1-w/Z_ss) - r*B
return [w,r,T_level]
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$d03ad22d-5543-4105-aa8d-dfe2ee180050„§cell_idÙ$d03ad22d-5543-4105-aa8d-dfe2ee180050¤codeÚÿmd"""
Overall, I feel this method provides a good compromise between the speed of the DAG method and the convenience of the Dynare-like method.
A downside compared to the former is that it is perhaps not as straightforward to use what ABRS call "Solved Blocks", although it can be adapted to some extent: If it were desirable to have one such "block" in a richer version of this model, one could e.g. add an additional set of inputs named `solved_out` to `G2_fun` and then proceed similar to the HA block.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$17da4c85-a1c8-4204-91df-001a17bd07c4„§cell_idÙ$17da4c85-a1c8-4204-91df-001a17bd07c4¤codeÚüfunction G_fun(Xlag,X,XPrime,het_out,RShock,np::NumericalParameters)
@unpack mp = np
@unpack μ, κ, B, θ_π, Z_ss, r_target = mp
#unpack aggregate variables
wp, rp, T_levelp, divp, ip, Yp , τp , πip = XPrime
w, r, T_level, div, i, Y , τ, πi = X
w_, r_, T_level_, div_, i_, Y_ , τ_, πi_ = Xlag
#unpack HA block outputs
L, A = het_out
#helper variable to get correct eltype for container below
ind_var = Yp + Y + Y_ + L + RShock
#container for equation residuals
eq_out = zeros(eltype(ind_var),length(Xlag))
#NK Philips curve
eq_out[1] += κ*(w/Z_ss - (1/μ)) + 1/(1+rp)*(Yp/Y)*log(1+πip) - log(1+πi)
#Asset market clearing
eq_out[2] += A - B
#labor market clearing
eq_out[3] += Y - Z_ss*L
#dividendeds
eq_out[4] += div - Y*(1-w/Z_ss)
#taxes
eq_out[5] += Ï„ - r*B
#net Transfers
eq_out[6] += T_level - (div - Ï„)
#nominal rate
eq_out[7] += i - (r_target + θ_π*πi + RShock)
#dynamics of real interest rates
eq_out[8] += r - ((1+i_)/(1+Ï€i) - 1)
return eq_out
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$0933d143-a15b-49d9-8ced-9ddc42526a48„§cell_idÙ$0933d143-a15b-49d9-8ced-9ddc42526a48¤codeÙÿbegin
#extract nominal rate response
r_dyn3 = AggVars_dyn3[301:600,:]
#plot nom. rate response
plot(1:20,10_000*r_dyn3[1:20,:], labels = labels, linewidth = 3.0, ylabel = "Basis Points", legend = :topright, title = "Real rate Response (G method 2)")
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$e7a61ca2-b9a3-4dae-97d6-e7d1046acaa2„§cell_idÙ$e7a61ca2-b9a3-4dae-97d6-e7d1046acaa2¤codeÙ#@btime get_jacobs_G2($Js,$SS_objs);¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$404e75c6-77d3-48df-9376-4eaae9cc9827„§cell_idÙ$404e75c6-77d3-48df-9376-4eaae9cc9827¤codeÙ&@btime get_jacobs_direct($Js,$SS_objs)¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$fb08ff71-d0f9-4065-99ec-e4a1f6479072„§cell_idÙ$fb08ff71-d0f9-4065-99ec-e4a1f6479072¤codeÚù#function to get GE jacobians
function get_GE_jacob_DAG(Js,SS_objs)
#unpack parameters and SS objects
@unpack np, w_ss, Y_ss, L_ss, r_ss = SS_objs
@unpack mp, T = np
@unpack B, μ, θ_π, κ, Z_ss = mp
#TxT sparse identity ad 0 matrices
I_sp = sparse(I,T,T) ; sp0 = spzeros(T,T)
#### relevant derivatives firm block
#inputs: Y, w ; outputs: d, L
dL_dY = 1/Z_ss
dL_dw = 0.0
dd_dY = 1 - w_ss*dL_dY
dd_dw = -L_ss
#### relevant derivatives mon pol block
#inputs: π, RShock ; output: r
#derivatives of real rate w.r.t. inflation
#since r = (1 + r^* + θ π_ + ϵ_i)/(1+π) - 1 , both current and future π shows up
dr_dπ = spdiagm(-1 => repeat([θ_π],T-1), 0 => repeat([-(1+r_ss)],T))
dr_dRShock = spdiagm(-1 => repeat([1],T-1))
#### relevant derivatives fisc block
#inputs: r ; outputs: Ï„
dτ_dr = B
#### derivatives of targets
## Target 1: NK Philipps curve (κ(w-1/μ) + 1/(1+r)π(+1) - π = 0)
#inputs: π, w, Y, r
dT1_dw = κ
dT1_Ï€ = spdiagm(0 => repeat([-1],T), 1 => repeat([1/(1+r_ss)],T-1))
dT1_Y = 0.0 ; dT1_r = 0.0
##Labor market clearing (L - ð’© = 0)
#inputs: L, ð’©
dT2_dL = 1.0
dT2_dð’© = -1.0
##Asset market clearing (A - B = 0 )
#inputs: A
dT3_dA = 1.0
#assemble aggregate derivatives
#for Target 1
F_Ï€_T1 = dT1_Ï€ ;
F_Ï€_T2 = dT2_dð’©*(Js[2,2] .- Js[3,2]*dÏ„_dr)*dr_dÏ€ ;
F_π_T3 = dT3_dA*(Js[2,1] .- Js[3,1]*dτ_dr)*dr_dπ ;
F_RShock_T1 = sp0
#for Target 2
F_Y_T1 = sp0
F_Y_T2 = dT2_dð’©*Js[3,2]*dd_dY .+ I_sp*dT2_dL*dL_dY
F_Y_T3 = dT3_dA*Js[3,1]*dd_dY
F_RShock_T2 = dT2_dð’©*Js[2,2]*dr_dRShock .+
dT2_dð’©*Js[3,2]*(-dÏ„_dr)*dr_dRShock
#for Target 3
F_w_T1 = (dT1_dw*I_sp)
F_w_T2 = dT2_dð’©*(Js[1,2] .+ dd_dw*Js[3,2])
F_w_T3 = dT3_dA*(Js[1,1] .+ dd_dw*Js[3,1])
F_RShock_T3 = dT3_dA*Js[2,1]*dr_dRShock .+
dT3_dA*Js[3,1]*(-dτ_dr)*dr_dRShock
#put the above together
F_x = [F_Ï€_T1 F_Y_T1 F_w_T1;
F_Ï€_T2 F_Y_T2 F_w_T2;
F_Ï€_T3 F_Y_T3 F_w_T3]
F_RShock = [F_RShock_T1; F_RShock_T2; F_RShock_T3]
return F_x, F_RShock
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$7097a48b-8906-4197-8012-1b3673ab1d78„§cell_idÙ$7097a48b-8906-4197-8012-1b3673ab1d78¤codeÚ>md"""
Two parameters will be calibrated internally below: The discount factor $\beta$ will be chosen to ensure bond market clearing at a target real interest rate of 0.5% and the labor disutility parameter $\varphi$ is set so that steady state effective labor supply (and thus output) equals 1.
The discretization of the model will work as in my *[previous notebook](https://mhaense1.github.io/SSJ_Julia_Notebook/SSJ_notebook.html)* notebook and thus I will not go into these details here. Below I define an additional structure to store some additional useful objects.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$2f8230fd-a3b6-42c9-8ebc-fed90f15733e„§cell_idÙ$2f8230fd-a3b6-42c9-8ebc-fed90f15733e¤codeSS_objs.np.mp¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$3e4d39b1-34b2-4a63-a081-49db00e00866„§cell_idÙ$3e4d39b1-34b2-4a63-a081-49db00e00866¤codeÚDmd"""
While implementing the DAG method "manually" as above is perfectly doable for this not-that-complicated model, the above function looks somewhat tedious and I found it to be quite easy to make mistakes (try implementing if from scratch yourself). Naturally, a richer model would exacerbate this.
Nevertheless, we can now get the aggregate dynamics in response to a monetary shock. The code below obtains the model IRFs of inflation to a 25 basis point nominal rate disturbance that follows an AR(1)-process with different levels of persistence, ranging from 0.2 to 0.9.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$90c792f4-eef4-4900-813d-b6dc29e8389c„§cell_idÙ$90c792f4-eef4-4900-813d-b6dc29e8389c¤codeÚfunction direct_HAjacob_objective(Ï€_path,Y_path,w_path,RShock_path,Js,SS_objs)
@unpack np,r_ss,T_level_ss, L_ss,Y_ss, w_ss = SS_objs
@unpack mp = np
@unpack θ_π, B, μ, κ, Z_ss = mp
#implied path by labor productivity
L_path = Y_path./Z_ss
#implied nominal rates
i_path = r_ss .+ θ_π*π_path .+ RShock_path
#implied real rates - first period nominal rate is given by r_ss
r_path = (1 .+ vcat(r_ss,i_path[1:end-1]))./(1 .+ π_path) .- 1
#paths for or r_{t+1}, π_{t+1} and Y_{t+1} - assumes economy back in SS after time T
rp_path = vcat(r_path[2:end],r_ss) ; πp_path = vcat(π_path[2:end],0.0)
Yp_path = vcat(Y_path[2:end],Y_ss)
#implied path of dividends
div_path = Y_path .- w_path.*L_path
#implied path of taxes
Ï„_path = r_path*B
#implied path of net transfers
T_level_path = div_path .- Ï„_path
#labor supply path implied by linearized HA block
ð’©_path = L_ss .+ Js[1,2]*(w_path .- w_ss) .+ Js[2,2]*(r_path .- r_ss) .+
Js[3,2]*(T_level_path .- T_level_ss)
#Asset demand path implied by linearized HA block
A_path = B .+ Js[1,1]*(w_path .- w_ss) .+ Js[2,1]*(r_path .- r_ss) .+
Js[3,1]*(T_level_path .- T_level_ss)
#residual NK Philipps curve
resid_1 = κ*(w_path./Z_ss .- 1/μ) .+
(1.0./(1 .+ rp_path)).*(Yp_path./Y_path).*log.(1 .+ πp_path) .-
log.(1 .+ π_path)
#residual labor market clearing condition
resid_2 = L_path .- ð’©_path
#residual assset market clearing condition
resid_3 = A_path .- B
return vcat(resid_1,resid_2,resid_3)
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$e5a48ca3-c3b2-423a-8823-dd672941fd58„§cell_idÙ$e5a48ca3-c3b2-423a-8823-dd672941fd58¤codeÚ°md"""
Finally, you may ask yourself that even though we now got the GE Jacobians for $Y$, $w$ and $\pi$, what about the other variables? Do we still need to do DAG-ing for that?
Of course not, we can again just write a function that gives us the remaining variables as functions of $Y$, $w$ and $\pi$ and the shocks, differentiate it and combine it with the GE Jacobian we already have. This is done by the two functions below:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$e42b4882-d822-43f7-8bec-4ebaffb5d17b„§cell_idÙ$e42b4882-d822-43f7-8bec-4ebaffb5d17b¤codeÚâ#Calibration of the model
@with_kw struct ModelParameters{T}
#HH parameters
β::T = 0.98 #Household discount factor - will be changed by calibration
φ::T = 0.8 #labor disutility - will be changed by calibration
σ::T = 2.0 #relative risk aversion
ν::T = 2.0 #inverse Frisch elasticity
Ïs::T = 0.966 #persistence of HH income process
σs::T = (1.0-Ïs^2)^0.5 * 0.5 #variance for household income process
#aggregates
μ::T = 1.2 #steady state markup 20%
κ::T = 0.1 #Slope of NK Phillips curve
B::T = 5.6 #government bond supply
θ_π::T = 1.5 #CB reaction to inflation
Z_ss::T = 1.0 #steady state labor productivity
#calibration target
r_target::T = 0.005
Y_target::T = 1.0
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$2e4d93d6-c328-4281-8844-c57a085d50e3„§cell_idÙ$2e4d93d6-c328-4281-8844-c57a085d50e3¤codeÚÄfunction G2_fun(Xlag,X,XPrime,het_out,RShock,np::NumericalParameters)
@unpack mp = np
@unpack μ, κ, B, θ_π, Z_ss, r_target = mp
#unpack stuff:
Ï€ip, Yp, wp = XPrime
Ï€i, Y, w = X
Ï€i_, Y_, w_ = Xlag
L, A = het_out
#helper variable to get correct eltype for container below
ind_var = Yp + Y + Y_ + L + RShock
#get nominal and real rates
i = (r_target + θ_π*πi + RShock)
rp = (1+i)/(1+Ï€ip) - 1
#container for residuals
eq_out = zeros(eltype(ind_var),length(Xlag))
#NK Philips curve
eq_out[1] += κ*(w/Z_ss - (1/μ)) + 1/(1+rp)*(Yp/Y)*log(1+πip) - log(1+πi)
#labor market clearing
eq_out[2] += Y - Z_ss*L
#Asset market clearing
eq_out[3] += A - B
return eq_out
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$12e89139-f672-4265-bd5a-5639e3f96a5f„§cell_idÙ$12e89139-f672-4265-bd5a-5639e3f96a5f¤codeÙÌbegin
function inv_dist(Î ::AbstractArray)
#Î is a Stochastic Matrix
x = [1; (I - Î '[2:end,2:end]) \ Vector(Î '[2:end,1])]
return x./sum(x) #normalize so that vector sums up to 1.
end
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$ca2de9d9-879f-46ac-a02e-b5af6ad20f6f„§cell_idÙ$ca2de9d9-879f-46ac-a02e-b5af6ad20f6f¤codeÙ|md"""
Finally, a function that assembles the "fake news" matrix $\mathcal{F}$ and the HA block Jacobian $\mathcal{J}$:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$7de8f598-dd30-428a-afb6-4cc372e1c4f7„§cell_idÙ$7de8f598-dd30-428a-afb6-4cc372e1c4f7¤codeÙ%@btime -($F_x1)\Matrix(($F_RShock1));¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$3708fb95-a65d-4a1f-bb5e-f4db9a8766b9„§cell_idÙ$3708fb95-a65d-4a1f-bb5e-f4db9a8766b9¤codeÙµ#compute expected marginal value function
function update_EVa(cpol::AbstractArray,r::T,np::NumericalParameters) where T<:Real
return (1.0+r).*((cpol.^(-np.mp.σ)))*np.Ps'
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$b18c850d-1edd-4d30-b67f-5c6a21f2a7fc„§cell_idÙ$b18c850d-1edd-4d30-b67f-5c6a21f2a7fc¤codeÙ2md"""
Let's briefly test this for some values:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$9c53d4a7-526d-4065-88b5-0ef5e04f9b62„§cell_idÙ$9c53d4a7-526d-4065-88b5-0ef5e04f9b62¤codeÙèmd"""
Next, the below gets us the objects $\mathcal{D}$ and $\mathcal{Y}$ for which we need to conduct the backwards iteration. This works as in the previous notebook, except that we have more inputs and outputs of the HA block.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$19ea0f40-4b3e-11ef-1820-4992f2ba942f„§cell_idÙ$19ea0f40-4b3e-11ef-1820-4992f2ba942f¤codeÚ Fmd"""
# SSJ Notebook II - General Equilibrium with and without DAGs
## by [Matthias Hänsel](https://mhaense1.github.io/)
This notebook serves as a sequel to my *[previous notebook](https://mhaense1.github.io/SSJ_Julia_Notebook/SSJ_notebook.html)* in which I demonstrated how to solve a neoclassical Heterogenous Agents (HANC) business cycle model using the Sequence Space Jacobian (SSJ) linearization method proposed by *[Auclert et al. (2021 - henceforth ABRS)](https://onlinelibrary.wiley.com/doi/full/10.3982/ECTA17434)*.
While the HANC model as in my original notebook is attractive for learning purposes due to its simplicity, it essentially abstracted from another challenge of the ABRS method, getting the General Equilibrium (GE) Jacobians.
ABRS propose a specific method for this, which is based on chaining partial equilibrium Jacobians along a Directed Acyclical Graph (DAG).
However, I personally found that just from reading the paper, it is not that easy to grap how that needs to be done in practice and while ABRS's comprehensive *[Python package](https://github.com/shade-econ/sequence-jacobian)* implements the DAG method very efficiently, its "black box" nature does not make it easy to figure it out by looking at their code.
Furthermore, once you actually understood it, it turns out that implementing the DAG method without a package automating everything can be somewhat tedious and error-prone (at least in my opinion). Thus, in this notebook I will not only implement the DAG method manually for a simple HANK model but also demonstrate **two alternative ways** to obtain the GE Jacobians:
1. **A Direct Method employing the HA Jacobians:**
This idea is briefly mentioned in the ABRS paper and while inefficient, I find it conceptionally much simpler than the DAG method.
2. **A "Dynare-like" method:**
It is also possible to get Aggregate Jacobians by writing the aggregate parts of the model down in a way similar as one would e.g. in [Dynare](dynare.org). This idea was inspired by insights in [Bhandari et al. (BBEG)](https://www.nber.org/papers/w31744), who develop an interesting alterntive to the SSSJ method.
All methods are implemented relying solely on *automatic differentiation* through the `ForwardDiff.jl` package.
The example model for this notebook is the One-Asset HANK model also used in the original ABRS paper, which is briefly presented below. While still relatively simple, it has a more involved "macro" block due the presence of a Forward-Looking Phillips curve, endogenous labor supply and so on.
Additionally, ABRS have a [Python notebook](https://github.com/shade-econ/sequence-jacobian/blob/master/notebooks/hank.ipynb) also implementing the model so we can check the results here against the "original".
The notebook assumes some familiarity with Julia and features the following parts:
1. Description of the model
2. Solving/Calibrating the steady state
3. Obtaining the Heterogeneous Agent Jacobian
4. Different ways to obtain the GE Jacobians
As previously, I tried to focus more on a *clear* and *simple* implementation instead one optimized for speed.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$b2897cf6-720a-45b5-860d-85044cdd7f28„§cell_idÙ$b2897cf6-720a-45b5-860d-85044cdd7f28¤codeÚœmd"""
Nevertheless, while the above worked and conveniently returns the Jacobians and time-paths for *all* model variables at the same time, that is not always desirable: Effectively, that gave us a bigger $F_X$-Matrix and inverting big matrices is costly. Here the difference is not so big, but if you have, say, 20 aggregate variables and a longer truncation horizon (say $T=500$), it will start to matter.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$09f07dd5-a950-44f7-8a20-6070a8fa45b3„§cell_idÙ$09f07dd5-a950-44f7-8a20-6070a8fa45b3¤codeÙŸmd"""
Since $F_X$ and $F_Z$ are different here as they relate to bigger number of variables, let us instead test whether we get the same results as before:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$53188374-85e0-41f5-aa3b-a89a339cd974„§cell_idÙ$53188374-85e0-41f5-aa3b-a89a339cd974¤codeÙ%@btime -($F_x3)\Matrix(($F_RShock3));¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$d9fd8c94-f1a2-4628-b93f-51361f6a3efa„§cell_idÙ$d9fd8c94-f1a2-4628-b93f-51361f6a3efa¤codeÚÎfunction assemble_G_Aggr(GEJ,SS_objs,np)
#define vectors of steady state variables
Xss = [0.0,SS_objs.Y_ss,SS_objs.w_ss]
het_out_ss = [SS_objs.L_ss,SS_objs.A_ss]
#derivatives of aggregate variables w.r.t variables
dAggr_X = ForwardDiff.jacobian(x -> AggVars(Xss,x,zeros(2),np),Xss)
dAggr_Xlag = ForwardDiff.jacobian(x -> AggVars(x,Xss,zeros(2),np),Xss)
dAgg_RShocks = ForwardDiff.jacobian(x -> AggVars(Xss,Xss,x,np),zeros(2))
#define the helper matrices
Im = sparse(I,np.T,np.T)
Im_ = spdiagm(-1=>ones(np.T-1))
Imp = spdiagm(1=>ones(np.T-1))
d_Aggr_X = kron(dAggr_X,Im) .+ kron(dAggr_Xlag,Im_)
G_Aggr = (kron(dAgg_RShocks[:,2],Im) .+ kron(dAgg_RShocks[:,1],Im_) .+ d_Aggr_X*GEJ)
return G_Aggr
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$e1ec7322-f8fc-428f-85db-04ccbabf023f„§cell_idÙ$e1ec7322-f8fc-428f-85db-04ccbabf023f¤codeÙâmd"""
Indeed, we see that the Jacobians obtained using the DAG method and the Direct Method with HA Jacobian are the same (up to some small numerical error). Let's also plot the inflation response as visual demonstration.
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$24bdcddd-af1f-430c-bed5-3cda6536f4a1„§cell_idÙ$24bdcddd-af1f-430c-bed5-3cda6536f4a1¤code§Js[1,1]¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$6f160a63-8fec-4f8b-a2ea-0dc691e30d30„§cell_idÙ$6f160a63-8fec-4f8b-a2ea-0dc691e30d30¤codeÙèmd"""
As the plot above confirms, the model responses to the nominal rate shocks are the same as for the DAG method (up to some Floating Point Error). And in contrast to the Direct method, getting $F_X$ and $F_Z$ is rather fast:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$cd3b00b6-cfc5-4096-8c63-601ae7cd214e„§cell_idÙ$cd3b00b6-cfc5-4096-8c63-601ae7cd214e¤codeÚîmd"""
That worked, nice!
For getting the steady state distribution, I again use the Young (2010) histogram method. Since this was already discussed in the *[previous notebook](https://mhaense1.github.io/SSJ_Julia_Notebook/SSJ_notebook.html)*, I don't go into details and the functions that build the necessary transition matrix $\Lambda$ and solve for the stationary distribution $D$ are relegated to the hidden code fields below (they are called `build_Λ` and `inv_dist`, respectively).
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$1e2773ef-0dd5-4c6e-ae9c-93001fd2e0ff„§cell_idÙ$1e2773ef-0dd5-4c6e-ae9c-93001fd2e0ff¤codeÙ1SS_objs = get_steady_state(NumericalParameters())¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$54402a9f-56df-439f-98b8-64d15ad0705e„§cell_idÙ$54402a9f-56df-439f-98b8-64d15ad0705e¤codeÙmd"""
We can check our calibrated parameter values for $\beta$ as well as $\varphi$ and see that they correspond to those obtained by ABRS:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$7ea684fb-e2a5-4935-b18e-f491b4c098b4„§cell_idÙ$7ea684fb-e2a5-4935-b18e-f491b4c098b4¤codeÚmd"""
## Solving and Calibrating the Steady State
We are now ready to calibrate/solve the model's steady state. As mentioned above, I will choose $\beta$ so as to induce an SS interest rate of $r=0.005$ and $\varphi$ to achieve $L=Y=1$. The below function implements this:
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$a40334ec-d57f-477e-97ad-9aa82b7ff46d„§cell_idÙ$a40334ec-d57f-477e-97ad-9aa82b7ff46d¤codeÚŽ#ojective function for solver to calibrate β and φ
function β_φ_objective(β,φ,p,np)
@unpack s_grid = np
w_ss, r_ss, T_level_ss = p
@set! np.mp.β = β
@set! np.mp.φ = φ
#solve HH problem
c_ss, a_ss, n_ss = solve_EGM_SS(w_ss,r_ss,T_level_ss,np)
#build transition matrix
Λ = build_Λ(a_ss, np)
#get invariant distribution
D = inv_dist(Λ)
#aggregate capital stock implied by HH savings
A_ss = sum(reshape(D,(np.na,np.ns)).*np.a_grid)
L_ss = sum((reshape(D,(np.na,np.ns))).*(n_ss.*s_grid'))
dists = [(A_ss/np.mp.B) - 1, (L_ss/np.mp.Y_target) - 1]
#return relative distance to target
return dists
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$a7d93065-0d0c-45ac-b497-79845fccf27a„§cell_idÙ$a7d93065-0d0c-45ac-b497-79845fccf27a¤code±TableOfContents()¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$2d6e16f7-a39f-400a-8645-91048ed7e120„§cell_idÙ$2d6e16f7-a39f-400a-8645-91048ed7e120¤codeÙ…md"""
(Yes, these responses look somewhat ugly. Feel free to convince yourself that the ones in the ABRS notebook look the same.)
"""¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÃÙ$20688c81-1b2c-4a1e-bfb6-38f65c0e2d6f„§cell_idÙ$20688c81-1b2c-4a1e-bfb6-38f65c0e2d6f¤codeÚJbegin
G_Aggr = assemble_G_Aggr(GEJ,SS_objs,np)
AggVars_dyn3 = G_Aggr*dRShock
#extract nominal rate response
i_dyn3 = AggVars_dyn3[1201:1500,:]
#plot nom. rate response
plot(1:20,10_000*i_dyn3[1:20,:], labels = labels, linewidth = 3.0, ylabel = "Basis Points", legend = :topright, title = "Nom. rate Response (G method 2)")
end¨metadataƒ©show_logsèdisabled®skip_as_script«code_foldedÂÙ$3dc4a1f0-7d85-42c9-a8e3-4de80f604b1b„§cell_idÙ$3dc4a1f0-7d85-42c9-a8e3-4de80f604b1b¤codeÙ@solve_EGM_SS(1/1.2,0.01,0.01,NumericalParameters(),print = true)¨metadataƒ©show_logsèdisabled®skip_as_script«code_folded«notebook_idÙ$2dbc6722-4078-11f0-3ded-2be85a7a1069«in_temp_dir¨metadata€ 