Micro- and Macroeconomic Implications of Very Impatient Households

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Introduction¶

Buffer stock saving models of the kind implemented in $\text{𝙲𝚘𝚗𝚜𝙸𝚗𝚍𝚂𝚑𝚘𝚌𝚔𝚃𝚢𝚙𝚎}$ say that, if a standard ‘Growth Impatience Condition’, holds:

then the ratio of asets $𝐚$ to permanent income $𝐩$, $a=𝐚/𝐩$, has a target value $\check{a}$ that depends on the consumer’s preferences (relative risk aversion $\rho$, time preference $\beta$) and circumstances (interest factor $𝖱$, growth factor $\mathrm{\Gamma }$, uncertainty about permanent income shocks ${\sigma }_{\psi }^2$).

If everyone had identical preferences and everyone were at their target $\check{a}$, then inequality in the level of $𝐚$ would be exactly the same as inequality in $𝐩$.

“The Distribution of Wealth and the Marginal Propensity to Consume” (Carroll, Slacalek, Tokuoka, and White 2017; hereafter: “cstwMPC”) shows that, when such a model is simulated and agents draw their idiosyncratic shocks (so, agents are ex post heterogeneous -- see the definition in Intro-To-HARK) -- asset inequality is indeed close to $𝐩$ inequality even though everyone is not always at exactly their target $a$.

But a large body of evidence shows that actual inequality in assets is much greater than actual inequality in permanent income. Thus, to make a model that qualifies as what cstwMPC call a ‘serious’ microfounded macro model of consumption (one that matches the key facts theory says should be first-order important), the model must be modified to incorporate some form of ex ante heterogeneity: That is, there must be differences across people in $\beta$ or $𝖱$ or $\rho$ or $\mathrm{\Gamma }$ or ${\sigma }_{\psi }^2$.

The most transparent and simplest of these to change is the time preference factor $\beta$. So that is what the paper does. The main results are:

1. The distribution of $\beta$ need not be particularly wide to match the extreme concentration of wealth: roughly 0.91 to 0.98 (annual); that is, the most impatient person discounts the future about 6 percentage points more per year than the most patient agent agent

2. With such a distribution of $\beta$, simulated agents’ (annual) marginal propensity to consume (MPC) from transitory income shocks to income matches large body of microeconomic evidence that typically finds evidence of MPC’s in the range of 0.2 to 0.6. This is much better than RA macro models that typically yield MPC’s in the range of 0.01 to 0.05.

While the most impatient agents in the cstwMPC model have fairly high MPCs (~0.6 annual), there is microeconomic evidence that a significant fraction of households have even higher MPCs than the model predicts, especially at the quarterly frequency. This group of households is commonly referred to as “hand-to-mouth” -- they consume most of their transitory shocks to income not too long after they receive them (mostly within a quarter). There are several reasons why a household could be hand-to-mouth, but one plausible explanation is that these households are even more impatient than estimated by cstwMPC for the most impatient agent.

Calibrating a Basic Version of cstwMPC¶

To get started, let’s reproduce a simplified version of the main results from cstwMPC.

In cstwMPC, the authors calibrated nearly all of the model parameters-- risk aversion, income shock process, etc-- to commonly used or previously estimated values. The only parameter to be estimated is the distribution of $\beta$. cstwMPC assumed that $\beta$ is uniformly distributed on $[\stackrel{`}{\beta }-\nabla ,\stackrel{`}{\beta }+\nabla ]$, approximated by a seven point distribution.

Their estimation procedure seeks the values of $\stackrel{`}{\beta }$ and $\nabla$ that generate a simulated distribution of wealth that best matches empirical U.S. data. Their definition of “best match” has two aspects:

1. The simulated aggregate capital-to-income ratio matches the true U.S. value.

2. The sum of squared distances between the simulated and empirical Lorenz curves (at the 20th, 40th, 60th, and 80th percentiles) is minimized (conditional on item 1).

cstwMPC’s target empirical moments are a capital-to-income ratio of 10.26 and cumulative wealth shares as given in the table below. Yes, you are reading the table correctly: The “poorest” 80 percent of households own 17.5 percent of wealth.

To reproduce their basic results, we must import an $\text{𝙰𝚐𝚎𝚗𝚝𝚃𝚢𝚙𝚎}$ subclass and define a dictionary with calibrated parameters identical to those in the paper.

• Start with importing the predefined dictionaries for agent types from the HARK toolkit. Then only show the parameters that are actually different from the imported dictionary.

Creating an ex-ante distribution of heterogeneous agents¶

Now let’s make several instances of our class of agents and give them different values of $\beta$, following cstwMPC’s estimated distribution. In our specification of interest, we will use $\stackrel{`}{\beta }=0.9855583$ and $\nabla =0.0085$.

NB: Reported parameter estimates in cstwMPC use a model with aggregate shocks and wage and interest rates determined dynamically (a heterogeneous agents DSGE model); this is the $\text{𝙰𝚐𝚐𝚂𝚑𝚘𝚌𝚔𝙲𝚘𝚗𝚜𝚞𝚖𝚎𝚛𝚃𝚢𝚙𝚎}$ in HARK. The estimated parameters are slightly different in this exercise, as we are ignoring general equilibrium aspects and only using the $\text{𝙸𝚗𝚍𝚂𝚑𝚘𝚌𝚔𝙲𝚘𝚗𝚜𝚞𝚖𝚎𝚛𝚃𝚢𝚙𝚎}$

Method 1: “Brute force” approach¶

There are two methods for accomplishing this. The first is a more “brute force” approach: We specify a uniform distribution and discretize it for the number of household types we would like to see in the model. From there, We place each of the household types with a different time preference factor in a list. Finally, a for-loop is used to compute the solution for each household’s optimization problem.

Method 2: The new AgentPopulation class¶

The second approach incorporates a new class defined in the HARK toolkit: AgentPopulation. This class will create a new iterable object AgentPopulation, which takes two arguments. The first is the dictionary which defines the AgentType. The second is a distribution over the parameter value for which the user would like to see ex-ante heterogeneity in. From there, the approx_distributions method is called on the object to discretize the distribution given the specified number of discrete points. Finally, the create_distributed_agents method is called to create the distribution of agents, for which the standard solution method calls can be used to solve each agents problem.

Solving and Simulating the Baseline Agents¶

Now let’s solve and simulate each of our types of agents. If you look in the parameter dictionary (or at any of the agent objects themselves), you will see that each one has an $\text{𝙰𝚐𝚎𝚗𝚝𝙲𝚘𝚞𝚗𝚝}$ attribute of 10000. That is, these seven ex ante heterogeneous types each represent ten thousand individual agents that will experience ex post heterogeneity when they draw different income (and mortality) shocks over time.

In the code block below, fill in the contents of the loop to solve and simulate each agent type for many periods. To do this, you should invoke the methods $\text{𝚜𝚘𝚕𝚟𝚎}$, $\text{𝚒𝚗𝚒𝚝𝚒𝚊𝚕𝚒𝚣𝚎\_𝚜𝚒𝚖}$, and $\text{𝚜𝚒𝚖𝚞𝚕𝚊𝚝𝚎}$ in that order. Simulating for 1200 quarters (300 years) will approximate the long run distribution of wealth in the population.

After solving the model, we will check that the aggregate level of capital (total assets held by all households) to income ratio equals what we expected it would be. This will serve as a verification that the code is written correctly. To do this, we combine the asset holdings of all types, take the mean, and compare it to the desired capital to income ratio of 10.26.

NB: Because there is no permanent income growth in this model, all shocks are mean one and idiosyncratic, and we have many agents, aggregate or average income is 1.0.

1. Solution for the “brute force” approach¶

Here is the output for the solution to the first approach to the problem, as well as a computation of the mean level of assets across the agent types. As you can see, the model does a good job of matching the aggregate capital-to-output ratio found in the data, as cited by Carroll et al. (2017).

2. Solution using the AgentPopulation class¶

We now conduct the same exercise for the new tool implemented in HARK. As you can see, both the out for the solution and the mean level of assets are nearly identical to the output when using the “brute force” approach.

Note: As an aside, the working theory for why these two approaches give nearly-but-not-perfectly-identical solution outputs is that, the “brute force” approach requires the user to specify the random seed when creating the discretized distribution of agents. The approach using AgentPopulation, on the other hand, specifies the random seed for the user during the implementation of the code. As you may know from the basics of programming in Python, the random seed’s main function is for reproducibility of results when using random number generators. In order for this theory to be tested, the AgentPopulation tool in HARK must be updated to allow for users to fix the random seed in this way.

Plotting the Lorenz Curve¶

Next, we conduct the exercise of matching the inequality in the observable distribution of wealth using the distribution of assets that arises from the model we’ve constructed with ex-ante heterogeneity in the time preference factor. As you can see from the plots below, the matching exercise is again nearly identical from the two approaches.

Calculating the Lorenz Distance at Targets¶

Now we want to construct a function that calculates the Euclidean distance between simulated and actual Lorenz curves at the four percentiles of interest: 20, 40, 60, and 80.

NB: For some reason, this step is not completed in the code. We leave this cell here, as this exercise should probably be performed as well.

The Distribution Of the Marginal Propensity to Consume¶

For many macroeconomic purposes, the distribution of the MPC $\kappa$ is more important than the distribution of wealth. Ours is a quarterly model, and MPC’s are typically reported on an annual basis; we can compute a (very) approximate annual MPC from the quraterly ones as ${\kappa }_Y\approx 1.0-{(1.0-{\kappa }_Q)}^4$

In the cell below, we retrieve the MPCs from our simulated consumers and show that the 10th percentile in the MPC distribution is only about 6 percent, while at the 90th percentile it is almost 0.5

1. MPC distribution from the “brute force” approach¶

Again, we first show the computation from the “brute force approach”. This is done in the following code block.

2. MPC distribution from the AgentPopulation approach¶

Next, we compare this distribution of marginal propensities to consume with the one from the analogous computation when the AgentPopulation class is used. Notably, the distribution of MPCs in this case is nearly identical, with the only difference being at the 90th percentile.

Adding Very Impatient Households¶

Now that we have some tools for examining both microeconomic (the MPC across the population) and macroeconomic (the distribution and overall level of wealth) outcomes from our model, we are all set to conduct our experiment.

In this exercise, we are going to add very impatient households to the economy in a very direct way: by replacing the most impatient consumer type with an even more impatient type. Specifically, we will have these agents have a discount factor of $\beta =0.80$ at a quarterly frequency, which corresponds to $\beta \approx 0.41$ annual.

In the code block below, we:

1. Replicate the list of agents using $\text{𝚍𝚎𝚎𝚙𝚌𝚘𝚙𝚢}$.

2. Set the $\beta$ of the most impatient type to 0.80 (for the copied set of agents).

3. Solve and simulate the most impatient type (for the copied set of agents).

We conduct this exercise for both approaches again. The code is essentially identical, other than the naming conventions used for variables in each case. Notice that the output from each of the final two code blocks are exactly identical!