A two agents model of inequalities.

Business Cycles and Fluctuations - AE2E6

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# uncomment the next lines, the first time you run
# import Pkg; Pkg.add(["DBnomics", "Dynare", "DataFrames"])

Preference for wealth and marginal propensity to consume

For now, we consider a single representative agent. She has the ability to buy a two periods bond, yielding 1 after one period. The price of the bond at any date is \(q\), hence its (riskfree) interest rate is \(r=1/q\).

Agent values consumption \(c_t\) and wealth \(b_t q_t\) so that she maximizes1:

\[\max \sum_t \beta^t \left( \frac{c_t^{1-\frac{1}{\sigma}}}{1-\frac{1}{\sigma}}+ \varphi \frac{ (1+b_t)^{1-\frac{1}{\eta}} } {1-\frac{1}{\eta}} \right)\]

under the budget constraint

\[c_t = y_t + b_{t-1} - b_t q_t\]

where \(y_t\) is exogenous income following AR1

\[(y_t-\overline{y})=\rho (y_{t-1}-\overline{y}) + \epsilon^y_t\]

using Dynare

  1. Write down the optimality condition for debt holdings.

  2. What are the equations defining the deterministic equilibrium?

  3. Inspect and run one_agent.mod model. Show that there is a unit root. Can you interpret it?

  4. What is the consumption response to a temporary income shock? To a permanent one? (with autocorrelation \(\rho=0.9\) and \(\rho=1.0\))

  5. In the modfile, add a preference for wealth term in the utility function and adjust the calibration of beta accordingly.

  6. Simulate the response to a temporary and a persistent shock. Given phi what is the effect of eta?

A two agents model

We now assume there are two agents: bottom and top earners. Top earners amount for a fraction \(\chi\) of the total population. Together they earn a fraction \(z\in[0,1]\) of the total production \(y\) which is an AR1 process as in the first part. The rest goes to the bottom earners.

Top earners can save by lending to bottom earners. We denote by \(B_t\) the total quantity of riskfree bonds, traded at \(q_t\). Note that debt per capita is \(\frac{B_t}{\chi}\) for top earners and \(\frac{B_t}{1-\chi}\) for bottom earners. Top earners have preference for wealth as in the first part, while bottom earners have standard preferences (with \(\varphi=0\))

  1. Write down the budget equations for both agents. What are the new Euler equations? Check that it is consistent with the two_agents.mod modfile. What are the per capita variables?

  2. What is qualitatively the effect of a permanent redistributive shock? (simulate the model)

Calibrating and simulating the model

The model in the modefile is pre-calibrated to match US data in 1983. Assume the model is in equilibrium for an initial level of debt \(B=0.91\) (which is the debt/gdp ratio in the us in 1983).

Taking \(\varphi=0.05\) as constant, for any given choice of \(\eta\), there is a unique value of \(\beta\) consistent with the equilibrium as in the one agent case.

Now we would like to calibrate the value of \(\eta\) so as to match the marginal propensity to save of top earners which was approximately 50% in 1983.

Since the two agents model is already calibrated for most variables, we reuse it rather than adapting the one agent model.

  1. In the model two_agents.mod replace the Euler equation of bottom earner by q = 1/rbar. Justify why, from the top earners perspective, the model is now equivalent to a single agent model.

@. Use the modified model to compute the marginal propensity out of a permanent income shock after 6 periods. Choose parameter eta so that this m.p.s. is approximatively 50%.

  1. What is the effect of a 10% permanent increase in inequalities? Over 30 years and in the long run?

Footnotes

  1. this is the “preference for wealth” specification↩︎