1. Human capital accumulation

Consider a worker living in two periods, $t \in \{1,2\}$.

In each period she decides whether to work ($l_t = 1$) or not ($l_t = 0$).

She can not borrow or save and thus consumes all of her income in each period.

If she works her consumption becomes:

where $w$ is the wage rate and $h_t$ is her human capital.

If she does not work her consumption becomes:

where $b$ is the unemployment benefits.

Her utility of consumption is:

Her disutility of working is:

From period 1 to period 2, she accumulates human capital according to:

where $\Delta$ is a stochastic experience gain.

In the second period the worker thus solves:

In the first period the worker thus solves:

where $\beta$ is the discount factor and $\mathbb{E}_{1}\left[v_2(h_2)\right]$ is the expected value of living in period two.

The parameters of the model are:

The relevant levels of human capital are:

Question 1: Solve the model in period 2 and illustrate the solution (including labor supply as a function of human capital).

Question 2: Solve the model in period 1 and illustrate the solution (including labor supply as a function of human capital).

Question 3: Will the worker never work if her potential wage income is lower than the unemployment benefits she can get? Explain and illustrate why or why not.

2. AS-AD model

Consider the following AS-AD model. The goods market equilibrium is given by

where $y_{t}$ is the output gap, $r_{t}$ is the ex ante real interest and $v_{t}$ is a demand disturbance.

The central bank's Taylor rule is

where $i_{t}$ is the nominal interest rate, $\pi_{t}$ is the inflation gap, and $\pi_{t+1}^{e}$ is the expected inflation gap.

The ex ante real interest rate is given by

Together, the above implies that the AD-curve is

Further, assume that the short-run supply curve (SRAS) is given by

where $s_t$ is a supply disturbance.

Inflation expectations are adaptive and given by

Together, this implies that the SRAS-curve can also be written as

The parameters of the model are:

Question 1: Use the sympy module to solve for the equilibrium values of output, $y_t$, and inflation, $\pi_t$, (where AD = SRAS) given the parameters ($\alpha$, $h$, $b$, $\alpha$, $\gamma$) and $y_{t-1}$ , $\pi_{t-1}$, $v_t$, $s_t$, and $s_{t-1}$.

Question 2: Find and illustrate the equilibrium when $y_{t-1} = \pi_{t-1} = v_t = s_t = s_{t-1} = 0$. Illustrate how the equilibrium changes when instead $v_t = 0.1$.

Persistent disturbances: Now, additionaly, assume that both the demand and the supply disturbances are AR(1) processes

where $x_{t}$ is a demand shock, and $c_t$ is a supply shock. The autoregressive parameters are:

Question 3: Starting from $y_{-1} = \pi_{-1} = s_{-1} = 0$, how does the economy evolve for $x_0 = 0.1$, $x_t = 0, \forall t > 0$ and $c_t = 0, \forall t \geq 0$?

Stochastic shocks: Now, additionally, assume that $x_t$ and $c_t$ are stochastic and normally distributed

The standard deviations of the shocks are:

Question 4: Simulate the AS-AD model for 1,000 periods. Calculate the following five statistics:

1. Variance of $y_t$, $var(y_t)$
2. Variance of $\pi_t$, $var(\pi_t)$
3. Correlation between $y_t$ and $\pi_t$, $corr(y_t,\pi_t)$
4. Auto-correlation between $y_t$ and $y_{t-1}$, $corr(y_t,y_{t-1})$
5. Auto-correlation between $\pi_t$ and $\pi_{t-1}$, $corr(\pi_t,\pi_{t-1})$

Question 5: Plot how the correlation between $y_t$ and $\pi_t$ changes with $\phi$. Use a numerical optimizer or root finder to choose $\phi\in(0,1)$ such that the simulated correlation between $y_t$ and $\pi_t$ comes close to 0.31.

Quesiton 6: Use a numerical optimizer to choose $\sigma_x>0$, $\sigma_c>0$ and $\phi\in(0,1)$ to make the simulated statistics as close as possible to US business cycle data where:

1. $var(y_t) = 1.64$
2. $var(\pi_t) = 0.21$
3. $corr(y_t,\pi_t) = 0.31$
4. $corr(y_t,y_{t-1}) = 0.84$
5. $corr(\pi_t,\pi_{t-1}) = 0.48$

3. Exchange economy

Consider an exchange economy with

1. 3 goods, $(x_1,x_2,x_3)$
2. $N$ consumers indexed by $j \in \{1,2,\dots,N\}$

3. Preferences are Cobb-Douglas with log-normally distributed coefficients

   $\begin{aligned} u^{j}(x_{1},x_{2},x_{3}) &= \left(x_{1}^{\beta_{1}^{j}}x_{2}^{\beta_{2}^{j}}x_{3}^{\beta_{3}^{j}}\right)^{\gamma}\\ \beta_{i}^{j} &= \frac{\alpha_{i}^{j}}{\alpha_{1}^{j}+\alpha_{2}^{j}+\alpha_{3}^{j}} \\ \boldsymbol{\alpha}^{j} &= (\alpha_{1}^{j},\alpha_{2}^{j},\alpha_{3}^{j}) \\ \log(\boldsymbol{\alpha}^j) &\sim \mathcal{N}(\mu,\Sigma) \\ \end{aligned}$

4. Endowments are exponentially distributed,

Let $p_3 = 1$ be the numeraire. The implied demand functions are:

where consumer $j$'s income is

The parameters and random preferences and endowments are given by:

Question 1: Plot the histograms of the budget shares for each good across agents.

Consider the excess demand functions:

Question 2: Plot the excess demand functions.

Question 3: Find the Walras-equilibrium prices, $(p_1,p_2)$, where both excess demands are (approximately) zero, e.g. by using the following tâtonnement process:

1. Guess on $p_1 > 0$, $p_2 > 0$ and choose tolerance $\epsilon > 0$ and adjustment aggressivity parameter, $\kappa > 0$.
2. Calculate $z_1(p_1,p_2)$ and $z_2(p_1,p_2)$.
3. If $|z_1| < \epsilon$ and $|z_2| < \epsilon$ then stop.
4. Else set $p_1 = p_1 + \kappa \frac{z_1}{N}$ and $p_2 = p_2 + \kappa \frac{z_2}{N}$ and return to step 2.

Question 4: Plot the distribution of utility in the Walras-equilibrium and calculate its mean and variance.

Question 5: Find the Walras-equilibrium prices if instead all endowments were distributed equally. Discuss the implied changes in the distribution of utility. Does the value of $\gamma$ play a role for your conclusions?