Problem Set 3
(Due Monday, November 11th, by 10AM, in Canvas. You can work in groups with up to three members. If you do so, write down the group members in the first page.)
1 Credit Markets and Inequality
Consider the following simple version of the model discussed in lectures 15 and 16. In particular, consider a small open economy with 3 types of individuals: workers, unproductive entrepreneurs, and productive entrepreneurs. The total population is N, with half of the population being workers N/2, and the rest entrepreneurs, N/4 of which are low productivity and N/4 high productivity. Types are fixed, i.e., perfectly persistent (in terms of the notation in the lecture notes, we assume that γ = 1). This economy can borrow or lend at a fixed world interest rate r. Labor is inmovil (i.e., there is no migration in and out of this economy, while capital can freely flow).
Workers supply inelastically a unit of labor and save a fraction s of their available resources. Their net-worth evolves according to the following law of motion
Entrepreneurs with productivity zi
, i = 1, 2, produce output yi using capital ki and labor li using a technology described by the following Cobb-Douglas production function
The amount of capital that they can invest is limited by the following collateral constraint
ki,t+1 ≤ λni,t+1
(to simplify the analysis, we take the leverage λ as given. See lecture 15 for a derivation of the equilibrium leverage in terms of more fundamental parameters). The law of motion of the net-worth of entrepreneurs with productivity zi evolves according to the following law of motion
where profits
are payments to the talent of entrepreneurs zi
.
1.1 Perfect Credit Benchmark
Assume that the leverage is sufficiently high, so that entrepreneurs can invest the optimal amount of capital irrespective of their net-worth. In this case, the equilibrium wage w and the profits of entrepreneurs of productivity zi
, πi
, are independent of the distribution of net-worth and equal to
and
where aggregate output
and
L = N/2.
To further simplify the analysis, in the rest of this exercise we are going to assume that the unproductive entrepreneurs are extremely bad, i.e., they can’t produce, i.e., z1 = 0 < z2.
1. (15 points) Graph the Lorenz curve of non-capital income (labor income for workers and profits for entrepreneurs) in this economy under the as-sumption that (1 − α − θ) 2 > θ. Interpret this condition.
2. (15 points) Solve for the steady state net-worth of workers and entrepreneurs under the assumption that s (1 + r) < 1.
3. (15 points) Graph the Lorenz curve of net-worth in this economy under the assumption that (1 − α − θ) 2 > θ and s (1 + r) < 1. Compare the Lorenz curves of income and wealth.
1.2 Imperfect Credit
Assume that the leverage is sufficiently low so that productive entrepreneur are constrained. As before, assume that the productivity of the unproductive entrepreneur is zero, z1 = 0. In this case, the equilibrium wage wt and the profits of productive entrepreneurs π2t and aggregate output Yt are a function of the net-worth of productive entrepreneurs and equal to
and
To further simplify the analysis, assume the following parameter values: λ = 1 (no credit), θ = 0.4, α = 0.3, z2 = 10, s = .1, r = 0.02, and δ = 0.06.
1. (10 points) Graph the Lorenz curve of non-capital income in this economy. How does the curve changes with ?
2. (10 points) Solve for the steady state net-worth of workers and entrepreneurs.
3. (10 points) Graph the Lorenz curve of net-worth in this economy. Compare your answer with that of exercise 1.1, using the same parameter values (other than the value of λ).
4. (10 points) Calculate the Gini coefficient of non-capital income and net-worth in the steady state of the two economies, i.e., the one with perfect credit market (exercise 1.1) and the one with λ = 1 (exercise 1.2), using the same parameter values (other than the value of λ).
5. (15 points) Calculate the GDP in the steady state of the two economies, i.e., the one with perfect credit market (exercise 1.1) and the one with λ = 1 (exercise 1.2), using the same parameter values (other than the value of λ).