Question

Assume the existence of a two-period Diamond Model. Individuals may live for up to two periods t and t + 1. All individuals both live and work in period t. Some proportion γ of individuals die at the end of the first period, and do not consume in the second period. The remaining proportion (1 − γ) will will live and consume in period 2. Expected lifetime utility is given by: 2 U(C1t , C2t+1) = ln(C1t) + (1 − γ)ln(C2t+1) It is assumed that there is no discount factor and that individual’s value second period consumption as much as they value first period consumption. The lifetime budget constraint, which is not directly effected by the possibility of death, is given by: C1t + C2t+1 1+rt+1 = wt 1 + rt+1 and wt denote the marginal products of capital and labour respectively. The production function in this economy is given by Yt = Kα t L. L denotes a fixed quantity of labour. That is, there is no population growth in the model. We assume that At = 1, and there is no growth in the state of technology. a) Solve for the optimal values of C1t and C2t+1 (4 pts). b) Find the optimal savings rate s in this economy. (2 pts) c) Assume that there is 100% capital depreciation from one year to the next, and next year’s capital stock is simply this years investment. Solve for the steady state level of capital and output, denoted by K∗ and Y ∗ (4 pts). d) Solve for the steady state levels of wt , 1 + rt+1, C1t ,C2t+1, and denote your solutions by w ∗ ,1 + r ∗ ,C ∗ 1 , and C ∗ 2 respectively (12 pts) e)Using your results from d), find ∂C∗ 2 ∂γ (6 pts). f) Is your result from e) positive, negative, zero, or are we able to tell? Explain the intuition behind this finding (8 pts).

Answer 1

The pure-exchange OLG model was augmented with the introduction of an aggregate neoclassical production by Peter Diamond.^[4]  In contrast, to Ramsey–Cass–Koopmans neoclassical growth model in which individuals are infinitely-lived and the economy is characterized by a unique steady-state equilibrium, as was established by Oded Galor and Harl Ryder,^[11] the OLG economy may be characterized by multiple steady-state equilibria, and initial conditions may therefore affect the long-run evolution of the long-run level of income per capita.

Since initial conditions in the OLG model may affect economic growth in long-run, the model was useful for the exploration of the convergence hypothesis.^[12]

Convergence of OLG Economy to Steady State

The economy has the following characteristics:^[13]

• Two generations are alive at any point in time, the young (age 1) and old (age 2).
• The size of the young generation in period t is given by N_t = N₀ E^t.
• Households work only in the first period of their life and earn Y_1,t income. They earn no income in the second period of their life (Y_2,t+1 = 0)
• They consume part of their first period income and save the rest to finance their consumption when old.
• At the end of period t, the assets of the young are the source of the capital used for aggregate production in period t+1.So K_t+1 = N_t,a_1,t where a_1,t is the assets per young household after their consumption in period 1. In addition to this there is no depreciation.
• The old in period t own the entire capital stock and consume it entirely, so dissaving by the old in period t is given by N_t-1,a_1,t-1 = K_t.
• Labor and capital markets are perfectly competitive and the aggregate production technology is CRS, Y = F(K,L).

Two-sector OLG modelEdit

The one-sector OLG model was further augmented with the introduction of a two-sector OLG model by Oded Galor.^[5] The two-sector model provides a framework of analysis for the study of the sectoral adjustments to aggregate shocks and implications of international trade for the dynamics of comparative advantage. In contrast to the Uzawa two-sector neoclassical growth model,^[14] the two-sector OLG model may be characterized by multiple steady-state equilibria, and initial conditions may therefore affect the long-run position of an economy