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Consider the following Diamond economy (Le., the Overlapping Generations Model: OLG). In every period t with t = 0, 1,2, ..., there always exist two types of individu- als, young and old who are continuously born or are continuously dying, respectively. L two-period-lived Individuals are born in period & and the population grows at a rate n (Li = (1 + n)Lt-1). Each young individual supplies one unit of labor when he or she is young and divides the resulting labor income between first-period consumption and saving. In the second period, the individual simply consumes the saving and any interest he or she earns. Let's assume that instantaneous (period) utility is given by the following constant-relative- risk-aversion (CRRA) utility function: (C) = 6: 030. (1) Show that the period utility function is increasing and concave in C. (2) Show that the coefficient of relative risk aversion (defined as - 2) is actually constant. Let Gir and Car denote the consumption in period : of young and old individuals, Thus the (lifetime) utility of an individual born at t, denoted , depends on Cu and Car+1 and is given by U = [Cu) + Itp Lu(Cuti), p>-1, where p is the rate of time preference. A production function Y, = F(Ki, AL,) , where Ke represents capital, A, the technology or the effectiveness of labor, and &, labor itself, is assumed. Suppose that F( ) follows the usual properties, i.e., it has constant returns to scale and satisfies the Inada conditions. Also assumed is At grows at exogenous rate g (so At = (1 + g)A-1). Markets are competitive; thus labor and capital earn their marginal products, and firms earn zero profits. (3) Define It = 7.17 the amount of capital per unit of effective labor, # = 7 7, output per unit of effective labor, and me = f(4) the production function in intensive form. Assuming no depreciation, show that the real interest rate is given by r = f'(k) and the wage per unit of effective labor is given by w = /(1) - byf(k).Economists use production functions to analyze the output or productivity of a system such as a manufacturing process; in a few words, production functions describe how inputs are transformed into outputs. In this project we examine a prototype model in which productivity depends on only two variables: labor and capital. Therefore our production function has the form P=AL, K), where [ > 0 and K > 0 represent long-term units of labor and capital, respectively. A basic assumption is that a manufacturer can produce the same number of goods using various combinations of labor and capital. For example, a circuit board company may produce 200 boards per hour (output) with 200 laborers and conventional tools (small capital outlay) or with 20 laborers and a crew of robots (high capital outlay). The output is the same with different amounts of labor and capital. 1. The productivity function P=AL, K) is a function of two variables, so it may be displayed as a surface or as a Capital set of level curves. The latter Curves of approach turns out to be more useful. constant productivity Figure I shows the level curves (called isoquants) of a typical production function. It says, for example, that P = 60 units may be P = 80 produced using a large capital outlay P = 60 and low labor, or a large number of - P = 40 P = 20 laborers and low capital outlay, or various other combinations of L and Labor K. Explain why the level curves Figure I increase in their productivity value as one moves away from the origin.Since my is the wage per unit of effective labor, Arty is labor income for the young who provide one unit of labor. Labor income is divided between consumption and saving, and they carry their saving forward to the next period (4) Show that the individual's (lifetime) budget constraint is 1 (5) Show the first-order condition of the Individual's maximization problem (the Euler equation) is given by Ca+1 1 + p (6) Briefly explain the implication of the Euler equation. (7) Using the budget constraint and the Euler equation above, show the fraction of income saved in the first period s(r) with the parameters & and p is given by (1 + r)(1-eye *(r) = 7 ( 1 +p) /+ (1+ r)(1-oy/0- The capital stock in period + + 1 is the amount saved by young individuals in period t. Thus, Consider now as a special case a Cobb-Douglas production function in intensive form f(k) = k" where a is the capital share of income. Suppose further that d = 1 in the individual utility function which then simplifies to a logarithmic utility function. (8) With these assumptions, show that capital per unit of effective labor in period + + 1, kt+1, is determined as follows *+1= (1+ ") (1+9) 2+ p -(1 - a)kp. (9) Taking by on the horizontal axis and ky+ 1 on the vertical axis, sketch /41 as a function of &, with the 45-degree line. Discuss the dynamics & based on the diagram. (10) Find, if any, the balanced-growth-path values of capital per unit of effective labor (1") and output per unit of effective labor (y*) as functions of parameters (n, g, p, and a). (11) Discuss how the economy would responds to shocks, such as a fall in the discount rate p, when the economy is initially on its balanced growth path
