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Exercise 1 (20 points) 2. (20 points) We consider the overlapping-generations model of Lecture 4, with 8 = 1/2: U = log(#) + ;log(c*+,) We denote the (net) real interest rate by r, and the wage by wt. The production function is Cobb-Douglas with a = 1/4: Y = K, L, . The labor force is constant so that L: = 1. Moreover: 6 = 1 = 100%. (a) (2 points) Use the "intuitive" method to find a relation between cit1, C, we and T't, coming from utility maximization. (Give the economic intuition.) (b) (2 points) Use the intertemporal budget constraint to solve for of, and cf. (c) (4 points) What is the law of motion for the capital stock? Compute the steady- state capital stock K, the (net) steady-state real interest rate r*. UCLA - Econ 102 - Fall 2019 7/ 11 (d) (4 points) Compute the Golden Rule net interest rate r, capital K, and wage wa. (e) (4 points) Compare the Golden Rule and steady-state levels of r* and K*.Exercise 2 (20 points) 3. (20 points) Consider the neoclassical labor market model. On the demand side, we assume a Cobb-Douglas production function for f(), such that: f(1) = All-a. On the supply side, we assume a linear utility for consumption as well as a power function of disutility for work U(c, !) = c - B . !'+/(1 + =). (a) (4 points) Assume that the price of consumption is p, and that the wage is w. Derive the labor demand curve assuming that firms maximize their profits pf (1) -wl. (b) (4 points) Derive the labor supply curve assuming that workers' budget constraint is given by pc = w/ (you can use whichever of the 4 methods you prefer). UCLA - Econ 102 - Fall 2019 9 / 11 (c) (4 points) Calculate the equilibrium quantity of labor Z. (d) (4 points) Calculate the equilibrium real wage w/p.UCLA - Econ 102 - Fall 2019 10 / 11 Exercise 3 (20 points) 4. (20 points) Consider the Solow growth model of Lecture 2, with however two small changes. Assume that the production function is given by F(K,, L,) = A, KPL)-, with A, such that A, = (1 + g)', and L, such that: L, = (1 + n)*. (a) (2 points) Write the law of motion for capital Ke. (b) (6 points) Define ke as: It = mint-7,; and write a law of motion for ke. Assume that n, and g are small in order to simplify this law of motion. Hint: if n and g are small then: (1 + g) 1/(1-@)(1+ n) = 1+ pag+n. UCLA - Econ 102 - Fall 2019 11 / 11 (c) (4 points) Compute k*, the steady-state of ke. Compute y" and c' corresponding to steady-state k* with: #t = guitar, and a = 1/(1-2)LExercice 1: The Keynesian Multiplier (20 points) 2. (20 points) Consider the closed economy goods market model where consumption is linear in disposable income with C(YD) = co + CYD, disposable income is income minus taxes, taxes are exogenous and equal to 7, and investment depends on output according to the Keynesian investment function, through I = by + bjY. However, government spending depends on the level of output. For example, the government systematically spends more when GDP is higher (it builds new roads, hires new teachers, etc.), and conversely when GDP is lower (it then stops construction projects, fires teachers, etc.). Thus, government spending is given by the following equation, with g1 > 0: G = go + g1Y (a) (4 points) Solve for equilibrium output. (b) (4 points) If ci + by + gi 0, is the multiplier higher or lower than when government spending does not depend on GDP (91 = 0)? What is the intuition for this? UCLA - Econ 102 - Fall 2019 9 / 11 (c) (8 points) Give both a graphical as well as an algebraic justification for the value of the multiplier.3. (20 points) Consider the closed economy goods market model where consumption is linear in disposable income with C(YD) = co + ciY'D, disposable income is income minus taxes, government spending and taxes are exogenous and equal to G and T respectively, and investment depends on output according to the Keynesian investment function, through I = bot bY. (a) (4 points) Solve for equilibrium output. (b) (4 points) Assume that there is a fall in autonomous spending given by Ago