ECO 303 Problem Set 1 Instructor: Chunzan Wu Due: Oct 9, 2017 in class Instructions: You need to show enough details of your work to receive full credit. When asked to draw a graph, label things clearly. (I recommend you to draw graphs by hand. Do not copy and paste any graphs from the lecture notes, otherwise you will receive zero credit.) 1 Math Review [10] 1. Use L’Hopital’s rule to derive the following limit: C 1− σ − 1 . σ →1 1 − σ lim 2. Let f ( x, y) = x log( x2 + y2 ), derive partial derivatives ∂f ∂x and ∂f ∂y . 3. Use the first order Taylor approximation around x0 = 0 to show: log(1 + x ) ≈ x, when | x | ≈ 0. 4. Use the method of Lagrange multipliers to solve the following constrained maximization problem: max 3 log x + 2 log y + log z ( x,y,z) s.t. 3x + 2y + z = 6. 2 The One-Period Model of Households [40] Consider a household who lives for only one period. The household is endowed with h units of time, which can be allocated between leisure l and time spent working. The 1 real wage of the household is w units of the consumption good per unit of working time (i.e., the consumption good is the numeraire). The household also owns some firm stocks, and hence receives dividend income π. Finally, the household pays a lump-sum tax T to the government. The household’s preferences over the consumption good and leisure are represented by the utility function U (C, l ), which is both monotonic and concave. 1. In the (l, C ) space, draw two of the households indifference curves with different utility levels. What are the properties of the indifference curves? Briefly explain why. Label clearly which one of the two indifference curves represents a higher utility level. 2. Write down the budget constraint of the household. 3. Are there other constraints that the household’s choice must obey? If so, what are them? 4. In the (l, C ) space, draw the set of feasible choices of the household (Hint: you should separate the cases when π > T and π < T). Label clearly the following: (1) the slope of the budget line; (2) The intercepts on the vertical and horizontal axes; (3) the areas of feasible and infeasible choices. 5. Write down the optimization problem of the household. Be clear about the following: (1) the objective function; (2) the choice variables; (3) the constraints if any. 6. Assume π > T and there is an interior solution. In the (l, C ) space, use indifference curves and the set of feasible choices to find the solution to the household’s optimization problem, i.e., the optimal choice of the household. 7. Based on your graph in the last question, write down the optimality conditions that must be satisfied at the household’s optimal choice. Briefly explain why they must hold. (Hint: you should have two conditions.) 8. Suppose both the consumption good and leisure are normal goods, discuss without any graphs what will happen to the household’s choice if there is (a) an increase in dividend income π; (b) an increase in wage w. 9. Suppose now only the consumption good is normal, and leisure is inferior. Repeat the last question. 2 From now on, let the utility function of the household be 1 U (C, l ) = log(C − (h − l )2 ), 2 and assume that π > T and there is an interior solution to the household’s problem. 10. Derive the marginal rate of substitution of leisure for consumption MRSl,C . How does it change with l? 11. Solve the household’s optimization problem using the method of Lagrange multipliers. That is, derive the formula for the household’s optimal choice. 12. From your answer to the last question, how does the household’s choice respond to (a) a decrease in the lump-sum tax T; (b) an increase in wage w. Interpret your results in terms of the income and substitution effects. 13. In the (l, C ) space, show graphically the effects of the two changes in the last question to the household’s choice. If applicable, be clear about which part of the change is due to the income effect and which part is due to the substitution effect. 3 The One-Period Model of Firms [25] Consider a firm with the production function: Y = zF (K, N ) where Y is the output, z is the total factor productivity, K is the capital stock, and N is the labor hired. Suppose this production function is well-behaved, i.e., it satisfies all the assumptions required in class and lecture notes. The price of output is normalized to be 1, and the real wage in the labor market is w. The firm is a price-taker, and its capital stock K is given (i.e., not chosen by the firm). 1. Write down the firm’s optimization problem. Be clear about the following: (1) the objective function; (2) the choice variables; (3) the constraints if any. 2. In the ( N, Y ) space, draw the output Y as a function of N holding other things constant. In the same graph, draw the total cost of production as a function of N. Graphically identify the solution to the firm’s optimization problem. 3 3. Based on your graph in the last question, write down the optimality condition that must be satisfied at the firm’s optimal choice. Briefly explain why it must hold. (Hint: you should have one condition.) From now on, let the production function of the firm be Y = zK α N 1−α , where α is a parameter between 0 and 1. 4. Verify that this production function has constant return to scale. 5. Derive the marginal product of labor MPN and marginal product of capital MPK . How does MPN change with N and K? 6. Solve the firm’s optimization problem analytically. That is, derive the formula for the firm’s optimal choice. 7. From your answer to the last question, how does the firm’s choice respond to (a) an increase in the real wage w; (b) an increase in the capital stock K; (c) an increase in the total factor productivity z. 4 Competitive Equilibrium: A Mini Study [25] Consider the standard one-period model of the macroeconomy presented in class and in the lecture notes. (If needed, check the competitive equilibrium part of the lecture notes for a quick refresh of your memory about the structure of the economy, constraints, notations, etc.) In this exercise, we use this model to study the effect of changes in government spending G to the economy. We start by solving for the competitive equilibrium in this economy. 1. At the competitive equilibrium, the representative household must be optimizing. Let the utility function of the representative household be U (C, l ) = log(C ) + log(l ). Assuming an interior solution, solve the household’s optimization problem with the standard budget constraint in the lecture notes. That is, derive the household’s consumption C and leisure l as functions of (h, w, π, T ). 4 2. At the competitive equilibrium, the representative firm must be optimizing. Let the production function of the representative firm be Y = zF (K, N d ) = zN d . Note that this implies the firm only uses labor to produce the output. (a) From the firm’s optimization problem, derive the optimality condition for the firm. What does it imply about the real wage w in the labor market? (b) What is the profits of the firm π given this wage w? 3. At the competitive equilibrium, the government budget constraint must hold. What does it imply about the lump-sum tax T? How is it related to government spending G? 4. Now using your results from question 2 and 3 about w, π and T, substitute (w, π, T ) in your formulas for household consumption C and l from question 1, and write C and l as functions of only exogenous variables (h, G, z). 5. Based on your answer to question 4, what are the equilibrium levels of labor supply N s , labor demand N d , and output Y? Now, we have solved the competitive equilibrium in this economy, i.e, we have denoted all the endogenous variables (C, l, w, π, T, N s , N d , Y ) as functions of exogenous variables (h, G, z). 6. A popular idea about government policy during recessions is that an increase of government spending can stimulate the economy. We can test this idea with our model. Based on your answers to question 4 and 5, what are the effects of an increase in government spending G on consumption C, leisure l, and output Y? 7. If we treat output Y in our model as the counterpart of GDP in the real economy, which is a common measure of economic performance, do you agree that an increase of government spending can stimulate the economy? 8. Given our model, do you think the government should increase government spending? Briefly explain why. (Hint: Is the representative household better or worse off after an increase of government spending?) 5