Intermediate Macroeconomics Problem Set-
1. Imagine a representative household with the following utility function:
u(C, l) = γ ln (C) + (1 - γ) ln (l),
where 0 < γ < 1, and C represents consumption of a consumption good (or basket), while l represents hours of leisure.
(a) Find the marginal rate of substitution at some arbitrary point, (C, l).
(b) We considered three assumptions that consumer's preferences must satisfy. Check whether this utility function satisfies the first two assumptions (namely, more is better and a taste for diversity) or not.
(c) Does this utility function satisfy the Inada conditions?
This is representative household has a budget constraint that looks like
C =wNs + π - T,
where w represents the real wages, Ns the amount of hours worked, T the lump-sum taxes government imposes, and π the profits of a representative firm which the representative household owns. Finally, the representative household has a total amount of h hours that she can allocate to working or enjoying leisure time; that is
l + Ns =h.
For the rest of this problem, suppose T < π.
(d) Show that the budget constraint can be rewritten in real terms (in terms of goods instead of currency) as
C + wl =wh + π - T.
(e) Draw the budget constraint in real terms and mark every important element in it.
(f) Write the maximization problem of the household. (Do not forget to identify the domain of the utility function, or non-negativity constraints.)
(g) Suppose that γ = 1/4, h = 30, π = 30, and T = 10. Solve the maximization problem of the representative household. Make sure to argue whether the boundary points can or cannot be the optima. What will be the optimal level of consumption, leisure, and labor, as functions of real wage, w?
(h) Are leisure and consumption good normal or inferior goods?
2. Consider preferences that can be represented by the following utility function;
U (C, l) = al + bC,
where a and b are positive constants. Leisure and consumption are said to be perfect substitutes for a consumer who has such preferences.
(a) Illustrate indifference curves that correspond to such utility function. Why do you think leisure and consumption good are said to be perfect substitutes in this utility function? Do you think it is likely that any consumer would treat consumption good and leisure as perfect substitutes?
(b) Given perfect substitutes, is more preferred to less? Do preferences satisfy the diminishing marginal rate of substitution property?
(c) If the budget constraint is given by
C = w (h - l),
determine, graphically and algebraically, what consumption bundle the consumer chooses. Show that the consumption bundle the consumer chooses depends on the relationship between a/b and w, and explain why.
3. Imagine a representative firm with the following production function:
Y = zF (K, N) = zKα(Nd)1-α,
where 0 < α < 1, and z represents the total factor productivity, K the amount of capital that the firm owns, and, finally, Nd the amount of labor it hires. Here, the representative firm is only interested in its profit, which is defined as
π =Y - wNd.
(a) Write the maximization problem of the firm. (Do not forget to specify the domain of the profit function, or the so-called non-negativity constraints.)
(b) Suppose that z = 30, K = 1, and α = 1/3. Solve the maximization problem of the representative firm. Argue whether the boundary point can or cannot be the optimal choice for the ?rm. What will be the optimal level of hours hired production, and profits of the firm, as functions of wage w?
(c) Find the labor demand curve of the representative firm, as a function of w. Is it an increasing or a decreasing function?
(d) Using your knowledge on comparative statics, determine the sign of the following derivatives (at a given wage, w);
dNd/dK and dNd/dz.
4. Consider the one-period simple economy that we studied in class. Assume the preferences of the representative consumer are represented by the following utility function,
U (C, l) = ln (C) + βl,
and the technology that the representative firm uses is represented by the following production function,
F (K, N) = zKαN1-α.
β > 0 and 0 < α < 1 are parameters. In addition, assume that the government levies a tax rate of T = G on the consumer.
(a) Define the competitive equilibrium in this economy. Specify which variables are endogenous and exogenous in your definition.
(b) Write down the consumer's and firm's problems, and solve them. (Keep in mind that consumer and firm are price-takers in our simple mode.)
(c) Use your answer in Part (b), and find the competitive equilibrium in this economy analytically. Make sure that you find consumption, leisure, employment, and real wage.
(d) Graphically, illustrate the steps you take in Parts (b) and (c). Find the competitive equilibrium using this graphical approach. (Your graphs need to be only approximations of the actual functional forms.)
(e) Show that competitive equilibrium of Part (c) is Pareto optimal.