MACROECONOMIC THEORY

Assignment #1: Optimization in Matlab

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1. Problem 1 (Utility maximization problem). Consider a consumer who lives for two period: young and old. The consumer works when young and retires when old. The consumer supplies 1 unit of time to labor market and earns w units of income when young. The consumer chooses a sequence of consumption to maximize the life-time utility maximization problem as follows:

? c1−σ c1−σ ? V(w,R) = max 1 + 2

 c1,c2 1−σ 1−σ s.t.

c1+c2 =w. R

ˆ (1.1) Find the analytical solution for optimal consumption and saving.

ˆ (1.2) Assume that σ = 2, w = 5 and R = 1.04. Solve the household problem numerically with the

fsolve function. Report the optimal choices.

ˆ (1.3) Now add labor-leisure choice to the problem

(?cθl1−θ?1−σ c1−σ ) V(w,R) = max 11 + 2

 c1,c2 1−σ 1−σ

s.t.

c1+c2 = w(1−l1), R

where l1 is leisure and (1−l1) is labor supply. Assume that σ = 3, θ = 13, w = 5 and R = 1.04. Solve the household problem numerically with the fsolve function. Vary the wage rate between 1 and 10 and plot the labor supply. Is the labor supply curve upward slopping? Why?

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2. Problem 2 (Cost minimization problem). Suppose a firm has a cost function C = qK + wL. The firm’s cost minimization problem is given by

V (q,w,y) = min{qK +wL} K,L

s.t. K1/3×L2/3 = y.

Assume that q = 1, w = 2, and y = 10.

ˆ (2.1) Solve the firm problem numerically with the fsolve function.

ˆ (2.2) Suppose that w and y are constant with w = 2, and y = 10, but q varies between 0.5 and 1.5. Plot the demand for K, K(q, y = 10). Is it an increasing or decreasing function? Why?

3. Problem 3 (Comparative statics analysis). Supposed a consumer has income I, pays income tax τII, and consumes two goods c1 and c2 with market prices p1 and p2. The consumer allocates his income between two consumption goods to maximize his utility as follows:

V(p,I) = max[acρ1+(1−a)cρ2]ρ1 c1, c2

s.t.

after−tax income z }|I {

p1c1+p2c2 = (1−τ )I .

Assumethata=0.5,ρ=2,I=10,p1 =1andp2 =2.

ˆ (3.1) Set τI = 10% and solve for the consumer’s optimal allocation using Matlab.

ˆ (3.2)VarytheincometaxrateτI between[0,0.95]. Plottheoptimalconsumptionrules: c1 ?p,(1−τI)I? and c2 ?p, (1 − τ I )I ? . Explain the effects of tax hikes on consumption.

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