Assume that Y1 and Y2 are not exogenous from the point of view of the individual. Assume

that Y1=W1*L1, where L1=1-R1, where L1 is hours worked in period 1, and R1 is hours of

leisure in period 1. 1=L1+R1 equals time endowment (total number of hours available) in

period 1 that is normalized to 1.

That is, we assume that the time endowment is not 24 hours but equals 1.

Assume also that Y2=W2*L2, where L2=1-R2, where L2 is hours worked in period 2, and R2

is hours of leisure in period 2 of life. 1=L2+R2 equals total number of hours available in

period 2 that are normalized to 1. We also assume that W1 (=nominal wage in period 1) and

W2 (=nominal wage in period 2) are exogenous from the point of view of the individual.

Assume that the price of current consumption equals 1: P1=1.

Assume: 1 U C C R R 1 2 1 2 =

Where the preference parameters, , , , 1 , all are assumed to be between

zero and 1.

a) Write up the intertemporal budget constraint of the individual.

b) Derive the optimal levels of C1, C2, R1, R2, L1, L2, and saving as functions of the

exogenous variables.

c) What happens to the optimal levels of C1, C2, R1, R2, L1, L2, (and Saving) if W2

increases?

d) What happens if r increases?

e) What happens if the pension in period 2 increases?

(Here you have to assume that the individual may receive non-labor income

In the second period.)

8. MORE ECONOMIC GROWTH: THE SOLOW MODEL WITH CONTINOUS

TECHHNOLOGICAL PROGRESS.

The questions are from the second chapter on economic growth in the textbook of Mankiw.

These problems are not required for B-macroeconomics students but they are required for

the economics growth students.

When doing problems,

assume that (1 ) Y t K t A t L t ( ) ( ) ( ( ) ( )) = y k = , where y Y AL = / .

Problem 8.1A: Solve for y as a function of the exogenous variables: s,n,g,and .

Problem 8.1B. A developed country has a saving rate of 28 percent and a population growth

rate of 1 percent per year. A less developed country has a saving rate of 10 percent and a

population growth rate of 4 percent per year. In both countries, g=0.02 and =0.04. Find the

steady state value of y for each country.

Problem 8.1C: What policies might the less-developed country pursue to raise its level of

income?

Problem 8.2: In the US, the capital share of GDP is about 30 percent; the average growth in

output is about 3 percent per year; the depreciation rate is about 4 percent per year; and the

capital-output ratio is about 2.5. Suppose that the production function is Cobb-Douglas, so

that the capital share in output is constant, and that the US has been in a steady state.

A. What must the saving rate be in the initial steady state? (Hint: Use the steady-state

relationship, sy n g k = + + ( ) ( ) .)

B. What is the marginal product of capital in the initial steady state?

C. Suppose that public policy raises the saving rate so that the economy reaches the Golden

Rule level of capital. What will the marginal product of capital be at the Golden Rule steady

state? Compare the marginal product of capital at the Golden rule steady state to the marginal

product of capital in the initial steady state?

D. What will the capital-output ratio be at the Golden Rule steady state? (Hint: For the CobbDouglas production function, the capital-output ratio is related to the marginal product of

capital.)

Problem 8.3: Prove each of the following statements about the steady state of the Solow

model with population growth and technological progress.

A. The capital-output ratio is constant.

B. Capital and labor each earn a constant share of an economy's income.

C. Total capital income and total labor income both grow at the rate of population growth

plus the rate of technological progress, n+g.

D. The real rental price of capital is constant, and the real wage grows at the rate of

technological progress, g.