Econ 5140: Problem Set #1 Fall 2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Fall 2022 Econ 5140:
Problem Set #1 : Simple Dynamic General Equilibrium Model and Solow Growth Model
Q1: Download China’s annual and quarterly real GDP data from 1979-2021 and use the Eview command to apply HP filer to the data and show the trend component and cyclical component. What do you find from the data?
Eview command: (for annual data, λ = 100)
% Example Eview code for hp−filter, gdp hp is the trended component;
Select all
gdp .hpf(lambda=1600) gdp hp
series gdp c = gdp − gdp hp
Q2: Consider a two period economy with production, where consumers have utility given by
U(C1 , C2 ) = ln C1 + η ln(1 _ H1 ) + β[ln C2 + η ln(1 _ H2 )] (0.1)
The consumer’s budget constrain is given by
C1 + K2 = W1 H1 + R1 K1
C2 = W2 H2 + R2 K2
(0.2)
where W is the wage rate, R is the gross capital rental rate, H is the labor supply
and K is the investment.
This says that in the first period, the consumer consumes and invests and earns wages from working and rents from renting her capital to the firm. In the second period, the consumer consumes all wages and rents. Since all consumers are alike we can think of the one consumer as representing ‘the whole economy’ .
a). The consumers will choose consumption, investment and labor supply in period 1 and 2 to maximize her utility, taking wage and rental rate as given. Describe the conditions that characterize the consumers’ optimal choice.
b). Explain the intuition of the intertemporal optimality condition (the Euler equa- tion) and the intratemporal optimality condition (the equation that relates the labor supply, wage and consumption).
c). Assume that the firms are perfectly competitive. What will be the com- petitive wage and return to capital in each period, if the production function is Y = ΘKα L1 −α in each period?
d). Now use the results from a and b to compute the equilibrium value of employ- ment and first period investment for the economy. Does employment depend on the productivity term Θ? Can you explain?
Q3: Take the Solve growth model with exogenous saving rate s, population growth rate n, depreciation rate 6, and rate of labor augmenting technical progress g . In addition, assume that the production function is of the Cobb-Douglas form:
Y = Kα (AL)1 −α (0.3)
a). Assume that factors of production are paid their marginal product. What is the expression for the wage and the return to capital, in terms of the intensive form of the production function?
b). Show that along the stead state, the return to capital will be constant, but the wage will be growing. At what rate will the wage grow?
c). Assume that the economy start off below the steady state capital per effective labor. Show that the rate of return to capital will be falling over time, but the wage will be growing at a faster rate than in the steady state.
d). Compute the saving rate that is necessary so that the steady state of the economy is below the golden rule level for the capital per effective labor.
e). Assume now that there is a government which spends a fraction z of GDP in every year, so that the government spending is zY . Using the national income identity Y = C + I + G to work out the new rule for capital accumulation in the Solow model. How will government spending affect the long run growth rate of output per capita and the steady state level of GDP?
Q4: Read Section 1.6 of Romer, answer the following question. Suppose there are two economies a and b, let y˜a = L(Y)a(a) and y˜b = Suppose that the production function in both economies are Cobb-Douglas, i.e., Y = Kα (AL)1 −α . Also suppose
that the market is competitive and the marginal product of capital is equal to the
rate of return to capital.
a). If = 10 and suppose the technology level (Aa = Ab ) is the same across two economies, what is the implied difference of rate of return in economy a and b?
Is the observed difference in rate of return on capital reasonable? Why?
b). Now suppose that = 10 still holds, but these two economies have different technology level (Aa Ab ). Usually the observed rate of return on capital between poor countries and rich countries is 3 _ 4 times, assume that this is the difference in rate of return on capital between country a and b, i.e = 4, what is the implied difference in technology in these two economies?
(Do not need to hand in.)Q5: Using the facts of growth and explain in what aspects the Solow growth model provides a theory of growth which is consistent with the data, and in what aspects it is not so successful.
2022-10-02
Simple Dynamic General Equilibrium Model and Solow Growth Model