Homework 3-

1. Suppose you receive $100 from your grandfather when you are born and you invest this $100 in an account that pays you 5% interest a year. Each year you reinvest the initial amount plus any interest you have earned. Assume the interest rate is paid at the end of each year.

a. What is the value of your account at the end of the first year?

b. What is the value of your account at the end of the fifth year?

c. What is the value of your account at the end of the tenth year?

d. According to the rule of 70, how many years will it take to approximately double the initial value of $100?

Now, suppose you instead invest this amount in an account that pays you 10% interest a year.

e. What is the value of your account at the end of the first year?

f. What is the value of your account at the end of the fifth year?

g. What is the value of your account at the end of the tenth year?

h. According to the rule of 70, how many years will it take to double the initial value?

Economic Growth:

2. You are given the following information about India and the United States (information taken from the CIA Factbook).

a. Provide an estimate of GDP per capita for 2008 (just use the numbers given) for these two countries.

b. Calculate GDP per capita in the United States for 2018 (ten years time) assuming that GDP and population grow at the rates given in the above table. Then, calculate the rate of growth of GDP per person over this ten year period of time.

c. Calculate GDP per capita in India for 2018 (ten years time) assuming that GDP and population grow at the rates given in the above table.

d. Given your calculations in parts (b) and (c), what is happening to India's GDP per capita relative to the United States' GDP per capita over this ten year period of time?

e. If these growth rates persist, what will happen in the very long run to United States' GDP per capita and India's GDP per capita?

3. Suppose you are told that an economy's aggregate production function is

Y = 20K^1/2L^1/2

where Y is real GDP, K is units of capital and L is units of labor. Suppose that initially this economy's stock of capital is fixed at 100 units.

a. Using Excel, fill in the following table. For your homework you can submit a truncated table giving the values in the table for L = 1 to 10 and L = 90 to 100. [Hint: the "Hide" function in Excel will make this easy.]

b. Plot the data you found in part (a) in a graph where Y, real GDP, is measured on the vertical axis and L, labor, is measured on the horizontal axis. [Hint: Excel's graphing tools will make this a relatively simple matter.]

c. Suppose L equals 25. What is the value of labor productivity?

d. Suppose L increases to 49. What is the value of labor productivity now? Explain what is happening to labor productivity as L increases from 25 to 49.

4. Suppose you are told that an economy's aggregate production function is

Y = 20K^1/2L^1/2

where Y is real GDP, K is units of capital and L is units of labor. Suppose that initially this economy's stock of capital is fixed at 25 units.

a. Fill in the following table.

b. Sketch a graph of this economy's aggregate production function measuring real GDP on the vertical axis and units of labor on the horizontal axis. Label the line on your graph carefully as "aggregate production function 1".

c. Now, suppose technology improves in this economy and the aggregate production function is now given by the equation

Y = 40K^1/2L^1/2

Fill in the following table. Assume there is no change in capital.

d. Now draw "aggregate production function 2" on your graph in part (b). Label this new line clearly. Describe in words the effect of a change in technology on the aggregate production function.

e. Suppose L is equal to 25 units. What happens to labor productivity in this economy as technology improves? Explain your answer.

5. Suppose you are told that an economy's demand for labor and supply of labor are given by the following equations where L is units of labor and W is the real wage rate.

Demand for Labor: W = 200 - 4L

Supply of Labor: W = 4L

Furthermore, you know that this economy's aggregate production function is

Y = 2K^1/2L^1/2

where Y is real GDP and K is units of capital. In this economy capital is fixed and equal to 9 units.

a. Given the above information, what is the equilibrium real wage and equilibrium level of labor in this economy?

b. When the labor market is in equilibrium, what is the level of aggregate production in this economy?

c. Given your answers in parts (a) and (b), calculate the value of labor productivity for this economy.

Now, suppose at every real wage employers now wish to hire 22 additional units of labor.

d. What is the equation for the new labor demand curve?

e. Given this new information, calculate the new equilibrium real wage, W', and the new equilibrium level of labor, L', in this economy.

f. Given your answer in (e), what is the new level of aggregate production, Y', for this economy?

g. Given your answers in (e) and (f), calculate the value of labor productivity for this economy.

h. Compare and contrast your answers in parts (c) and (g). Provide a verbal explanation for what is happening to labor productivity and why this is happening.

6. Suppose you know a country's aggregate production function and you also know that this aggregate production function exhibits diminishing marginal returns (i.e., as the level of factor usage increases holding all other factors constant, output increases but it increases at a diminishing rate).

a. Suppose the level of labor used in the economy increases. What happens to capital productivity and to the level of real GDP in this economy? Provide a verbal explanation as well as a sketch or sketches to illustrate your answer.

b. Suppose the level of labor used in the economy increases. What happens to labor productivity and to the level of real GDP in this economy? Provide a verbal explanation as well as a sketch or sketches to illustrate your answer.

c. Suppose the level of capital used in this economy increases at the same time that there is a technological advance in the economy. What happens to capital productivity and to the level of real GDP in this economy? Provide a verbal explanation as well as a sketch or sketches to illustrate your answer.