Homework 2-

1. Suppose that the production function for an economy is given by Y = F(K, N, A) = AK1/2N1/2  where Y is output, K is capital, N is labor, and A is the level of technology.

a. Suppose that the value of A is equal to 1.  Fill in the table below based on this information and the given production function.

b. Suppose that the value of A increases to 10.  Fill in the table below based on this information and the given production function.

c. Draw a picture of the two production function relationships you computed in parts (a) and (b) placing output on the vertical axis and labor on the horizontal axis.  Verbally explain the effects of a change in technology on the aggregate production function.

d. In the tables you computed the MPN using a derivative and the MPN using a formula for discrete data (the change in output/the change in labor).  Provide an explanation for why these two measures would not be equivalent and use a graphical explanation to explore what would make these two measures numerically closer to one another.

2. Suppose that the production function for an economy is given by Y = F(K, N, A) = K^1/2N^1/2A where Y is output, K is capital, N is labor, and A is the level of technology.

a. If K equals 100, N equals 400, and technology equals 1, what is output (Y) and output per worker (Y/N)?

b. What is the value of the marginal product of capital using the information in (a)? Show your work.

c. What does it mean to say that the marginal product of capital is proportional to output per unit of capital?

d. What is the value of the marginal product of labor using the information in (a)? Show your work.

e. What does it mean to say that the marginal product of labor is proportional to output per worker?

f. What is the total wage bill in the economy described in (a)?

g. What is the total return to capital in the economy described in (a)?

h. What is the ratio of labor income to capital income in the economy described in (a)?

i. If capital and labor both increase by 100% while technology is constant, what will the new levels of output and output per worker be in this economy? (Round your answer to the nearest whole number. Hint: you really don't need to round at all with this one...just be alert to what's happening in this question!)

j. If capital increases by 50% from its initial level, labor increases by 25% from its initial level, and technology is constant, what will the new levels of output and output per worker be in this economy? (Round your answer to the nearest whole number.)

3. You are given the following information for this set of questions.

Furthermore you are told that the market basket for purposes of computing the CPU is defined as 100 Pizzas, 30 cokes, and 40 salads.

a. Calculate the CPI and for this economy and put your answer in the table below.

b. From the information you found in part (a), calculate the rate of inflation for the following periods:

4. Suppose you are told that government spending equals 100 and net taxes equal 50.  You are also told that the level of government spending and the net taxes is independent of the level of the real interest rate. Furthermore, you are told that when the real interest rate is 20% private saving, SP , equals 200 and when the real interest rate falls to 10% , private saving, SP , is equal to 100. You also know that the relationship between private saving and the real interest rate is linear.

a. Draw three graphs that are horizontally aligned and then use these graphs to depict government saving, SG, in the first graph; private saving, SP, in the second graph, and national saving, NS, is the third graph.

b. Write an equation for NS using the information provided.

c. Suppose I = 300 - 750r. Graph this investment function on a graph along with the NS you found in part (a). 

d. What is the equilibrium level of investment and real interest rate in this economy?

5. This question is from Chapter 6 in your text: you should be able to do this question from the information that is provided to you in the question.  But, you may find the text helpful.

We can define the Labor Force, L, as the sum of employed workers, E, plus unemployed workers, U. For this problem assume the labor force is a constant.

Suppose initially there are 80,000 employed workers and 20,000 unemployed workers.

a. What is the labor force equal to in this economy and the initial unemployment rate?

Now, suppose that the rate of job separation each month, s, changes and is now equal to 1% of the previous month's employment.  Furthermore, suppose that the rate of job finding each month, f, changes and is now equal to 3% of the previous month's unemployment.  Use this information to answer the following questions.  Note: your labor force is constant and should always sum to the answer you provide din part (a).  If necessary, round to the nearest whole number.

b. Calculate the following values for the end of the first month after these changes to s and f:

Unemployment Rate at the end of the first month =

Calculate the following values for the end of the second month after these changes to s and f:

Unemployment Rate at the end of the second month =

Calculate the following values for the end of the third month after these changes to s and f:

Unemployment Rate at the end of the third month =

c. Generalize the pattern you have when you consider your answers if (c), (d), and (e)?

d. The unemployment rate will neither fall nor rise when sE = fU.  An alternative way to convey this idea is to say that the steady state of unemployment, (U/L)SS, equals s/(s + f). In this example, what is the steady state of unemployment? If this economy is at its steady state of unemployment, what do E and U equal?

e. Using the figures you gave in part (d), verify by filling in the table below that this really is a steady state level of unemployment?

Unemployment Rate in steady state =