Homework 3-

1. Suppose you are given the production function

Y = F(K, L) = AK^1/4L^3/4

Where Y is real GDP, K is capital, L is labor, and A is a parameter expressing the level of technology available in this economy.

a. Suppose K and L are both tripled in this economy. Does this production function exhibit constant returns to scale? Provide a proof of your answer.

b. Rewrite the above production function in terms of output per worker, y. That is, normalize the function so that it can be written as y = f(k) where k is capital per worker.  Provide a verbal interpretation of your mathematical result.

c. Using the expression you found in part (b) and assuming labor is equal to 16 units, fill in the following table. Recall that k, the capital to labor ratio (K/L), measures capital intensity: that is, it provides a measure of how much capital each unit of labor has to work with in the economy. Assume that A equals 1 in this problem. Hint: this might be a good time to pull out your Excel spreadsheet program.

d. Use your Excel spreadsheet program to graph k and y, placing k on the horizontal axis and y on the vertical axis.

e. Let's define s as the saving rate per worker, i as the amount of investment per worker, and c as consumption per worker.  Suppose that you are told that the saving rate in this economy is constant and equal to .2. Use this saving rate and the values you found in part (c) to fill in the following table.  Note that i equals sy and y = c + i.

f. Suppose you are also told that depreciation of the capital stock, δ, is equal to 8% each year.  Using this information, plus the information you have previously been given or have calculated compute the change in the capital intensity, Δk, in this economy. Find Δk for each level of K in the economy by filling in the table below. Recall that the change in the capital intensity is equal to the difference between the level of investment per worker and the amount of capital per worker that depreciates during the given time period.

g. Now, graph the amount of depreciation, δk; the level of investment per worker, i (which is also equal to sf(k)); and the capital intensity, k.  Place k on the horizontal axis and δk and sf(k) on the vertical axis.

h. Suppose we define k* as the steady-state level of capital intensity or the level of capital per worker where the amount of capital depreciation per worker equals the level of investment per worker in each period.  At k* the capital intensity is constant over time.  Calculate the value of k* for your economy. Does your calculation agree with your findings in part (g)?

i. What would happen to k* if s increased, holding everything else constant?  Draw a sketch illustrating the initial situation and the effect of an increase in s. [Hint: why not run your Excel program again but this time with a higher value for s (I used .25)?]

j. What would happen to k* if δ increased, holding everything else constant? Draw a sketch illustrating the initial situation and the effect of an increase in δ? [Hint: why not run your Excel program again but this time with a higher value for δ (I used .1)?]

2. Suppose you are asked to derive an aggregate demand (AD) cure using the Quantity Theory of Money.  Furthermore, suppose the velocity of money is constant and equal to 4, while the money supply is set at $10,000.

a. Using an Excel spreadsheet provide at least five potential combinations of real output, Y, and aggregate price level, P, that are on this AD curve. Insert your table of values into your answer sheet.

b. Using Exel graph this economy's AD curve placing P on the vertical axis and Y on the horizontal axis.

c. Provide a graph illustrating what happens to the AD curve if the velocity of money increases to 10, holding everything else constant.

d. Provide a graph illustrating what happens to the AD curve if the velocity of money is at its initial level but the money supply increases to $20,000.

3. Suppose you are given the following information about a long-run economy. The money supply equals 10,000, the velocity of money is constant and equal to 2, the production function is Y = F(K, L) = K^1/2L^1/2, capital (K) equals 100 units and labor (L) equals 400 units.  Assume prices are fully flexible.

a. Graph the AD curve and the AS curve for this economy on the same graph. Graph this for prices ranging from 0 to 500.

b. Calculate the equilibrium aggregate price level and real GDP for this economy.  Do your calculations agree with your findings in part (b)?

c. Suppose the Federal Reserve engages in monetary policy and reduces the money supply to 8000.  What is the new equilibrium aggregate price level and real GDP for this economy?

d. Does the money supply affect the level of output in this model? Explain your answer.

4. Suppose you are given an AD/AS model with the following properties:

• In the long-run prices are completely flexible;
• In the short-run prices are constant and equal to P1; and
• In the long-run, real GDP is determined by the level of capital and labor in the economy and the amount of these resources is fixed.

a. Draw a graph illustrating a long-run equilibrium situation in this model that is also a short-run equilibrium.  Label your axes, any curves, any points of intersection, and any equilibrium levels clearly in your drawing. Hint: why not try drawing this diagram using a program like "Paint"?

b. Suppose there is an increase in velocity.  Provide a sketch of the new short-run equilibrium compared to your initial equilibrium.  Clearly label all aspects of your diagram. 

c. Explain your diagram from part (b).

d. Redraw your figure from part (b), but include in your figure the long-run adjustment process. 

e. Explain your diagram from part (d).

f. In the long-run, what is the effect of an increase in the velocity of money on real GDP and the aggregate price level?