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1. Hula hoop fabricators cost \$100 each. The Hi-Ho hula Hoop Company is trying to decide how many of these machines to buy. HHHHC expects to produce the following numbers of hoops each year for each level of capital stock shown.

Number of Fabricators                                                                 Number of Hoops

Produced per Year

0                                                                                                                                                                     0

1                                                                                                                                                                     100

2                                                                                                                                                                     150

3                                                                                                                                                                     180

4                                                                                                                                                                     195

5                                                                                                                                                                     205

6                                                                                                                                                                     210

Hula hoops have a real value of \$1 each. HHHHC has no other costs besides the cost of fabricators.

a.       Find the expected future marginal product of capital (in terms of dollars) for each level of capital. The MPKf for the third fabricator, for example, is the real value of the extra output obtained when the third fabricator is added

(a)   This chart shows the MPKf as the increase in output from adding another fabricator:

 # Fabricators Output MPKf 0 0 — 1 100 100 2 150 50 3 180 30 4 195 15 5 205 10 6 210 5

b.      If the real interest rate is 12% per year and the depreciation rate of capital is 20% per year, find the user cost of capital (in dollars per fabricator per year). How many fabricators should HHHHC buy?

(b)  uc = (r + d)pK = (0.12 + 0.20)\$100 = \$32. HHHHC should buy two fabricators, since at two fabricators, MPKf = 50 > 32 = uc. But at three fabricators, MPKf = 30 < 32 = uc. You want to add fabricators only if the future marginal product of capital exceeds the user cost of capital. The MPKf of the third fabricator is less than its user cost, so it should not be added.

c.       Repeat Part (b) for a real interest rate of 8% per year

(c)  When r = 0.08, uc = (0.08 + 0.20)\$100 = \$28. Now they should buy three fabricators, since MPKf = 30 > 28 = uc for the third fabricator and MPKf = 15 < 28 = uc for the fourth fabricator.

d.      Repeat Part (b) for a 40% tax on HHHHC’s sales revenues.

(d)  With taxes, they should add additional fabricators as long as (1 - t)MPKf > uc. Since t = 0.4, 1 - t = 0.6. They should buy just one fabricator, since (1 - t)MPKf = 0.6 ´ 100 = 60 > 32 = uc. They shouldn’t buy two, since then (1 - τ)MPKf = 0.6 ´ 50 = 30 < 32 = uc.

e.       A technical innovation doubles the number of hoops a fabricator can produce. How many fabricators should HHHHC buy when the real interest rate is 12% per year? 8% per year? Assume that there are no taxes and that the depreciation rate is still 20% per year.

(e)  When output doubles, the MPKf doubles as well. At r = 0.12, they should buy three fabricators, since then MPKf = 60 > 32 = uc; they shouldn’t buy four, since then MPKf = 30 < 32 = uc.

At r = 0.08, they should buy four fabricators, since then MPKf = 30 > 28 = uc; they shouldn’t buy five, since then MPKf = 20 < 28 = uc.

2. An economy has full-employment output of 6000. Government purchases, G, are 1200. Desired consumptions and desired investment are

Cd =3600 – 2000r + 0.10Y, and

Id = 1200 – 4000r,

Where Y is output and r is the real interest rate.

a.       Find an equation relating desired national saving, Sd, to r and Y

(a)   Sd = Y - Cd - G

= Y - (3600 - 2000r + 0.1Y) - 1200

= -4800 + 2000r + 0.9Y

b.      Using both versions of the goods market equilibrium conditions, Eqs. (4.7) and (4.8), find the real interest rate that clears the good market. Assume that output equals full-employment output.

(b)  (1) Using Eq. (4.7): Y = Cd + Id + G

Y = (3600 - 2000r + 0.1Y) + (1200 - 4000r) + 1200

= 6000 - 6000r + 0.1Y

So 0.9Y = 6000 - 6000r

At full employment, Y = 6000. Solving 0.9 ´ 6000 = 6000 - 6000r, we get r = 0.10.

(2) Using Eq. (4.8):

Sd = Id

-4800 + 2000r + 0.9Y = 1200 - 4000r

0.9Y = 6000 - 6000r

When Y = 6000, r = 0.10.

So we can use either Eq. (4.7) or (4.8) to get to the same result.  Because they are the same I only graded that you did at least one.

c.       Government purchases rise to 1440. How does this increase change the equation describing desired national saving? Show the change graphically. What happens to the market-clearing real interest rate?

(c)  When G = 1440, desired saving becomes Sd = Y - Cd - G = Y - (3600 - 2000r + 0.1Y) - 1440 =  -5040 + 2000r + 0.9Y.
Sd is now 240 less for any given r and Y; this shows up as a shift in the Sd line from S1 to S2 in the Figure

Figure

Setting Sd = Id, we get:

-5040 + 2000r + 0.9Y = 1200 - 4000r

6000r + 0.9Y = 6240

At Y = 6000, this is 6000r = 6240 - (0.9 ´ 6000) = 840, so r = 0.14. The market-clearing real interest rate increases from 10% to 14%.

3. Suppose that the economywide expected future marginal product of capital is MPKf = 20 – 0.02K, where K is the future capital stock. The depreciation rate of capital, d, is 20% per period. The current capital stock is 900 units of capital. The price of a unit of capital is 1 unit of output. Firms pay taxes equal to 50% of their output. The consumption function in the economy is C= 100 + 0.5Y-200r, where C is consumption, Y is output, and r is the real interest rate. Government purchases equal 200, and full-employment output is 1000.

a. suppose that the real interest rate is 10% per period. What are the values of the tax-adjusted user cost of capital, the desired future capital stock, and the desired level of investment?

(a)   r = 0.10

uc/(1 - τ) = (r + d)pK/(1 - t) = [(.1 + 0.2) ´ 1]/(1 - 0.15) = 0.35.

MPKf = uc/(1 - t), so 20 - 0.02K = 0.35; solving this gives K = 982.5.

Since K - K-1 = I - dK, I = K - K-1 + dK = 982.5 - 900 + (.2 ´ 900) = 262.5.

b. Now consider the real interest rate determined by goods market equilibrium. This part of the problem will guide you to this interest rate.

i. Write the tax-adjusted user cost of capital as a function of the real interest rate r. also write the desired future capital stock and desired investment as functions of r.

ii. Use the investment function derived in Part (i) along with the consumption function and government purchases, to calculate the real interest rate that clears the goods market. What are the goods market-clearing values of consumption, saving, and investment? What are the tax-adjusted user cost of capital and the desired capital stock in this equilibrium?

(b)  i.  Solving for this in general:

uc/(1 - t) = (r + d)pK/(1 - τ) = [(r + .2) ´ 1]/(1 - 0.15) = .235 + 1.176r.

MPKf = uc/(1 - t), so 20 - 0.02K = 0.235 + 1.176r; solving this gives K = 988.25 - 58.8r.

I = K - K-1 + dK = 988.25 - 58.8r - 900 + (0.2 ´ 900) = 268.25 - 58.8r.

ii.  Y = C + I + G

1000 = [100 + (.5 ´ 1000) - 200r] + (268.25 - 58.8r) + 200

1000 = 1068.25 - 258.8r, so 258.8r = 68.25

r = 0.264

C = 100 + (0.5 ´ 1000) - (200 × 0.264) = 547.2

I = 268.25 - (58.8 × 0.264) = 252.7 = S

uc/(1 - t) = 0.235 + (1.176 ´ 0.264) = 0.545

K = 988.25 - (58.8 × 0.264) = 972.7

4.  Use the saving-investment diagram to analyze the effects of the following on national saving, investment, and the real interest rate. Explain your reasoning.

a.       Consumers become more future-oriented and thus decide to save more.

b.      The government announces a large, one-time bonus payment to veterans returning from a war. The bonus will be financed by additional taxes levied on the general population over the next five years.

This problem can be looked at in a very complex way with MPC and comparing vets to the general population.  The easiest way, however, is an increase in wealth reduced saving and the change in taxes will also decrease saving.

c.       The government introduces an investment tax credit (offset by other types of taxes, so total tax collections remain unchanged).

Encourages investment!

d.      A large number of accessible oil deposits are discovered, which increases the expected future marginal product of oil rigs and pipelines. It also causes an increases in expected future income.

The increase in expected future income decreases current desired saving, as people increase desired consumption immediately. The rise of the future marginal productivity of capital shifts the investment curve to the right. The result, as shown in Figure, is that the real interest rate rises, with ambiguous effects on saving and investment.

5.  A country loses much of its capital stock to a war.

e.      What effects should this event have on the country’s current employment, output, and real wage?

Less capital stock

f.        What effect will the loss of capital have on desired investment

Because the capital stock is lower, the marginal product of capital will be higher, so desired investment will increase.  This is tricky, but I encourage to think about why this is true.  Would this be true in every factory or is this just a macro answer?

g.       The effects of desired national saving of the wartime losses are ambiguous. Give one reason for desired saving to rise and one reason for it to fall.

(c)  Since current output declines, desired saving declines, because people do not want to reduce their consumption. On the other hand, since future output is also lower, people desire to save more today to make up for the loss of future income.

h.      Assume that desired saving doesn’t change. What effect does the loss of capital have on the country’s real interest rate and the quantity of investment?

6.  In a small open economy, output (gross domestic product) is \$25 billion, government purchases are \$6 billion, and net factor payments from abroad are zero. Desired consumption and desired investment are related to the world real interest rate in the following manner:

World Real Interest Rate                     Desired Consumption                   Desired Investment

5%                                                          \$12 billion                                            \$3 billion

4%                                                          \$13 billion                                            \$4 billion

3%                                                          \$14 billion                                            \$5 billion

2%                                                          \$15 billion                                            \$6 billion

For each value of the world real interest rate, find national saving, foreign lending, and absorption. Calculate net exports as the difference between output and absorption. What is the relationship between net exports and foreign lending?

The following table calculates key variables for this question for different values of the real interest rate. The column for S is calculated by the equation S = Y - (Cd + G). The column headed S - I is foreign lending. Absorption (A) is Cd + Id + G. Net exports (NX) are output (Y) minus absorption (A). Every column except r consists of dollar amounts in billions.

 r Cd Id S S - I A NX 5% 12 3 7 4 21 4 4% 13 4 6 2 23 2 3% 14 5 5 0 25 0 2% 15 6 4 –2 27 –2

Net exports and foreign lending are identical.

3.     All variables but interest rates are in billions of dollars.

(a)  S = 10 + (100 ´ 0.03) = 13

A = C + I + G

= 27 + 12 + 10

= 49

(b)  S = 13, as before.