1.  A more extensive aggregate production function makes sense when studying a number of problems. The Cobb-Douglas production function in the text can be expanded to include 3 factors of production (capital, labour, and energy). It can be written as:

The data for the Rubble Economy are given below. Fill in the missing cells, denoted "???".  (20 points)

---	Y	A	K	L	E
Alphas	---	---	alpha1 = 0.3	alpha2 = 0.4	alpha3 = 0.3
1950	450	???	100	23	94
1960	560	???	120	54	102
1970	550	???	159	78	120
1980	640	???	154	97	123

1950-60	0.2444	???	???	???	???
1960-70	???	???	???	???	???
1970-80	???	???	???	???	???

1950	---	---	1.35	???	???
1960	---	---	???	???	???
1970	---	---	???	???	???
1980	---	---	???	???	???

 

2. Consider a closed economy with two time periods, denoted 1 and 2.  In each time period, GDP is given by Y₁=Y₂=300. This question studies the effects of fiscal policy in the first time period. Taxes in the first period are denoted T₁ and taxes in the second period are denoted T₂, while r is the real interest rate. Assume that the interest rate is fixed at 20%.  Suppose that consumption depends on current and future after-tax income, as follows:

a. If G₁=30, T₁=30 and T₂=0 solve for C₁, S_pvt and S_govt. (3 points)

b. Now suppose instead that the government has cut taxes to T₁=15.  The government financed its deficit by borrowing 15, and then introduces a tax in the second period of T₂=15(1+r) in order to pay off its loan. Solve for C₁ under the new tax policy. (3 points)

c. As a third scenario, suppose that the government instead spent G₁= 50, and raised taxes in the first period to finance the increased spending. What is the effect on C₁, S_pvt, and S_govt of this change? (3 points)
 

3. [Based on material found in Chapter 8] This question explores how wealth, future income, and the interest rate
affect consumption and saving.  Suppose Joseph's assets are denoted a, his income is denoted y, and his expected future income is denoted y^f. His current consumption is c and his future consumption is c^f. The present value of his consumption is limited by the present value of his resources:

Furthermore, he tries to smooth his consumption over time, so that c =c^f.

a. If a=0, r=0.05, y=40 and y^f=40 then what is his current consumption and saving? (2 points)

b. What is his consumption and savings if instead y^f =60? (2 points)

c. Now suppose that, at this higher level of expected future income, the interest rate rises to r=0.08. What is his consumption and savings? Do changes in the interest rate have a large effect on his saving? (4 points)

d. Joseph's parents bequeath him a gift of a=10. At y^f=60 and r=0.08 what is the effect on his consumption and saving? Are the effects consistent with the predictions in the textbook? (4 points)
 

4.  Now suppose once again that a=0, r=0.05, y=40, y^f=40 and Joseph continues to smooth consumption (c = c^f).
Now add the realistic feature that Joseph must pay taxes.  Suppose that taxes are lump-sum in each period. Now the consumption plan must satisfy:

a. Find consumption and savings if t=t^f=10. (2 points)

b. Suppose that the government raises the current tax to a value of t= 12, without changing its current or planned spending. What do you predict will be the effect on consumption and savings?  Explain your assumptions. (Hint: see pp 602-605) (6 points)
 

5. A firm has current and future marginal productivity of capital given by MPK=10,000-2K + N, and marginal productivity of labour given by MPN = 50 - 2N + K. The price of capital is $5,000, the real interest rate is 10%, and capital depreciates at a 15% rate. The real wage rate is $15.

a. Calculate the user cost of capital. (5 points)

b. Find the firm's optimal amount of employment and the size of the capital stock. (5 points)
 

6. Consider a large open economy in equilibrium with zero current account balance. What are the effects on world real interest rate, national saving, investment, and the current account balance in equilibrium if:

a. future income rises? (4 points)

b. business taxes fall? (4 points)

c. government purchases decline temporarily? (4 points)

d. the future marginal product of capital rises? (4 points)
 

7. Suppose Canada has the following balance of payments data (in billions):

 Merchandise exports = 276
 Merchandise imports = 233
 Foreign travel in Canada = 12
 Foreign purchases of Canadian services (excluding travel) = 27
 Canadian travel abroad = 15
 Canadian purchases of foreign services (excluding travel) = 33
 Investment income receipts from assets = 18
 Investment income payments on assets = 21
 Net transfers from other countries to Canada = 2
 Increase in home country's ownership of assets abroad = 36
 Increase in foreign ownership of assets in home country = 30
 Increase in home reserve assets = 5
 Increase in foreign reserve assets = 4

Find the following (show all calculations):

a. Merchandise trade balance (3 points)

b. Net exports (3 points)

c. Current account balance (3 points)

d. Official settlements balance (3 points)

e. Statistical discrepancy (3 points)
 

8. Suppose the economy has a marginal product of labour given by MPN = 500 - 0.5 N, where N is aggregate employment. Labour supply can be described by NS = 400 + 8(1-t)w, where w is the real wage and t is the marginal tax rate.

a. Find the real wage and employment when t = 0.5.  (5 points)

b. Find the real wage and employment when t = 0.25. Explain the effect of the income-tax cut on the labour market equilibrium. (5 points)

1. Given the production function:

---	Y	A	K	L	E
Alphas	---	---	alpha1 = 0.3	alpha2 = 0.4	alpha3 = 0.3
1950	450	8.25	100	23	94
1960	560	6.75	120	54	102
1970	550	5.00	159	78	120
1980	640	5.35	154	97	123

---	Y	A	K	L	E
1950-60	0.244	-0.183	0.200	1.348	0.085
1960-70	-0.018	-0.258	0.325	0.444	0.176
1970-80	0.164	0.068	-0.031	0.244	0.025

---	---	---	K	L	E
1950	---	---	1.35	7.826	1.436
1960	---	---	1.400	4.148	1.647
1970	---	---	1.038	2.821	1.375
1980	---	---	1.246	2.639	1.561

 

2. Given the equation:

a. If G₁=30, T₁=30 and T₂=0,

C₁ = 0.45*[(300-30) + (300/1.2)] = 234

Y = C + S + T so for period 1, S = Y - C - T = 300 - 234 - 30 = 36

S = S_pvt + S_govt = S_pvt + (G-T) = S_pvt = 36

b. If T₁=15 and T₂=15(1+r),

C₁ = 0.45*[(300-15) + ((300/1.2) - 15)] = 234

The same as in part a. (This is an example of Ricardian equivalence.)

c. If G₁=50, the government would incur a deficit of 20 in period 1. Thus, in order to finance the new spending, the government must set T₁=50 (holding T₂constant).

Thus, C₁ = 0.45*[(300-50) + (300/1.2)] = 225

Since S = Y - C - T,
S = 300-225-50 = 25.

S = S_pvt + S_govt = S_pvt + (G-T) = S_pvt = 25

In words, the increase in (tax-financed) government spending has caused consumption in period 1 to fall, along with private savings. Government savings are still zero since the budget is balanced. (Incidentally, it also is predicted to affect the interest rate in a large economy -- which we have ignored by fixing r = 0.10).

3. The present-value of consumption is:

with c =c^f.

a. If a=0, r=0.05, y=40 and y^f=40,

c + (c/(1+0.05)) = 40 + 40/(1+0.05)
(2.05)*c = (2.05)*40

c = 40
s = y - c = 0

b. If instead y^f = 60,

c + (c/(1+0.05)) = 40 + 60/(1+0.05)
(2.05)*c = (1.05)*40 + 60

c = (102/2.05) = 49.76
s = y - c = 40 - 49.76 = -9.76

c. If the interest rate rises to r=0.08, (y^f=60)

c + (c/(1+0.08)) = 40 + 60/(1+0.08)

c = 49.62
s = y - c = -9.62

The change in the interest rate increases his saving (the income and substitution effects work in the same direction for borrowers) but the effect is small.

d. Now a=10, y^f=60 and r=0.08, so

c + (c/(1+0.08)) = 10 + 40 + 60/(1+0.08)
(2.08)*c = 50*(1.08) + 60 = 114

c = 54.80
s = -14.80. An increase in wealth increases consumption and reduces saving, as discussed in chapter 4.

4. Now a=0, r=0.05, y=40, y^f=40 and c = c^f, with,

a. If t=t^f=10,

c + (c/(1+0.05)) = (40-10) + [(40-10)/(1+0.08)]
c = 30
s = y - c - t = 40 - 30 - 10 = 0

b. If Joseph understands the government's budget constraint (as Ricardian equivalence) then he will not change his consumption. He will borrow 2 to pay his additional taxes, and repay it with interest from next period's tax cut of 2.1. But this assumes that (i) he can borrow at the same interest rate as the government and (ii) that he is not myopic and (iii) that the tax cut will occur while he is still alive. If these conditions do not hold then he may reduce his current consumption when taxes increase with no change in government spending.

5. MPK=10,000-2K + N, MPN = 50 - 2N + K. Pk = $5,000, real interest rate, r = 10%, capital depreciation, d = 15% rate, and wage rate, w = $15.

a. UC = r*Pk + d*Pk = (r + d)*Pk = (0.1 +0.15)* Pk = 0.25*Pk = $1,250

b. The firm's desired capital stock equilibrates the user cost with expected future MPK:

UC = $1,250 = 10,000 - 2K + N = MPK^f

which implies that N = 2K - 8750 (a). Assuming that the firm can hire as much labour as it wants at $15 a worker, we want MPN = w, or,

15 = 50 - 2N + K
K = 2N - 35 (b).

Substituting (a) into (b),

K = 2(2K - 8750) - 35 = 4K - 17,535
K = 5845

From (a), we have,

N = 2(5845) - 8750 = 2940

6. Consider a large open economy in equilibrium with zero current account balance. [Use graphical analysis similar to that found in Section 5.4]

a. r(w) rises, S falls, I falls, CA falls.

b. r(w) rises, S rises, I rises, CA falls.

c. r(w) falls, S rises, I rises, CA rises.

d. r(w) rises, S rises, I rises, CA falls.
 

7. From the balance of payments data,

a. Merchandise trade balance = Merchandise exports - Merchandise exports = 276 - 233 = 43.

b. Net exports = (Merchandise exports + Foreign travel in Canada + Foreign purchases of Canadian services) - (Merchandise imports + Canadian travel abroad + Canadian purchases of foreign services) = (276 + 12 + 27) - (233 + 15 + 33) = 315 - 281 = 34.

c. Current account balance = (Merchandise exports + Foreign travel in Canada + Foreign purchases of Canadian services + Investment income receipts from assets + Net transfers from other countries to Canada) - (Merchandise imports + Canadian travel abroad + Canadian purchases of foreign services + Investment income payments on assets) = (315 + 18 + 2) - (281 + 21) = 335 - 302 = 33.

d. Official settlements balance = Increase in home reserve assets - Increase in foreign reserve assets = 5 - 4 = 1.

e. Statistical discrepancy:

CA + KA + SD = 0
KA = Increase in foreign ownership of assets in home country - Increase in home country's ownership of assets abroad = 30 - 36 = -6.

SD = -CA - KA = -33 - (-6) = -27
 

8. MPN = 500 - 0.5 N, NS = 400 + 8(1-t)w, where w is the real wage and t is the marginal tax rate.

a. The equilibrium of the labour market is found where Ns = Nd. Rewrite MPN, the demand for labour, as w = 500 - 0.5N, and labour supply as N = 400 +8(1-t)w. When t=0.5, supply becomes:

Supply: N = 400 + 8(0.5)w = 400 + 4w
Demand: w = 500 - 0.5N

Substitute supply into demand as follows:

w = 500 - 0.5*(400 + 4w) = 500 - 200 - 2w
w = 100

Then N = 400 + 8(0.5)*100 = 800

b. When t = 0.25,

Supply: N = 400 + 6w
Demand: N = 500 - 0.5N

Proceed as in part (a) so as to get:

w = 75
N = 850

The tax reduction has caused workers to supply more labour at every given wage, since the quantity of labour supplied is assumed to decrease as the rate of taxation rises. The labour supply curve shifts "out" while labour demand remains the same. Consequently, the labour market equilibrium shows a higher level of unemployment with a lower market-clearing wage level.