ECON 222Macroeconomic Theory I

Fall Term 2014/15Section 002Instructor:Bill Dorval

Midterm Exam - ANSWER KEY

PART A: Long Questions.

Question A.1: Equilibrium in the Labor Market (30 Marks)

(a) (7 Marks) First we are asked to find the labor demand. In order to do that, we need to compute themarginal product of labor and equate it to the real wage: MPN = w.

First, notice that we can collect terms in the production function:

Y = Kα (AN)β

+ Lγ (AN)β

= (Kα + Lγ) (AN)β

Then we compute the derivative of the production function with respect to labor and impose the opti-mality condition:

MPN = βA (Kα + Lγ) (AN)β−1

= w

Finally, solving the equation for Nd (with the wage on the right hand side) gives the labor demand:

Nd (w) =1

A

[w

βA (Kα + Lγ)

] 1β−1

=1

A

[βA (Kα + Lγ)

w

] 11−β

(b) (9 Marks) To find the equilibrium wage w∗ we need to equate the labor demand to the labor supply:

Nd (w) = Ns (w)

1

A

[βA (Kα + Lγ)

w

] 11−β

= w1

1−β

(1

A

)1−β [βA (Kα + Lγ)

w

]= w

βAβ−1A (Kα + Lγ) = w2

w2 = βAβ (Kα + Lγ)

w∗ =[βAβ (Kα + Lγ)

] 12

Now we can plug w∗ in either the supply or in the demand of labor to find the equilibrium value of em-

ployment, getting N∗ =[βAβ (Kα + Lγ)

] 12(1−β) . The following figure represents graphically the equilibrium.

(c) (7 Marks) The following figure represents graphically the equilibrium. Notice that the labor demandshifts down, because in the graph the inverse supply and demand functions are represented. The newequilibrium is achieved with both a lower wage and a lower level of employment. This happens because thedecrease in labor demand creates a gap between demand and supply of labor at the old equilibrium wagew∗. More precisely, an excess supply of labor arises, because less firms are now willing to hire workers fora given wage, compared to the situation before the change. If wages are perfectly flexible, then the labor

market moves quickly from the old to the new equilibrium, with a decrease in the wage rate eliminating theexcess supply, and with a lower number of workers being employed because less people are willing to workat the lower wage.

(d) (7 Marks) With the given parameters, the production function in the short run (as a function oflabor only, because capital and land are fixed) is Y =

(40.5 + 2

)N0.5 ≈ 4∗N0.5. It follows that the marginal

product of labor is dY (N)dN = 2 ∗N−0.5. The equilibrium then is w∗ ≈

√0.5 ∗ 1 ∗ 4 =

√2 ≈ 1.414 and N∗ = 2.

The average labor productivity is APL∗ = Y∗

N∗ =4√2

2 = 2√

2 ≈ 2.818.

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Question A.2: Intertemporal consumption (30 Marks)

Consider a two-period model of consumption. An individual needs to decide how much to consume to-day, c1 and how much to consume tomorrow, c2. He begins his economic life in period 1 with some wealth,W , and receives income in both periods, y1 and y2. This person can borrow or lend at rate r. The marginalrate of substitution between c1 and c2 is:

−∂u∂c1∂u∂c2

= −β c2c1

a) (7 Marks) Derive an algebraic expression for the optimal present and future consumption, c1 and c2, asa function of the present value of lifetime resources (PVLR).

First we need to set the MRS equals to the slope of the budget constraint:

−β c2c1

= −(1 + r)⇒ c2 =1

β(1 + r)c1

Then we need to plug the latter in the intertemporal budget constraint:

c1 +c2

1 + r= W + y1 +

y21 + r

= PV LR⇒ c1 +1

β

(1 + r)

1 + rc1 = PV LR⇒ c1 =

β

1 + βPV LR

c2 =1

1 + β(1 + r)PV LR

b) (7 Marks) Assume β = 11/7, W = 0, y1 = 50, y2 = 42, and r = 5%. Using the formulas found in a), findthe numerical values for c1 and c2.

c1 =β

1 + βPV LR =

11/7

1 + 11/7

(50 +

42

1 + 0.05

)= 55

c2 =1

1 + β(1 + r)PV LR =

1

1 + 11/7(1 + 0.05)

(50 +

42

1 + 0.05

)= 36.75

c) (4 Marks) Is the agent a lender, a borrower or is he at the no-borrowing-no-lending point? Explain.

The agent is a borrower. His income in period 1 is 50 but he wants to consume 55. Therefore he needto borrow 5.

d) (5 Marks) Now assume that, because of new bank regulations, this agent is no longer allowed to borrow.What will be the new equilibrium. Explain.

Since before the borrowing constraint the agent was a borrower, he will try to consume as much as hecan in period one. Therefore the new equilibrium will be at the no-borrowing-no-lending point, i.e. c1 = y1.

e) (7 Marks) Now assume that borrowing is allowed. Also, the agent knows that he will receive a be-quest of 42 in period 2. In other words, in period 2 the agent receives some wealth, W2 = 42, that he canborrow against in period 1. Find the new equilibrium c1 using the values of part b).

Because the agent will now have a bequest in period 2, the PVLR becomes:

y1 +y2

1 + r+

W21 + r

Therefore:

c1 =β

1 + βPV LR =

11/7

1 + 11/7

(50 +

42

1 + 0.05+

42

1 + 0.05

)= 79.4

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Question A.3: Goods Market (30 Marks)

Assume that the production function for next period (i.e., the future period) of a closed economy is givenby:

Yt+1 = 80Kt+1 − 4K2t+1 + 10Nt+1 − 2N2t+1where Kt+1 is the capital stock in period t+1 and Nt+1 stands for the labour input in period t+1.

a) (5 Marks) Find the marginal product of capital for period t+1. This is MPKf .

MPKf =∂Yt+1∂Kt+1

= 80− 8Kt+1

b) (5 Marks) Assume that there is no tax on capital, τ = 0, that capital at rate d = 10%, and that the priceof capital, Pk, is 1. Find an expression for Kt+1 in terms of the interest rate, r.

MPKf = 80− 8Kt+1 =uc

1− τ=r + d

1− τPk =

r + 0.1

1− 0(1)

⇔ Kt+1 =79.9− r

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c) (5 Marks) Use your result from b) to derive the investment function in terms of the interest rate, r.Assume that the capital stock in period t, Kt, is 1 and the capital’s depreciation rate is the same as in partb).

We need to use the gross investment equation:

It = Kt+1 − (1− d)Kt

and plug in the formula for Kt+1 found in part b):

It =79.9− r

8− (1− 0.1) ∗ 1 = 72.7− r

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Now assume that consumption in this economy is given by:

Ct = 0.6(Y − T )− 18r

where Y = 40, G = 20 and the government budget is balanced (T = G).

d) (5 Marks) Find the equilibrium interest rate.

Saving in a closed economy is given by S = Y − C −G. By substituting the values for Y , C and G = T inthe latter we get:

S = Y − 0.6(Y −G) + 10r −G = 0.4(Y −G) + 18r = 0.4(20) + 18r = 8 + 18r

. Finally. the goods market equilibrium for a closed economy is S = I:

8 + 18r =72.7− r

8→ r = 0.06

e) (5 Marks) Assume that a new invention raises the expected future marginal product of capital. Thisresults in the following investment function:

It =80− r

8

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Also assume that the government decides to change its spending, while keeping a balanced budget. Theobjective of the government is to keep the interest rate at its equilibrium value found in d). By how muchG has to change? Note: Y is fixed at Y = 40.

We know that

It =80− r

8= 9.9925 ≈ 10

andS = 0.4(Y −G) + 18r

ThenI ≈ 10 = S = 0.4(40−G) + 18(0.06) = 17.7

So G has to decrease by 2.3 (from 20 to 17.7).

f) (5 Marks) Finally assume that this economy is now a small open economy and that the world inter-est rate is 4%. Is this economy a net borrower or a net lender? Draw in a graph the current account.

The world interest rate is under the closed economy’s interest rate, therefore this economy is borrowingfrom the rest of the world.

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Question B.1: Ricardian Equivalence and Consumption-Savings Decisions (20 Marks)

(a) There are two channels that affect the response of current consumption. The first channel, is thedirect effect of the interest rate increase. Since both the income effect and the substitution effect move inthe same direction, this leads unambigously to a reduction in current consumption. At the same time, theinterest payments arising from the public deficit increase. This will eventually lead the government to sethigher future taxes (compared to a situation without the interest rate change). This makes the consumer’spresent value of lifetime resources lower, leading to a further decrease in current consumption, but also toa decrease in future consumption. Overall, for current consumption the two channels reinforce each other,potentially leading to a large drop in its value.

(b) The fact that the cuts in the government expenditure haven’t been implemented together with thetax cuts has sparked a widespread belief of future tax increases. As a consequence, Italian households aresaving most of the tax cuts to be able to pay the higher future taxes, and they are not increasing theirconsumption expenditures. This is not necessarily the correct interpretation of these recent events, but it isindeed consistent with the key predictions of Ricardian Equivalence.

Question B.2: Taxes and Investment in Two Large Open Economies (20 Marks)

a) A sudden increase in corporate taxes leads the Investment curve in China to shift to the left. Forany interest rate, Chinese firms want to undertake less investment, as it’s now less profitable. The Chinesecurrent account is no longer compatible with the American one, so there is an adjustment in the interestrate, which decreases to restore the sum of current accounts equal to zero. This leads to an even largercurrent account surplus for China and a larger current account deficit for the U.S. See also the Figure below.

b) In this extreme case, the economies become closed economies. The Investment and Saving schedulewould remain the same, there would be two domestic interest rates, obtained when I = S in each countrytaken separately. In particular, the interest rate in the US economy would increase, to eliminate the excessdemand of investment.

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Question B.3: Investment and Desired Capital (20 Marks)

(a)1) MPKf = 24T 0.5 = (0.07 + 0.01) ∗ 100→ T = 9. (4 Marks)2) MPKf = 24T 0.5 = (0.07 + 0.01) ∗ 300→ T = 1. (3 Marks)3) (1− τ) ∗MPKf = (1− 0.5) ∗ 24T 0.5 = (0.07 + 0.01) ∗ 150→ T = 1. (3 Marks)

(b)The user cost of capital remains the same, so the answers are unchanged.1) MPKf = 24T 0.5 = (0.05 + 0.01 + 0.02) ∗ 100→ T = 9. (4 Marks)2) MPKf = 24T 0.5 = (0.05 + 0.01 + 0.02) ∗ 300→ T = 1. (3 Marks)3) (1− τ) ∗MPKf = (1− 0.5) ∗ 24T 0.5 = (0.05 + 0.01 + 0.02) ∗ 150→ T = 1. (3 Marks)

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