Queen’s UniversityDepartment of Economics

ECON 222Macroeconomic Theory I

Winter Term 2011/12

Section 002Midterm Examination

Please read all questions carefully. Record your answers in the answer booklet provided. You areencouraged to draw diagrams to support your answers. Please label the axis and lines or curves on yourdiagrams.

The exam has two parts. Marks will be awarded also on the basis of the logical arguments given tosupport your answers.

Part A consists of short answer questions. Do FOUR of the six questions. Each question is worth 10marks for a total of 40 marks.

Part B consists of long questions. Do all of the questions. Each question is worth 30 marks for a total of60 marks.

The exam is 80 minutes long. Budget your time carefully. Hand calculators (non programmable) arepermitted for this exam. Upon completion of your exam, hand in the answer booklet clearly labelled withyour student number and class section, as well as the exam. Any cheating attempt will be sanctioned withthe toughest possible punishment.

PART A: Short Questions

Answer any FOUR of the following six questions. Each question is worth 10 marks for a total of 40marks. Answers without any explanations will receive zero marks.

Question A.1Consider the two period model used in class. Explain the consumption-savings decision of a lender when theinterest rate decreases. Clearly differentiate between the income (wealth) and substitution effects in yourexplanation.

Solution: Lenders have a negative income affect which decreases current and future consumption hold-ing all else equal. The decrease in future consumption implies an increase in savings. Lenders also face asubstitution effect; current consumption is cheaper relative to future consumption. The effect is to increasecurrent consumption and decrease savings. Combining the two effects we see that the impact on savings isambiguous. By definition so is the impact on current consumption. However future consumption decreasesunambiguously.

The above explanation is sufficient for full marks. Consider the graph below. The movement from IC1

to IC2 on a line with the slope of the new budget constraint represents the substitution effect to cheapercurrent consumption. The movement from IC2 to IC3 represents the income effect as the other two levels ofconsumption are no longer affordable. The graph has been drawn in a way such that the substitution effectdominates the income effect. Current consumption decreases and savings increases.

Question A.2

MPN = (1/2)K1/2N−1/2 (1)

Ns = w2 (2)

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MPN is the marginal product of labour and Ns is the labour supply function. Find an expression for theequilibrium real wage. Then assume K = 64. What is the equilibrium real wage and employment?

Solution:

N−1/2 = 2K−1/2w

N =

[(2w)

K1/2

]−2

Nd =

[K

(2w)2

]

Set labour demand equal to labour supply

Nd = Ns

K

(2w)2= w2

K/4 = w4

w =[K4

]1/4andK = 64

w = [16]1/4

w = 2

Using the value for the real wage in the expression for labour supply we find employment:

Ns = w2

Ns = 22

N = 4

Question A.3What is Ricardian Equivalence? Suppose government is planning a lump-sum tax cut funded by a governmentdeficit (bond issuance) in a closed economy. You may assume output is fixed. Compare and contrast theimpact on the Goods Market (Investment - Savings) when Ricardian Equivalence holds and when it doesnot hold. What happens to the level of investment?(You do not need a math proof. A written explanation of your intuition is sufficient although a supportinggraph is highly recommended)

Solution:

Ricardian Equivalence (R.E.) is based on rational expectations. Households, when faced wtih a lumptax cut, expect government to raise taxes in the future to pay for the bond issuance. Thus households areentirely indifferent to the lump sum tax cut and save the entire amount to pay for future tax increases. Thehouseholds save the same amount the government dissaves. So if R.E. holds then national savings does notchange. If R.E does not hold then Households believe they have more income in the current period. Thiscauses a wealth (income) effect and consumption goes up as does private savings . However the governmentdeficit increases by the lump tax cut. The government savings is larger than the private savings which impliesnational savings goes down. This is analogous to a shift to the left in the Savings demand curve. Investmentis ”crowded out” by the amount that consumption went up. The graph below corresponds to the change insavings-investment when R.E. does not hold.

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Question A.4We discussed a number of government policies that could be useful in increasing productivity. Identify twopolicies and discuss under what conditions (if any) they would be effective.

Solution: See the textbook pages 212-216 for a complete description of the five policies

The five policies are:Improving infrastructureBuilding human capitalResearch and developmentIndustrialMarket

Question A.5

Table 1: Question A.5 - National Economic AccountsLabour Income 80,000 Unincorporated Business Income 15,000Indirect Taxes 25,000 Net Income From Abroad -5,000Corporate Profits 25,000 Gross Capital Stock 550,000Net Exports 8,000 Investment 33,000

What is Gross Domestic Product (Y) and Gross National Product (GNP)? Assume the governmentdeficit/surplus is zero. What is national savings ?

Solution: you must use the income-approach to calculate GDP

GDP = 80, 000 + 25, 000 + 22, 000 + 15, 000

= 142, 000

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and GNP:

GDP = GDP +NFP

= 142, 000− 5, 000

= 137, 000

and national savings:

Sd = Id + CA

= 33, 000 + 8, 000− 5, 000

= 36, 000

Question A.6

Unemployment can be categorized into three types: frictional, structural and cyclical. Briefly describeeach one. Which of these is associated with long periods of unemployment? Provide a current or historicalexample of this type of unemployment.

Solution: the three types of unemployment are frictional, cyclical and structural

Frictional results from the creation and destruction of jobs. It is typically temporary and short.

Cyclical comes from business cycles. This is measured as deviations from the natural rate of employ-ment.

Structural unemployment (or chronic unemployment) results from a lack of the right skills, living in ageographical depressed area or from industry downsizing. Current and historical examples include the New-foundland fisheries in the 1990’s and the more current but less drastic downsizing of the Canadian/Americanautomobile industry. Structural unemployment tends to be frequent and last for long periods of time. Ittakes time for workers to learn new and marketable skills. As such this form of unemployment represents adifficult challenge for government policy.

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PART B: Long Questions

Answer the two following questions. Each question is worth 30 marks for a total of 60 marks. Answerswithout any explanations will receive zero marks.

Question B.1: Consumption-Savings (30 Marks)

Suppose the productions side of a closed economy is characterized by the following equations:

MPKf = 6−[

1

100

]Kf

Y = 700

K = 490

δ = 0.2

pK = 1

I = Kf − (1− d)K

where MPKf is next periods expected marginal product of capital, Y is full employment output, K isthe current capital stock, δ is the rate of depreciation of capital, pK is the price of capital and I is grossinvestment. Additionally, define r as the real rate of interest. Assume there are no taxes on the marginalproduct of capital.

(a) What is the desired (optimal) gross investment decision as a function of the real rate of interest?First (i) Derive the desired future capital stock Kf as a function of r. (ii) Use Kf to find the desired grossinvestment which will also be a function of r.

ANSWER:Capital:

uc = pK(δ + r)

MPK = uc

6−[

1

100

]Kf = 0.2 + r

Kf = (5.8− r)100

Kf = 580− 100r

(b) We want to determine the investment demand function Id. Use your expression for the optimal futurecapital stock Kf and the gross investment equation to find investment demand as a function of r.

Investment:

Id = Kf − (1− δ)K= 580− 100r − (1− 0.2)490

= 188− 100r

(c) Aggregate consumption in the economy is defined by the equation

Cd = 0.6(Y − T )− 200r

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Government expenditure are G = 230, and taxes paid by households are T = 200. You may assume there areno interest payments on the national debt. Using this additional information, find an equation for desirednational savings in terms of the real rate of interest (r).

ANSWER:

Sd = SPV T + SGOV

= (Y − T − Cd) + (T −G)

= Y − Cd −G= 700− (700− 200)(0.6) + 200r − 230

= 170 + 200r

(d) Given the desired investment and savings derived in parts (a) and (b), define the goods market equi-librium. That is, find the market clearing real rate of interest and use this to find the equilibrium levels ofsavings and investment.

Next, assume this country is able to trade with the rest of the world. The country has no influence onthe world real interest rate, rw, and takes prices as given (i.e. this country is a small open economy). Theprevailing world real interest rate is rw = 0.03. Is the economy a net lender or a net borrower? Show orexplain why.

ANSWER:In equilibrium in a closed economy it must be that Sd = Id:

Sd = 170 + 200r = 188− 100r = Id

300r = 18

r =18

300r = 0.06 = 6%

Then

Sd = 170 + 200(0.06)

= 182

Id = 188− 100(0.06)

= 182

Which we know is correct since in a closed economy it must be that Sd = Id

The economy is open and the real rate of interest is rw = 0.03 while the implied dometic rate is 6%. Thisimplies the CA is negative (in deficit) and investment will be higher than savings. Why? The rate at whichmoney can be borrowed is cheaper after the economy is opened. This spurs investment while discouragingsavings. Calculating the new level of investment and savings is not necessary.

Sd = 170 + 200(0.03)

= 176

Id = 188− 100(0.03)

= 185

CA = 176− 185 = −9

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(e) Suppose that the small open economy described in part c has a Current Account (CA) deficit.Represent this on a properly labelled graph. Identify the implied domestic equilibrium and the open economylevels of savings (S) and investment (I).

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Question B.2: Neo-Classical Growth (Solow-Swan) Model (30 Marks)

Consider a closed economy with no government spending and an infinite number of time periods. Thepopulation and work force grow at rate n per period and the capital equipment depreciates at rate d. Theproduction function is:

Y = ZN1−α(AK)α (3)

where Y is aggregate output, K is aggregate capital, N is aggregate labour, A is the productivity of capitaland Z is total factor productivity. The per worker production function is:

y = Z(Ak)α (4)

where y and k are in per worker ratios.

Part a) Derive the Golden Rule capital to labour ratio kg. This is the ratio of capital to labour thatmaximizes consumption per worker.

Solution: there are two methods to find the golden rule capital-labour ratio. The first is to formally defineconsumption as output less savings, replace savings with investment since this is a closed economy andmaximize. The second is maximize output with respect to capital and set the MPK equal to the marginalcost of capital which is n+d.

c = Z(Ak)α − sZ(Ak)α

Sd = Id

c = Z(Ak)α − (n+ d)k

Whether we maximize the above expression for c or directly set the MPK equal to its marginal cost (n +d) we get:

αZAαkα−1 = n+ d

kα−1 =

[n+ d

αZAα

]kg =

[αZAα

n+ d

] 11−α

Part b) There now exists an exogenous savings rate s (0 < s < 1) that denotes the fraction of outputworkers save each period (sy = sZ(Ak)α). Suppose:

A = 16

Z = 1

n = 0.05

d = 0.2

α = 0.5

What is the savings rate required to sustain the Golden Rule kg per capita ratio of capital as a steadystate?

Solution: first solve for kg

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kg =

[αZAα

n+ d

] 11−α

=

[(0.5)160.5

0.25

] 11−0.5

=

[(0.5)(4)

0.25

]2= 82

kg = 64

and solve for the savings rate (s) by equating the fraction of output saved to the total cost of maintainingthe capital stock:

sZAαkαg = (n+ d)kg

s =(n+ d)kgZAαkαg

s =(0.2 + 0.05)64√

16√

64

=(0.25)(64)

(4)(8)

s = 0.5

Part c) What is the key determinant of long-term growth in this model and why? Graphically illustrate theeffect an increase in this determinant has on per capita output (y), consumption (c) and capital (k) using a(properly labelled) graph.

Solution: productivity improvements (increases) are the key to long term growth. The other twodeterminants, the savings rate and population growth are limited. Savings cannot be more than 100% andpopulation growth rates should be moderate but non-negative. However productivity improvements offeran unlimited potential for growth.

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Part d) Imagine this economy is at the steady state when a terrible pestilence kills half the population andworkforce. Assume the rate of population growth (n) is unchanged. What happens to per workerconsumption, output and capital in the short-run? What happens in the long-run?

Solution: The capital to labour ratio doubles pushing the economy to higher levels of output per capita,consumption per capita and savings per capita however the aggregate variables decrease because theaggregate production function has fewer workers. None of the curves shift - the movement is represented bya move along the output and savings curves. These higher levels of per worker ratios are unsustainable as asteady state with the savings rate and population growth held constant. This is because the amount ofsavings sZ(Ak)α is insufficient to meet the maintenance cost of capital (n+ d)K: depreciation andadditional capital for next period’s new workers. In time the economy will return to the original steadystate equilibrium and aggregate variables will grow at the same rate as the population.

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