Economics 200 Macroeconomic Theory

Economics 200Macroeconomic Theory

Midterm Examination Number 1February 19, 1996

You have 1 hour to complete this exam. Answer any four questions you wish.

1. Suppose that an increase in consumer confidence raises consumer's expecationsof future income and thus the amount that they wish to consume today. How would thisaffect investment and the real interest rate.

2. Is the US saving rate too low? By what standard? Would a rise in the USsaving rate permanently increase the rate of economic growth as measured by the annualpercentage increase in per capita GDP? What are the costs and benefits of a rise in the USsaving rate? 3. The chart below plots average annual growth rates over the period 1960 to1985 against GDP per person in 1960. What are the implications of this data for theconvergence hypothesis?

Growth and Initial GDP per Person

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 2000 4000 6000 8000 10000 12000

GDP per person in 1960

Ave

rage

ann

ual g

row

th ra

te 1

960

to 1

985

4. Consider an economy that obeys the Solow growth model (with notechnological progress) and has the aggregate production function 250 ..25 .75

(a) Show that the per capita production function is 250 .25

(b) Suppose that the economy has a saving rate of .2 and that the rates ofpopulation growth and depreciation are 1% and 4% respectively. Give the equationdescribing how changes over time.

(c) Find the steady-state amounts of capital and output per person.

5. Consider an economy whose (per-capita) production function is where and are output and capital per worker respectively and is a positive constant. Use

the “law of motion” for from the Solow model (ignoring technological progress) toshow that, in this economy, a rise in the saving rate will permanently increase the rate ofgrowth. Compare and contrast (as appropriate) this with the usual Solow model that westudied in class.

6. During the decade from January 1981 to December 1990, the average annualrate of growth of GDP was 2.3%. Over the same period the labor force grew at anaverage annual rate of 1.0% and the capital stock at an average annual rate of 2.0%. Ifcapital's share of output was an average 30% during this period, decompose the growth inGDP into the portions due to growth in total factor productivity, the capital stock and thelabor force. Be careful to state any assumptions that you make.


Solutions to Midterm Examination Number 1February 19, 1996

1. The supply of loanable funds is given by . For anygiven value of a rise in consumption for each level of disposable income reduces saving. Thusthe supply of loanable fund shifts to the left as shown below. The demand for loanable funds,

, does not change.

The new equilibrium has a higher real interest rate and lower investment.

2. Compared to the Golden Rule saving rate, which is equal to capital's share of output(about 30% for the US), the current average US saving rate of about 17.5% is too low. In theSolow model a rise in the saving rate produces a rise in the growth rate only during the transitionto the new steady state. Once that point is reached the growth rate of per capita GDP is again thesame as the rate of technological progress. Raising the saving rate would put the economy on atransition path to a steady state with higher per capita consumption than at present. The cost is alower level of per capita consumption now because a higher saving rate implies lowerconsumption until the higher savings result in a sufficiently large increase in the capital stock togive higher consumption.

3. The convergence hypothesis is the claim that, eventually, all countries will have thesame level of per capita income. It is motivated by the observation that, in the Solow growthmodel, the steady state level of income per capita is independent of the starting level. Theimplication for cross-country data on levels of per capita GDP and subsequent growth rates is thatthe two ought to be negatively related. The poor countries need to grow faster than the rich ifthey are to catch up. This is illustrated in the diagram over.

Thus, a plot of the average growth rate over some time period against level of per capita GDP atthe beginning of the period ought to display a negative relationship between these two variables.The data in the chart are inconsistent with this prediction and thus with the convergencehypothesis. If, however, we were to look at countries that are sufficiently similar such as those inthe OECD or at the states of the US we find that the data support this implication of theconvergence hypothesis.

4. (a) 250 250.25 .75 250 250 .25.25 .75 .25 .75

.25 .75

250 .25

(b) We have .2, 01 and .04. Given that the growth rate of thecapital stock is given by we have .2 .01 .04250

50 .05..75

(c) In the steady state 0 so 50 .05 0 50 .05.75 .75 .75

.05 5050 .05

.75 .251000 1000 10000 250 10000 2500. Thus, the43

steady-state values of capital and output per person are 10000 and 2500 respectively.

5. The law of motion for the per capita capital stock can be written as. In this problem so the law of motion becomes .

This model is shown in the chart below. An increase in the saving rate from to shifts the line up and permanently increases the growth rate.

The key difference between this model and the standard Solow model is the absense ofdiminishing returns to capital which drive the economy to a steady state as the rising capital stockbrings constant increments to “depreciation” but declining increments to output and thus grossinvestment. In the case at hand, growth continues unabated as the capital stock rises.

6. Suppose that the aggregate production function is . Assuming that isconstant returns to scale and that factors are paid their marginal products (competition) thegrowth rate of GDP can be decomposed as where is capital'sshare of output We measure growth in total factor productivity using the Solow residualcomputed as .Thus, of the 2.3 percent average growth in GDP, 1.0 percent was due to growth in total factorproductivity (technological progress), .6 percent (.3 2.0) was due to growth in the capitalstock, and .7 percent (.7 1.0) was due to growth in the labor force.


Midterm Examination Number 2April 8, 1996

You have one hour to complete any four of the six questions below. Do not attempt towrite on both sides of the page at the same time.

1. Consider the chart below. The "inflation differential" plotted on the horizontal axis isthe indicated country's average inflation rate over the period 1970 to 1991 less that in the UnitedStates. The "percentage change in the nominal exchange rate" is the average annual percentagechange in the country's exchange rate (measured as units of that country's currency per US$) overthe same period.

Describe the phenomenon you observe in this chart and use a theory or model to explain it. Notethat, the exchange rate as defined here is the inverse of that defined in class so that if a currencyfalls in value relative to the US$ the exchange rate rises. In other words, in class we definedexchange rates from the US point of view (with the US as the domestic country) as number ofunits of foreign currency per US$ (that is, per unit of the domestic currency). Here we aretraeting the US as the foreign country so the vertical axis shows % where is the nominalexchange rate defined as number of units of foreign currency per unit of domestic currency.

2. Why are the fiscal and trade deficits sometimes referred to as “twins”?

3. Why does the aggregate demand curve slope downward?

4. What is the “classical dichotomy"? Describe a model that does exhibit thisnot property.

5. In a newspaper article in the early 1980s, a reporter wrote

“A new recession is feared because higherPentagon spending is raising interest rates. Thosehigher rates are discouraging housing purchases andplant and equipment investment.”

Comment on the reporter's claim.

6. Suppose that US firms are given much greater access to Chinese and Japanese markets.Analyze the long-run effects on the US GDP, trade balance and real exchange rate.

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n an announcement more importantIto academics than the apprehension of theUnabomber, Professor Paul Johnson ofVassar College today revealed the answersto the second Economics 200 midterm.

Professor Johnson

rofessor Johnson revealed that thePanswers were:

. The chart shows that there is a1strong tendency for the currencies ofcountries with average inflation rates greater(smaller) than that in the US to depreciate(appreciate) relative to the US$. Treating,for example, Italy as the domestic countrywe know that the percentage changes in theexchange rates (nominal and real) and therates of inflation in Italy and the US (the“foreign” country here)% % % %where is measured in units of US$ per Lira.So, if the cross-country variation in % issmall relative to that in the other terms, wehave % % % whichcan be written as

% % % . Thispositive relationship between % and% % is observed in the

chart. Note however, that the observedrelationship is not exactly is predicted as itwould be if all of the points were to lie onthe line added to the chartbelow.

In particular, note that all but one country(Canada) lie below the line implying that formost of them % . Why this is so isone of the topics discussed in Economics346.1

. Beginning with the identities2 and

we can write. Thus

. Given that,particularly over long periods of time, and move together, much of the variation in

comes from that in So, if thereis a large fiscal deficit therewill also tend to be a large current accountdeficit . This relationship, and itsgraphic illustration during the last 15 yearsor so, has prompted some to refer to thedeficits as “twins”.

1You were not required to observe this phenomenonnor explain it to obtain full credit for this question. Ijust could not resist the opportunity to point this outand advertise one of the other classes that I teach!

Economics News April 8, 1996

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. The aggregate demand curve3slopes downward because, holding all elseconstant, a higher price level implies a lowerlevel of real balances, shifting the LM curveto the left. In the diagram below the pricelevel is greater than so the LM curveassociated with lies to the left of thatassociated with . The new short-runequilibrium point has a higher interest rate(and thus lower investment) and a lowerlevel of GDP demanded. Thus, all elseconstant, there is a negative relationshipbetween the price level and the quantity ofGDP demanded. In other words we havefound two points and ) onthe AD curve as shown. This is not to implythat a higher price level reduces GDP butonly that, all else constant, a higher pricelevel implies a reduction in the aggregatequantity of goods and services demanded.

. A model of the economy exhibits4the classical dichotomy when the realvariables (quantities and relative prices) canbe determined without reference to thenominal variables (for example, the pricelevel and the money supply). One way tobreak the dichotomy is to postulate thatconsumption depends on the level of realbalances, , and write the consumptionfunction as . The idea isthat real balances are a form of wealth andhigher wealth increases consumption at eachlevel of disposable income. Let the demandfor real balances be given by

where is the nominalinterest rate which can be written as


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, where is the expectedinflation rate. Assume that expectedinflation is given by , where is therate of growth of the money supply.Assembling all of this under the assumptionthat the demand for and supply of realbalances are equal, we can write theconsumption function as

. The supply ofloanable funds is now given by

.We discovered in problem sets 4 and 5 that,for each value of , a rise in increases thenominal interest rate, reducing the demandfor real balances and hence the quantity ofreal balances that are held. This reductionresults in lower consumption and hencehigher saving implying a rightward shift inthe supply of loanable funds curve. Thisshift in the savings function moves themarket to a new equilibrium with a lowerreal interest rate and higher saving andinvestment. The fall in the real interest rategiven a rise in the rate of growth of themoney supply violates the classicaldichotomy.

. As the reporter is concerned5about a recession, a model of the short-runbehavior of the economy is appropriate.Consider the effects of a rise in governmentspending in the IS-LM model illustratedbelow. The rise shifts the IS curve to theright by an amount equal to the rise times theautonomous spending multiplier (the lengthof the arrow). GDP does not, however,increase by this amount as that would implythat the economy was off the LM curve withthe demand for real balances exceeding thesupply. To keep the money market inequilibrium interest rates must rise as thereporter suggests and this will indeed reduceinvestment. The result is that GDP increasesfrom to rather than to . Note that

the amount of “crowding out” ( )equals the fall in investment times themultiplier. However, for the reporter's fearof a recession to be well founded, thereduction in investment would have toexceed the rise in government spending.2

2This would require a negatively sloped LM curve.To see this, draw an IS-LM diagram with anegatively sloped LM curve steeper than the IS curveand then increase government spending. Whatwould give such an LM curve?


Page 4

. The appropriate model is that of6a large open economy. The greater access toChinese and Japanese markets increasesexports for each value of the real exchangerate and so shifts the function to theright. as shown below.

Nothing in the loanable funds marketchanges so NCO is unchanged. With thenew function this means that the realexchange rate must rise. The intuition is thatthe relative price of domestic goods andservices rises as the demand for themincreases. Domestic GDP is unchangedbecause it depends on the available quantitiesof capital and labor and the level oftechnology, none of which have changed.

tudent reaction to the answers isSunknown at this time. “I hope that theseanswers are not news,” said Johnson, “butwe will have to wait until I grade the exampapers before we know for sure.”


Final ExaminationMay 13, 1996

You have two hours to complete this exam. Answer any 8 questions.

1. Consider the Cobb-Douglas production function where all symbolshave their standard meanings. Let and . (a) Show that . (b) Using the production function in (a), show that, in the Solow growth modelwith no technological progress, the steady-state level of is given by . (c) Show that the golden-rule level of in this economy is given by

. HINT: For the production function , . (d) Use the results in (b) and (c) to conclude that the golden rule saving rate isgiven by . Interpret this result.

2. While relaxing this summer you pick up a copy of the Among the letters toEconomist.the editor you notice an item criticizing a recent report on the closed economy of Ferme in whichit was claimed that investment as a fraction of GDP rose from 10% during the 1970s to 15%during the 1980s while the real interest rate rose from 2.5% to 4%. The writer of the letter statesthat this claim “is obviously absurd as everybody knows that investment falls when real interestrates rise”. What is your view?

3. Why does the aggregate demand curve slope downward?

4. Consider an economy with the Phillips curve .5 .06 . (a) What is the natural rate of unemployment for this economy? (b) Graph the short-run relationship between inflation and unemployment. (c) What unemployment rate is required to reduce inflation by 1 percentage point? (d) If Okun's Law is 2 .06 , what is the sacrifice ratio?

5. Suppose that during the summer you are retained as a consultant by the central bank ofa small open economy that places no restrictions on the flow of capital across its borders. Thecentral bank is considering a change in the exchange rate regime from its current fixed exchangerate to a freely floating exchange rate. You are told that most of the shocks in the economyoriginate in the goods market and that the citizens value stability in the level of GDP. Would youadvise for or against the change? Why?

6. Analyze the long-run effects on the US GDP, real interest rate, real exchange rate andcurrent account balance of an increase in taxes.

7. Analyze the short-run effects of a rise in the world real interest rate on the GDP, realinterest rate, real exchange rate and current account balance of a small open economy with afloating exchange rate and perfect capital mobility.

8. Most of our discussions of the IS–LM model assumed that expected inflation was zeroso the real and nominal interest rates were equal. In this question we will modify the model toallow for nonzero, but exogenous, expected inflation. One way to do this is to write theinvestment and money demand function as and respectively, where is theexpected inflation rate. The IS and LM functions become and

respectively. Your task is to use a diagram and careful explanation to indicatethe short-run effects on , , and of a rise in beginning from and a position of fullemployment, . Note that even when we continue to assume that is fixed in theshort run. HINT: Put on the vertical axis in the IS LM diagram.

9. What are the implications of the convergence hypothesis for cross-country or cross-region data on levels of per capita GDP and growth rates? What evidence is there supporting orrejecting these implications?

10. The following chart plots decade average inflation rates in the United States for the1870s through the 1980s against the corresponding money growth rates. Describe therelationship between these two variables and use a model to explain that relationship.

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Economics 200 Macroeconomic Theory