14.02 Principles of Macroeconomics

Problem Set 1 - Solutions

Posted: Wednesday, February 14, 2001

Due: Friday, February 23, 2001

Part 1:  True/False Questions: Decide whether each statement is true of false and justify your answer with a short argument. (5 points each, 35 points total)

1.  The higher the share of youth in the population, the lower the non-accelerating inflation rate of unemployment.

FALSE - As shown in class, the higher the share of youth in the population, the higher the NAIRU tends to be (see slide 8 from lecture number 2).  This makes sense since youth spend more time on average ‘voluntarily’ unemployed.

2.  The full employment unemployment rate has typically been about 4% in the post-war period.

FALSE - See, for example, slide 18 from lecture number 2.

3.  As a rule of thumb, current conditions suggest that for every point that unemployment is below the full employment level, inflation will increase by 0.5 percentage points per year.

TRUE - See slides 19 and 21 from lecture number 2.

4.  The Federal Reserve’s only priority is attaining full employment in the economy.

FALSE - The Fed’s is also concerned with controlling inflation; in fact, this is probably its primary priority.

5.  Exports can be larger than GDP.

TRUE - This is true for several small open economies such as Singapore and Hong Kong.  While their exports are larger than GDP, so are their imports.  Recall that GDP is equal to total value added: these countries tend to import intermediate goods, add some value, and export final goods.

6.  If the government cuts spending to reduce the deficit, the Fed will most likely raise interest rates to make sure the cut in spending doesn’t lead to inflation.

FALSE - A cut in government spending will tend to reduce GDP, increasing unemployment, so inflation is usually not much of a threat.  Rather, the fall in spending would tend to cause inflation to decrease.  The Fed may cut interest rates to help offset the negative effects of the cut in spending.  This is what happened earlier in the 90's.

7.  Fiscal policy is more effective when consumers’ marginal propensity to consume is higher.

TRUE - The multiplier is proportional to the MPC, and the effectiveness of fiscal policy depends on the size of the multiplier.  With a higher multiplier, any given change in G or T will have a larger effect on equilibrium income, Y.

Part 2: Unemployment vs. Changes in Inflation (10 points each, 30 points total)

1.  Assume that the rate of change of wages, RW, follows the following equation:

RW = RP-1 + 2 - 1.5(U-Uvol)

where RP-1 is the past year’s rate of change in prices, U, is total unemployment, and Uvol is voluntary unemployment (all of the variables are measured in percentage points).

Why does the rate of change of wages depend on changes in the price level?  Why is there a constant in the equation?  Explain why the rate of change in wages depends on the difference between the unemployment rate and the voluntary unemployment rate.

The rate of change in wages depends on changes in the price level because workers care about their real income, i.e. how much they can buy with the money they receive.  (Also, note that the rate of wage change depends on last period’s change in prices because wages are set in advance, and we assumed that they use past inflation as an estimate of what inflation will be in the future.  You did not need to mention this point to get full credit for your answer.)  The constant can reflect many things not covered by the other terms, in particular, productivity growth.  We expect that wage growth depends on the amount of involuntary unemployment because it affects workers’ bargaining power.  Intuitively, if there are a lot of people looking for work and unable to find it, you’re less likely to demand a big raise.

2.  Assume that prices are set by firms according to the following equation:

P = K(W/A)

RK = - 0.5(U-Uvol)

RA = 2

where K is the markup.

What is W/A?  Is it a realistic assumption to assume that prices are a markup over only labor costs? Why or why not? Is the markup pro-cyclical or counter-cyclical?  Why might this be the case?

W/A is the unit labor cost: it is the cost of the labor required to produce a unit of output.  It is realistic to assume that prices are a markup over labor costs only if the cost of other inputs is a constant fraction of all costs.  If the cost of some input rises suddenly relative to the wage, for example, firms might raise the markup to cover the additional costs.  The markup is pro-cyclical: it is bigger when the economy is booming and there is little involuntary unemployment.  This could reflect the fact that when demand is high, firms are able to charge higher prices for their goods.

3.  Solve for the change in price inflation as a function of involuntary unemployment.  Is inflation in this economy more or less sensitive to excess demand than the U.S.?  If RP=12 and U=Uvol=5, what will the total unemployment rate have to be in the next year to get inflation down to 4%?  What unemployment rate is needed to achieve the inflation target in two years instead of one?  Do you think it would be better to reduce inflation in one year or two?

Recalling that P=K(W/A) implies RP = RK + RW - RA and substituting yields RP - RP-1 = - 2(U-Uvol).  This economy is more sensitive to excess demand than the U.S., where the coefficient is about one-half instead of two (i.e., inflation changes by more for a given change in involuntary unemployment).  Setting RP=4, RP-1=12, and Uvol=5 we find that U=9: To get inflation to fall by eight percentage points, we’ll need to have 4 percentage points of involuntary unemployment, so the total unemployment rate will have to be 9%.  Or, we could have two percentage points of involuntary unemployment for two years, so the total unemployment rate would be 7% for two years.  It might be better to reduce inflation quickly because on average we might expect that individuals who lose their job will be unemployed for a shorter period of time.  Longer spells of unemployment are bad if they cause workers to become discouraged and stop looking for work, or if workers loose some of their skills.  It might be better to reduce inflation more slowly, on the other hand, because too large a negative shock on the economy might cause additional problems like bank failures.  (Note that either answer is OK, as long as you provide an explanation.)

Part 3.  Fiscal Policy (Parts 1, 2, and 3, 5 points each; parts 4 and 5, 10 points each; 35 points total)

Consider the following model of the economy:

C = 50 + 0.6(Y-T)

I = 10 +0.1Y - i

G = 100

X = M = 0

where C is consumption, Y is income, T is taxes, I is investment, i is the interest rate (measured in percentage points, i.e. 5 instead of 0.05), G is government spending, X is exports and M is imports.

1.  State the equilibrium condition for GNP (national income) and give a brief explanation of what it means.  Solve for national income as a function of the unknown variables, i and T.

The equilibrium condition is Y=C+I+G or Y=C+I+G+X-M.  It is an accounting identity: income is equal to total spending on domestic production (every purchase is a sale).  This is the sum of spending by consumers, investment by firms, and government spending.  If the economy is open to trade, we have to subtract off the spending on foreign goods and services and add the amount foreigners spend on domestic goods and services.  Substituting the values given and solving for Y yields Y=(10/3)(160-.6T-i).

2.  Now assume that the government budget is balanced, and write income as a function of the interest rate.  Plot this curve in i-Y space.  State and interpret the slope of this curve.

If the budget is balanced, G=T so T=100.  This gives us Y=(10/3)(100-i).  The graph of this curve is a downward sloping line, with Y on the horizontal axis.  The slope is -0.3 and tells us that for every percentage point that interest rates increase, equilibrium income decreases by 3 1/3.  (Note that investment falls by one unit for each point increase in interest rates and the multiplier is 3 1/3.)

3.  Assume that i=10.  What is the value of autonomous spending? What is the value of the multiplier?  Interpret the multiplier.

If i=10, then autonomous spending, 100-i, is 90, the multiplier is 10/3, and equilibrium income is 90 times 10/3 or 300.  The multiplier tells us the change in equilibrium income for a one-unit change in autonomous spending.

4.  The government decides to increase spending by 10.  If it doesn’t raise taxes, what will the new values of autonomous spending, the multiplier, and equilibrium income be?  Give a brief explanation of why income changed by as much or as little as it did.  If the government raises taxes at the same time to maintain a balanced budget, what will the new values of autonomous spending, the multiplier, and equilibrium income be?  Give a brief explanation of why income changed by as much or as little as it did.

When G increases to 110, autonomous spending increases by 10 to 100, the multiplier doesn’t change, and thus equilibrium income increases by 10 times 10/3, or 33 1/3.  Income changed by more than the amount of the change in government spending because of the feedback effects in the economy: increased government spending led to increased income.  Increased income caused consumers to spend more and firms to invest more.  This increase in consumption and investment, further increasing income,  etc.  If T also increases to 110, then autonomous spending increases by 10-.6(10)=4, the multiplier is unchanged, and thus equilibrium output increases by 4 times 10/3 or about 13.3.  You might have expected that equilibrium output would be unchanged since this is just a transfer of ten dollars from consumers to the government, but while consumers spend 60 cents of every dollar they receive, the government spends all of the money it receives in our model (100 cents on the dollar) so there is an increase in equilibrium income.

5.  Now, instead of assuming that the government collects a fixed amount of taxes, assume that it collects a fixed percentage of national income: T=tY.  Assuming the tax rate, t, is one-third, solve for equilibrium income, autonomous spending, and the multiplier.  Explain any differences with your answers to part 2.  Is the government budget balanced?  What happens now if the government increases spending by 10?  State and briefly explain the changes in equilibrium income and the government budget deficit.

Now C=50+0.6(Y-1/3Y)=50+0.4Y, so C+I+G=160+.5Y-i, and thus Y=2(160-i).  With i=10, autonomous spending is 150, the multiplier is 2 and equilibrium income is 300, as before.  (Also note that tY=100, the same starting point as before.)  The multiplier is lower: before when a consumer received an extra dollar of income, she spent 60 cents of it.  Now, when she receives an extra dollar of income she gives a third of it to the government, so her disposable income only rises by 66 2/3 cents.  She then spends 60% of that, which is 40 cents.  For an extra dollar of income, consumption spending is only increased by 40 cents instead of 60 cents and this means a smaller feedback effect.  The government budget is balanced.  If the government increases spending by 10, equilibrium income rises by 20.  This means 20/3 or 6 2/3 more is being collected in taxes, so the government is now running a deficit.