Economics 12 – Problem Set 3By Gerardo Zampaglione

Problem 11. Suppose that an economy can be described by the following three equations.

a. What is the natural rate of unemployment for this economy?The natural rate of unemployment for this economy is 5%, because the Phillips curve = π t−π t−1=α(ut−un), and because un in the above equation = 5%, the natural rate of unemployment in this economy is 5%.

b. Suppose that the unemployment rate is equal to the natural rate and that the inflation rate is 8%. What is the growth rate of output? What is the growth rate of the money supply?If inflation is constant, then according to the Phillips curve this implies that unemployment = the natural rate of unemployment and that the economy is in the medium run. Okun’s Law thus yields that gyt=g y; therefore, in the medium run, output

must grow at its normal rate of growth, gy. In the above scenario, the growth rate of

output is therefore 3%. The growth rate of the money supply, gm, is equal to gm=π+g y. Therefore, the growth rate of the money supply must equal 11%.

c. Suppose that conditions are as in (b), when, in year t, the authorities use monetary policy to reduce the inflation rate to 4% in year t and keep it there. Given this inflation rate and using the Phillips curve, what must happen to the unemployment rate in years t, t + 1, t + 2, and so on? Given the unemployment rate and using Okun’s law, what must happen to the rate of growth of output in years t, t + 1, t + 2, and so on? Given the rate of growth of output and using the Aggregate Demand equation, what must be the rate of nominal money growth in years t , t + 1, t + 2, and so on?In the short run, the following table shows what would happen to the unemployment rate, output growth rate, and nominal money growth in years t, t + 1, and t + 2.

Statistic Symbol T – 1 T T + 1 T + 2Output Growth Rate gyt 3% -7% 3% 3%Unemployment Rate ut 5% 9% 5% 5%Inflation π t 8% 4% 4% 4%Nominal money growth

gm 11% -3% 7% 7%

In year t – 1 (the original year), all the values are as described in problem: unemployment is at its natural rate of 5%, the inflation to be remedied is at 8%, output growth is at its natural rate of 3%, and therefore nominal money growth is at 11%. In year t, the central bank reduces inflation to 4%, triggering unemployment to rise by 4%

(as α=1) to 9%, growth to fall by 10% to -7%, and nominal money growth to fall by 8% to -3%. The next year (t + 1), unemployment returns to its natural rate, because in the medium run, “the unemployment rate returns to the natural rate.” Likewise, the output growth rate returns to its normal rate of 3% in t + 1, and the nominal money growth rate rises to a value that is permanently lower than before. For both t + 1 and t + 2, nominal money growth rate and inflation are both permanently lower than before, in t – 1, meaning that a temporary increase in unemployment buys a permanent decrease in both inflation and nominal money growth rate.

Problem 22. Suppose the central bank wishes to reduce inflation by 10%.

a. Calculate the point years of excess unemployment needed to reduce inflation by 10% for each of the following values of α ; 1.5, 1.4, 1.15, and 1.0.

Phillips Curve: π t−π t−1=α(ut−un). Sacrifice ratio = 1α×10

α Point-Years of Excess Unemployment

1.0 101.15 8.6961.4 7.1431.5 6.667

b. What happens to the point years of excess unemployment as α decreases in size. Explain.As α decreases in size, the point-years of excess unemployment increases in size. This is because the sacrifice ratio, or the number of point-years of excess unemployment

needed to achieve a decrease in inflation of 1%, is equal to 1α

. As α decreases, then the

ratio approaches 1, which means that every percentage-point decrease in inflation will yield a percentage point more in unemployment. Therefore, the point-years of excess unemployment will increase as α decreases.

c. Given the goal of reducing inflation by 10%, can the central bank influence the number of point years of excess unemployment calculated in (a)? Explain.The central bank cannot influence the number of point years of excess unemployment as calculated in part (a), because exogenous and static α controls the number of point years of excess unemployment.

d. Can the central bank choose the distribution of excess unemployment over time? If so, give three examples of the choices available to the central bank.The central bank can choose the distribution of excess unemployment over time, but it cannot change the total number of point-years of excess unemployment. Therefore, it can spread out its inflation-reduction target over a number of years, and hence minimize the damage to growth over time. For example (1), if the central bank decides

to decrease the 10% inflation over a 10-year period, with α=1, the economy will suffer from unemployment equal to 1% more than the natural rate of unemployment (un+1) for 10 years, which would, according to Okun’s Law, decrease the rate of output growth

by ½% each year: ut−ut−1=−0.4 (g yt−0.03 )→ 0.01−0.4

+0.03=−0.005. (2) If the

central bank decides to decrease the 10% inflation over a 5-year period, with α=1, the economy will suffer from unemployment equal to 2% more than the natural rate of unemployment (un+2) for 5 years, which would, according to Okun’s Law, decrease the rate of output growth by 2% each year:

ut−ut−1=−0.4 (g yt−0.03 )→ 0.02−0.4

+0.03=−0.02. (3) If the central bank decides to

decrease the 10% inflation over a one year period, with α=1, the economy will suffer from unemployment equal to 10% more than the natural rate of unemployment (un+10) for 1 year, which would, according to Okun’s Law, decrease the rate of output growth

by 22% for that year: ut−ut−1=−0.4 (g yt−0.03 )→ 0.1−0.4

+0.03=−0.22, a

catastrophic scenario for any central bank.

Problem 33. Assume πe

t = πt-1 and that the Phillips curve for the United States is given by the following: πt - πt-1 = -1.5 ( ut – un ). Suppose ut = 0.07, un = 0.06, and πt-1 = 0.08.

a. Calculate πt . Is πt greater than, less than or equal to πt-1?π t=−1.5 (0.07−0.06 )+0.08=0.065→6.5%. Therefore, π t<π t−1.

b. Calculate πt for each of the following values of ut : 0.08, 0.09, 0.10.

ut π t

0.08 π t=−1.5 (0.08−0.06 )+0.08=0.05→5%

0.09 π t=−1.5 (0.09−0.06 )+0.08=0.035→3.5%

0.10 π t=−1.5 (0.1−0.06 )+0.08=0.02→2%

c. What happens to the change in inflation in t for each 1% increase in ut?π t decreases linearly by α (or 1.5%) for every 1% increase in unemployment rate above the natural rate of unemployment.

d. As ut increases, what happens to inflation in period t?Like above, as ut increases, the inflation in period t, π t, decreases linearly by α (or 1.5%).

Problem 44. Assume the typical consumer in the United States and in Mexico buys only two types of

goods: (1) food; and (2) durable goods. Use the information provided below to answer the following questions:

Price of food Quantity of food consumed

Price of durables Quantity of durables consumed

United States $2 4000 $4 8000Mexico 2 pesos 2000 20 pesos 1000

a. Calculate U.S. consumption in dollars ($).$2×4000 ( food→US )+$ 4×8000 (durables→US )=$ 40,000

b. Calculate Mexican consumption in pesos.MXN 2×2000 ( food→MX )+MXN 20×1000 (durables→MX )=MXN 24,000

c. Assume the exchange rate is 0.10 ( i.e., $0.10 per peso). Using the exchange rate method, calculate Mexican consumption per capita in dollars. What is the relative consumption per capita in Mexico compared to that in the United States?

MXN 24,000×$0.10MXN 1

=$2 ,400. Using the exchange rate method, the relative

consumption per capita in Mexico compared to that of the United States is equal to 240040000

=0.06, or 6% of America’s consumption per capita.

d. Use the purchasing power parity method to calculate Mexican per capita consumption in dollars. Using this method, what is the relative consumption per capita in Mexico compared to that in the United States?$2×2000 ( food→MX )+$4×1000 (durables→MX )=$8,000. Using purchasing power parity, the relative consumption per capita in Mexico compared to that in the

United States is equal to 800040000

=0.2, or 20% of America’s consumption per capita.

e. To what extent does the relative standard of living (or, in this case, consumption) between these two countries depend on the method used to obtain measures of Mexican consumption?Using an exchange rate calculation of Mexican consumption, Mexico’s per capita consumption equals 6% of America’s per capita consumption, whereas using purchasing power parity, Mexico’s per capita consumption equals 20% of America’s per capita consumption. Purchasing power parity is a much more accurate way of measuring the per capita consumption (or relative standard of living) between countries because it measures the consumption (or production) of identical consumption baskets, giving them uniform dollar amounts, rather than relying on fickle exchange rates, which, as the example of China shows, can often be misleading. Hence, using the two methods can have significant effects on comparisons of relative consumption and relative standards of living.

Problem 55. Use the graph provided below to answer the following questions:

a. What do the distance AB, AC, AD, CB, and CD represent?

The distance…

Represents…

AB Output per workerAC Investment per worker/private savings per workerAD Depreciation per workerCB The level of consumptionCD The amount by which the capital stock increases

b. What will happen to K/N and Y/N over time? Is the economy in steady state at K0N

? What will be the final levels of K/N and Y/N? Explain. Note that I combined all three of the sections into this because they can be condensed easily.Over time, K/N will converge to the “steady state” point at the intersection of the

investment-per-worker line (sf (K tN

)) and the depreciation-per-worker line (δK tN

),

when output per worker and capital per worker are constant and thus investment =

depreciation. Starting at point K0N

, the capital per worker will increase until that point;

to the right of that point, the capital per worker will decrease until it reaches equilibrium. Likewise, Y/N will converge to the point along the output per worker line (

f (K t

N)) which has an X-axis value of the steady state of K/N: the X-axis value at which

investment per worker = output per worker. See below graph for specific plotting of the steady state equilibriums of Y/N and K/N.

Steady State Equilibrium (of Y/N)

Steady State Equilibrium (of K/N)