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Department of Economics Bo ğ aziçi University EC 205 Macroeconomics I Fall 2018 Problem Set 1 Solutions Q1 Abby consumes only apples. In year 1, red apples cost $1 each, green apples cost $2 each, and Abby buys 10 red apples. In year 2, red apples cost $2, green apples cost $1, and Abby buys 10 green apples. a. Compute a consumer price index for apples for each year. Assume that year 1 is the base year in which the consumer basket is fixed. How does your index change from year 1 to year 2? b. Compute Abby’s nominal spending on apples in each year. How does it change from year 1 to year 2? c. Using year 1 as the base year, compute Abby’s real spending on apples in each year. How does it change from year 1 to year 2? d. Defining the implicit price deflator as nominal spending divided by real spending, compute the deflator for each year. How does the deflator change from year 1 to year 2? e. Suppose that Abby is equally happy eating red or green apples. How much has the true cost of living increased for Abby? Compare this answer to your answers to parts (a) and (d). What does this example tell you about Laspeyres and Paasche price indexes?
A
Thus, the implicit price deflator suggests that prices have fallen by half. The reason for this is that the deflator estimates how much Abby values her apples using prices prevailing in year 1. From this
If we take CPI in 2000 as 100 (since 2000 is the base year), then CPI in 2010 must be 1.6*100=160. b. There is no clear-cut answer to this question. Ideally, one wants a measure of the price level that accurately captures the cost of living. As a good becomes relatively more expensive, people buy less of it and more of other goods. In this example, consumers bought less bread and more cars. An index with fixed weights, such as the CPI, overestimates the change in the cost of living because it does not take into account that people can substitute less expensive goods for the ones that become more expensive. On the other hand, an index with changing weights, such as the GDP deflator, underestimates the change in the cost of living because it does not take into account that these induced substitutions make people less well off. Q3. What is a market-clearing model? When is it appropriate to assume that markets clear? A3. A market-clearing model is one in which prices adjust to equilibrate supply and demand. Market-clearing models are useful in situations where prices are flexible. Yet in many situations, flexible prices may not be a realistic assumption. For example, labor contracts often set wages for up to three years. Or, firms such as magazine publishers change their prices only
Q5. Use the neoclassical theory of distribution to predict the impact on the real wage and the real rental price of capital of each of the following events: a. A wave of immigration increases the labor force. b. An earthquake destroys some of the capital stock. c. A technological advance improves the production function. A5. a. According to the neoclassical theory of distribution, the real wage equals the marginal product of labor. Because of diminishing returns to labor, an increase in the labor force causes the marginal product of labor to fall. Hence, the real wage falls. b. The real rental price equals the marginal product of capital. If an earthquake destroys some of the capital stock (yet miraculously does not kill anyone and lower the labor force), the marginal product of capital rises and, hence, the real rental price rises. c. If a technological advance improves the production function, this is likely to increase the marginal products of both capital and labor. Hence, the real wage and the real rental price both increase. Q6. Suppose that an economy’s production function is Cobb–Douglas with parameter α =0.3. a. What fractions of income do capital and labor receive?
b. Suppose that immigration increases the labor force by 10 percent. What happens to total output (in percent)? The rental price of capital? The real wage? c. Suppose that a gift of capital from abroad raises the capital stock by 10 percent. What happens to total output (in percent)? The rental price of capital? The real wage? d. Suppose that a technological advance raises the value of the parameter A by 10 percent. What happens to total output (in percent)? The rental price of capital? The real wage? A6.
MPL=dY/dL=3/4K1/4L-1/ c. dMPK/dK=-3/16K-7/4^ L3/4<0.Hence, MPK is diminishing. dMPL/dL=-3/16K1/4L-5/4<0. Hence, MPL is diminishing Q8- Consider an economy described by the following equations: Y=C+I+G Y= G= T= C=250+0.75(Y-T) I=1000-5000r a. In this economy, compute private saving, public saving and national saving. b. Find the equilibrium interest rate. c. Now suppose that G rises to 1250. Compute private saving, public saving and national saving. d. Find the new equilibrium interest rate. What happens to investment? What is the A8 a. Private saving is the amount of disposable income, Y – T , that is not consumed: S private^ = Y – T – C = 5,000 – 1,000 – (250 + 0.75(5,000 – 1,000)) = 750. Public saving is the amount of taxes the government has left over after it makes its purchases: S public^ = T – G = 1,000 – 1,
Total saving is the sum of private saving and public saving: S = S private^ + S public = 750 + 0 = 750. b. The equilibrium interest rate is the value of r that clears the market for loanable funds. We already know that national saving is 750, so we just need to set it equal to investment: S = I 750 = 1,000 – 5000 r Solving this equation for r , we find: r = 5%. c. When the government increases its spending, private saving remains the same as before (notice that G does not appear in the S private^ above) while government saving decreases. Putting the new G into the equations above: S private= 750 S public^ = T – G = 1,000 – 1, = –250. Thus, S = S private^ + S public = 750 + (–250)
b. Private Saving —The increase in taxes decreases disposable income, Y
- T , by $100 billion. Since the marginal propensity to consume ( MPC ) is 0.6, consumption falls by 0.6 × $100 billion, or $60 billion. Hence, ΔPrivate Saving = – $100 b – 0.6 ( – $100 b ) = – $40 b. Private saving falls $40 billion. c. National Saving —Because national saving is the sum of private and public saving, we can conclude that the $100 billion tax increase leads to a $60 billion increase in national saving. Another way to see this is by using the third equation for national saving expressed above, that national saving equals Y – C ( Y – T ) – G. The $100 billion tax increase reduces disposable income and causes consumption to fall by $60 billion. Since neither G nor Y changes, national saving thus rises by $60 billion. Q10- Suppose in 2006, the CPI equals 100. That year, John borrows a nominal value of $1,000 from the bank. Suppose the expected rate of inflation is 3%. Both parties agree the loan will be repaid in one year along with a real interest rate of 2%. a. What is the CPI in 2007, the year in which the loan is repaid –if expectations are realized? b. What is the nominal payment John must make to the bank in 2007? c. What is the real payment John must make to the bank in 2007 (take 2006 as the base year)? d. Suppose actual inflation in 2007 unexpectedly realized 7%. What is the nominal payment that John should make to the bank in 2007 if the inflation were known to be 7%? e. Who, if anyone, benefits from the unexpected inflation? f. What would be the macroeconomic effects of such unexpected inflation? Is this socially costly? If so, why?
A10- CPI={Q1xP1(current)+Q2xP2(current)+...}/{Q1xP1(base) +Q2xP2(base)+...} CPI(2006)=100 Eπ=3% r=2% 1000$ a. We should check price level in 2007 by using expected inflation rate. CPI x (1+Eπ) = 100 x (1+0.03) = 103 (CPI in 2007) b. i = r + Eπ = 2% + 3% = 5% , i = 5% 1000 x (1+0.05) = $1050 (nominally) c. 1000 x 1.02 = $ d. i = 2% + 7% = 9% 1000 x (1.09) = $ e. Borrower, whose real cost of borrowing is at the rate of 0% is better off, and lender is worse off. f. Unexpected inflation induces arbitrary transfer of wealth between borrowers and lenders, and such uncertainty is not desirable. Q11 -Consider the long-run classical theory studied in Chapter 3. Suppose that the production technology in the economy is given by the functional form:
Y K 0.5^ L 0.5 KL
. Long-run capital in the economy is fixed at K^ ^100 and labor is fixed at L^ ^25. The consumption in the economy follows the following functional form: C^ ^10 ^ 0.75(^ Y^ ^ T ) where T^ is the lump-sum tax in the economy, given by T^ ^10. Further, government expenditure is given by G (^5) , and investment demand function is given by I 10 100 r where r (^) is the real interest rate. a. Determine which of K L Y C I G T ,^ ,^ ,^ , ,^ ,^^ and r^ are exogenous and which of them are endogenous. b. What is the level of total output in this economy? c. What is the level of consumption, investment and real interest rate? d. What is the value of private saving, public saving and total saving?
Q12 - Assume an economy has a production function: Y=AF (K,L ), where A is productivity, K is capital, L is labor. The consumption function of the economy is given by C=MPC*(Y-T ), where Y is the output, T is tax and 0<MPC<1. Also assume this is a close economy, which means the demand of the economy is composed by consumption, investment, and government spending: Y=C+I+G , where investment is a function of interest rate: I=I(r). What is the effect of an increase in A on equilibrium amount of saving, investment, and interest rate? Use graphs to show your results. A12- See next page
