1 Chapter 1 Extra Questions and Answers

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1 1 Chapter 1 Extra Questions and s Question 1. What does GDP stand for? Write down and then define (that is, explain) the four major expenditure components of GDP. GDP stands for Gross Domestic Product. The four major expenditure components of GDP are Consumption, Investment, Government Spending, and Net Exports. (Private) Consumption is anything that is produced today that gives us utility today, but no utility at any point in the future. An example is electricity or a message. (Private) Investment is anything we produce that does not give us utility today, but allows us to produce goods and services in the future. Government spending are consumption and investment goods and services that are purchased by the government. Net Exports is the value of our exports less the value of our imports. If negative, it implies a trade deficit. Question 2. Please fill in the empty cells of the following table. Make sure to use exactly the procedures discussed in the book. Show work as best as possible. Apples Bananas Nominal Real GDP Annual Growth Rates in % Year Quantity Price Quantity Price GDP (constant $2005) Real GDP Inflation $ $ $ $ $ $1.12 NA NA Expenditure shares for 2005 are used for growth-rate computations and expenditure shares for 2006 are used for for growth-rate computations: Apples Bananas Nominal Real GDP Annual Growth Rates in % Year Quantity Price Quantity Price GDP (constant $2005) Real GDP Inflation $ $1.00 $25.00 $ % 1.80% $ $1.05 $28.52 $ % 2.63% $ $1.12 $32.44 $ NA NA Other answers for inflation: If inflation is computed as growth of nominal GDP divided by growth in real GDP, the inflation rate will be 1.86% in 2005 and 2.69% in 2006 (I view this as correct as well). 1

2 If the inflation rate is computed as the growth rate of nominal GDP less the growth rate of real GDP, which is not really correct, the results will be 2.08% in 2005 and 2.98% in Question 3. Households in Minneapolis pick apples a and bananas b from trees each period. For 2000 and 2001, data on apples picked a, bananas picked b, and the price of apples p a and the price of bananas p b in Minneapolis is Year a p a b p b $ $ $ $2.566 a. What is nominal GDP in Minneapolis in 2000 and 2001? b. What is the growth rate of real GDP in Minneapolis from 2000 to 2001? c. What is the inflation rate in Minneapolis from 2000 to 2001? d. Suppose that households in Minneapolis have preferences for apples and bananas of What do you think φ is? φln (a) + (1.0 φ) ln (b) a. Nominal GDP in 2000 is 25 $1+30 $2.5 = $100. Nominal GDP in 2001 is 26 $ $2.566 = $ b. The growth rate of real GDP is the measured expenditure share on apples ( φ) times growth in apples plus the measured expenditure share on bananas (1 φ) times growth in bananas less 1.0: φ a ( a φ ) b b ( ) ( 2000 ) so the growth rate of real GDP is 3.5 percent c. The inflation rate is the measured expenditure share on apples ( φ) times growth in apple prices plus the measured expenditure share on bananas (1 φ) times growth in banana prices 2

3 less 1.0: φ p a, p a,2000 ( ( 1 φ ) ) p b, p b,2000 ( ) so the inflation rate is 2.48 percent d. When household preferences are of the form φln (a) + (1.0 φ) ln (b) Then households optimally spend φ fraction of their income on apples and (1.0 φ) fraction of their income on bananas. If GDP (the value of aggregate output) is equal to the value of aggregate income, and if people are optimizing, then I would expect φ = φ = Question 4. You are given the following data: Nominal Expenditure Apple Banana Year GDP Share, Apples Prices Prices 2001 $ $5.00 $ $ $5.20 $ $ $5.41 $10.61 Copy the following table to your blue book and complete the table. Use four decimal points for all prices and quantities. Show all work. expenditure share method). (HINT: Compute growth rates of inflation and real GDP using the Apples Bananas Growth Rates Level of Real GDP Year Price Quan. Price Quan. Real GDP Inflation const. $2002 const. $ NA NA Apples Bananas Growth Rates Level of Real GDP Year Price Quan. Price Quan. Real GDP Inflation const. $2002 const. $ $ $ NA NA $ $ % 3.25% $ $ % 3.28%

4 Question 5. Citizens of the country of Canada produce apples and bananas. You have been told i. The expenditure share on apples is 20 percent in 2005, 22 percent in 2006, and 23 percent in ii. Nominal GDP in Canada is $500 in the year 2005, $525 in the year 2006, and $560 in the year You have also been given the following data on the price of one pound of apples p a and bananas p b in Canda: p a p b 2005 $ $ $ $ $ $ Determine (a) the inflation rate and (b) the growth rate of real GDP in Canada between 2005 and 2006 and again between 2006 and Make sure to show work and use 4 decimals everywhere! The inflation rate in year t can be computed as the sum, across all goods, of the expenditure share in year t times the growth rate of prices between t and t : 0.20 (2.040/2.000) (10.200/10.000) 1.0 = 2.000% 2006 : 0.22 (2.083/2.040) (10.553/10.200) 1.0 = 3.163% The growth rate of real GDP rate between in year t can be computed as the sum, across all goods, of the expenditure share in year t times the growth rate of quantities between t and t + 1. First, you have to use the information given to get expenditures and then quantities: Nominal Apples Bananas Year GDP Share Expenditures Price Quantity Share Expenditures Price Quantity (1) (2) (3)=(1)*(2) (4) (5)=(3)/(4) (6) (7)=(1)*(6) (8) (9)=(7)/(8) 2005 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Now compute the growth rate of real GDP: 2005 : 0.20 (56.618/50.000) (40.147/40.000) 1.0 = 2.941% 2006 : 0.22 (61.834/56.618) (40.860/40.147) 1.0 = 3.413% An equally acceptable approach (for the purposes of this exam) to compute the growth rate of real GDP would be to divide the growth rate of nominal GDP (5% in 2005 and 6.67% in 2006) by the 4

5 inflation rate. 2005: / = 2.941% 2006: / = 3.396% Question 6. In Fredonia, apples and bananas are produced, and Fredonians have the following preferences for apples and bananas: 0.4ln (a) + 0.6ln (b). Between 2007 and 2008 the price of apples increased by 2 percent but the overall price level increased by 5 percent. What happened to the price of bananas in Fredonia between 2007 and 2008? (NOTE: assume that the measured expenditure share on apples is equal to the optimal expenditure share). We know from the utility function that the optimal expenditure share on apples is 40 percent and the optimal expenditure share on bananas is 60 percent. This means the overall inflation rate will be computed as 0.05 = 0.40 (1.02) x = 0.60 x 1.07 = x The inflation rate for bananas in Fredonia between 2007 and 2008 was 7 percent. Question 7. The country of Fruitcake produces apples and bananas. Fruitcakes have timeinvariant preferences for pounds of apples a and pounds of bananas b of 0.2ln (a) + 0.8ln (b) a. You have been told that nominal GDP in Fruitcake is $100 in the year 2005, $105 in the year 2006, and $110 in the year Assuming households maximize utility, and all apples and bananas are consumed in each year, what are nominal expenditures on apples in dollars in 2005, 2006, and 2007? Make sure to justify your answer. b. You have given the following data on the price of one pound of apples p a and bananas p b in Fruitcake: p a p b 2005 $ $ $ $ $ $

6 Given the answer to part a., determine the inflation rate and the growth rate of real GDP in Fruitcake between 2005 and 2006 and again between 2006 and Make sure to show work. a. We know from the utility function that the optimal expenditure share on apples is 20 percent and the optimal expenditure share on bananas is 80 percent. This enables us to compute expenditures on apples and bananas given data on nominal GDP. Expenditures On: Nominal GDP Apples Bananas 2005 $100 $20.00 $ $105 $21.00 $ $110 $22.00 $88.00 b. Since we know expenditures on apples and bananas and prices of apples and bananas, we can compute apples and bananas quantities. With this, and with data on prices, we can compute the growth rate of real GDP and the inflation rate using the expenditure share method (answer below). Apples: Bananas: Growth Rates: Expen. Prices Quan. Expen. Prices Quan. Real GDP Prices 2005 $20.00 $ $80.00 $ % 2.20% 2006 $21.00 $ $84.00 $ % 3.40% 2007 $22.00 $ $88.00 $ NA NA Question 8. NIPA Table shows income accruing to various sources in the U.S. A simplified version of that table for 2009 looks as follows: Source Amount Unambiguous Capital Income $32 Unambiguous Labor Income $63 Proprietor s Income $5 Total Income $100 Using the method described in the book and the class, calculate capital s share of income. Show work. 6

7 Propriteor s income is ambiguous income because proprietors use both capital and labor to produce output. So we use the formula from the book: α = α = Unambiguous Capital Income Total Income - Ambiguous Income = Question 9. A German friend named Dirk gives you a table of income accruing to various sources that he has put together for Germany in By Dirk s reckoning, German national income in 2007 can be attributed to various sources as Source Amount Capital Income $25 Labor Income $60 Ambiguous Income $15 Total Income $100 Calculate capital s share of income in Germany implied by Dirk s data. Use two decimals and show work. We use the formula in the book to compute capital s share as α = Capital Income Total Income - Ambiguous Income α = $25 $100 $15 α = Question 10. In New Zealand, bananas and cherries are produced. Between 1920 and 1921, the expenditure share on bananas was 25 percent and the price of bananas increased by 30 percent. The overall price level between 1920 and 1921 increased only by 10 percent, however. What happened to the price of cherries in New Zealand between 1920 and 1921? Make sure to show work. The inflation rate in year t can be computed as the sum, across all goods, of the expenditure share in year t times the growth rate of prices between t and t + 1. For New Zealand, given a 30 percent increase in the price of bananas but only a 10 percent increase in the overall price level, this means: 0.25 (1.30) x 1 =

8 Solving this yields x = Meaning the price of cherries increased by 3.333% between 1920 and Question 11. In Fredonia, apples and bananas are produced. Between 1920 and 1921, the expenditure share on apples was 20 percent and the overall price of apples increased by 50 percent. The overall price level between 1920 and 1921 increased only by 5 percent, however. What happened to the price of bananas in Fredonia between 1920 and 1921? Denote growth in the price of bananas between 1920 and 1921 as π b. The overall increase in the price level, 5 percent, can be written as: 5% = % π b. Thus π b = 5%/0.8 = 6.25%. 8

9 2 Chapter 2 Extra Questions and s Question 1. Explain why the following thinking an example of Marx s labor theory of value as written on the web site does not follow from our model of production: A worker in a factory is given $30 worth of material, and after working 3 hours producing a good, and using $10 worth of fuel to run a machine, he creates a product which is sold for $100. According the Marx, the labor and only the labor of the worker increased the value of the natural materials to $100. The worker is thus justly entitled to a $60 payment, or $20 per hour. In addition to the materials and fuel (which are themselves created by technology, labor and capital), the firm combines technology with labor (the worker) and capital (the factory) to make output. Labor and capital are both scarce, and both require payments. The factory owner does not build a factory unless he/she gets a return if the return to building a factory were truly zero, then the factory owner would choose to consume rather than build the factory. Question 2. Write down a Cobb-Douglas production function. Derive the marginal product of cpaital and the marginal product of labor. Show that the marginal product of capital is declining in the amount of capital, and that the marginal product of labor is declining in the amount of labor. The Cobb-Douglas production function is Y = zk α L 1 α. The Marginal Product of Capital is the derivative of this function with respect to K, ( ) L 1 α MPK = αzk α 1 L 1 α = αz K Given 0 < α < 1, this function is declining in K holding z and L fixed. The Marginal Product of Labor is the derivative of this function with respect to L, ( ) K α MPL = (1 α) zk α L α = (1 α) z L Given 0 < α < 1, this function is declining in L holding z and K fixed. Question 3. Define the average product of labor, or productivity. Why do wages rise with productivity? 9

10 The average product of labor (productivity) is Y/L. Wages rise with productivity because when firms maximize profits, they employ labor up to the point where the marginal product of labor is equal to the wage. (1 α) zk α L α }{{} Marginal Product of Labor = w }{{} Wage The marginal product of labor is proportional to the average product of labor: (1 α) zk α L α = (1 α) zk α L α L L = (1 α) zkα L 1 α = (1 α) Y L L Thus wages rise with productivity because of profit-maximization by firms and competitive labor markets. Question 4. Assume the economy produces output according to the Cobb-Douglas production function Y t = z t K α t L 1 α t. Assume capital share in production is 0.32 and the economy-wide ratio of capital to output is What is the pre-tax and pre-depreciation rental rate on a unit of capital? Show work and use four decimals. We have shown in the book and in this homework that when firms profit maximize, the following equation holds r = α Y K = = Question 5. Assume that the following about India: (a) The depreciation rate on capital is 6 percent; (b) capital s share of production is 30 percent; (c) the after-tax and after-depreciation rate of return on capital is 6 percent; and (d) the capital income tax rate (inclusive of bribes and corruption) is 70 percent What is the economy-wide ratio of capital to output in India? 10

11 We know that r = αy/k and ˆr = (1 τ k ) (r δ). This means that ˆr 1 τ k + δ = α Y K = 0.30 Y K 0.26 = 0.30 Y K = Y K (This also means that r = = 0.26) = K Y Question 6. The economy-wide ratio of capital to output is 1.8. Assume capital share in production is What is the pre-tax and pre-depreciation rental rate on a unit of capital? Since r = αy/k, then r =.27 (1/1.8) = Question 7. (HARD!) Consider a Cobb-Douglas production function with capital share of α = You have been told the marginal product of labor is and the marginal product of capital is What is the ratio of K/L? MPK = α z K α 1 L 1 α ( K = 0.27 z L ( ) K α = z L ) α 1 MPL = (1 α) z K α L α ( ) K α = 0.73 z L ( ) K α = z L ) α = z ( K L z ( ) K α 1 = K L = 2.5 L 11

12 Question 8. Prove that if the worldwide after-tax and after-depreciation rate of return on capital is fixed (at say 6 percent), and the depreciation rate on capital is also fixed (at also, say 6 percent), that the steady-state growth rate of capital is equal to the steady-state growth rate of output. Because the after-tax rate of return is fixed, and because the tax rate and depreciation rate is fixed, then the pre-tax and pre-depreciation rate of return, r, is also fixed. When (a) output is produced using a Cobb-Douglas aggregate of capital and labor, and (b) firms take wages and the rental rate on capital as given, and (c) firms maximize profits, then the pre-tax rental rate on capital is equal to the marginal product of capital, or r = αy/k. Since r is fixed over time, Y and K must be increasing at the same rate. Question 9. Suppose that Y = zk α L 1 α. Prove that when the labor input is fixed, the growth rate of output is equal to the growth rate of z plus α times the growth rate of K. Because of Cobb-Douglas production, we can express growth in output between two periods as: Y t+1 = z ( ) α ( ) 1 α t+1 Kt+1 Lt+1 Y t z t K t L t Assume the labor input is fixed (such that L t+1 = L t ), and take logs of the above: ( ) ( ) ( ) Yt+1 zt+1 Kt+1 ln = ln + α ln Y t z t K t ln(1 + g Y ) = ln(1 + g z ) + αln (1 + g K ) g Y = g z + α g K Note that the second equation follows from the first because X t+1 /X t = (X t /X t ) (1 + g X ), by definition of the growth rate of X, g X. Question 10. Write down the growth accounting equation discussed in the book. Then, using the framework of the growth accounting equation, describe why real GDP per capita has increased rapidly in China since Please limit your answer to ideas discussed in the book. The growth accounting equation expresses the growth rate of output as a function of the growth rates of the inputs to production: g y = g z + α g k + (1 α) g l China s output has increased rapidly since 1973 because all three of its inputs have increased rapidly: 12

13 g z has increased because it has allowed in foreign technology and expertise g k has increased because it has allowed foreign investment and increased its own investment in capital g l has increased because the work effort of its population has shifted from relatively unproductive farm work to relatively productive factory work. Question 11. Suppose a representative firm produces output each period according to the Cobb- Douglas production function described in class, Y t = z t K α t L 1 α t. Holding the labor input constant constant, why is it that technology growth is required for sustained increases to per-capita real GDP? That is, why is it impossible for a country to sustainably increase real GDP per-capita through the accumulation of capital alone? We know (from the book) that g Y = g z + αg K + (1 α) g L. Consider the case where g z = 0. Also consider the case of a stable population such that g L = 0. Then g Y = αg K. This implies that output is increasing less rapidly than capital because α < 1. When Y is increasing at a less rapid rate then K, interest rates fall: When firms profit maximize they set r = αy/k. Eventually r falls to the point where no one is willing to invest any more: They would rather consume. Thus g z > 0 is a requirement for growth. Question 12. Suppose that output in period t is produced according to the following function: Y t = Kt α (z tl t ) 1 α and suppose that firms pay r t for each unit of capital and w t for each unit of labor a. Define firm profits. b. Show that if r t is constant over time, that profit-maximization by firms implies that Y t and K t increase at the same rate. c. Denote the growth rate of Y t as g y, the growth rate of K t as g K, the growth rate of L t as g L, and the growth rate of z t as g z. Assume that r t is constant over time, and then show why g y = g z + g L. 13

14 a. Firm profits are K α t (z t L t ) 1 α r t K t w t L t b. Profit maximization take the derivative of the profit function with respect to K and set to zero (which sets the marginal benefit of an additional unit of capital equal to its marginal cost): αk α 1 t (z t L t ) 1 α = r t αk α t (z t L t ) 1 α = r t K t αy t = r t K t α (Y t /K t ) = r t If r t is constant, then profit-maximization implies that Y t /K t is constant, meaning Y t and K t increase at the same rate c. To do growth accounting, note that growth in output between t and t + 1 has the expression: Y t+1 Y t = ( Kt+1 K t ) α ( ) zt+1 L 1 α t+1 z t L t Now take logs and use the growth rate notation: ln(1 + g y ) = α ln (1 + g K ) + (1 α) ln (1 + g z ) + (1 α) ln (1 + g L ) Use the first-order Taylor series approximations to get: g y = α g K + (1 α) g z + (1 α) g L If r t is constant then g y = g K (see the answer to part b). Inserting this: g y = α g y + (1 α) g z + (1 α) g L Divide by (1 α) to get: (1 α) g y = (1 α) g z + (1 α) g L g y = g z + g L Question 13. Suppose that in any year t, output (Y t ) is produced according to the following production function: Y t = z γ t K α t L β t 14

15 where z t is technology, K t is capital used in production, and L t is labor used in production. Determine the annual growth rate of Y t, call it g Y, as a function of the annual growth rate of technology, g z, the annual growth rate of capital in production, g K, and the annual growth rate of labor in production, g L. Show work. To do growth accounting, note that growth in output between t and t + 1 has the expression: Y t+1 Y t = ( zt+1 ) γ ( ) α ( Kt+1 Lt+1 z t K t L t ) β Now take logs and use the growth rate notation: ln (1 + g y ) = γ ln (1 + g z ) + αln (1 + g K ) + β ln (1 + g L ) Use the first-order Taylor series approximations to get: g Y = γ g z + α g K + β g L 15

16 3 Chapter 3 Extra Questions and s Question 1. Suppose people have preferences for two goods, c 1 and c 2. The price of c 1 is r and the price of c 2 is 1.0. The utility from these two goods is ln (c 1 ) + β ln (c 2 ). People have total income of I to spend on these two goods. a. Suppose the budget constraint, written the appropriate way, is I (1.0 + r) c c 2 = 0 Write down the function we maximize to find the optimal bundle of c 1 and c 2. b. Take the derivative of the function we should maximize with respect to c 1 and set the derivative to zero. c. Take the derivative of the function we should maximize with respect to c 2 and set the derivative to zero. d. Given your answer to b and c, what does c 2 /c 1 equal? a. The function to maximize is ln (c 1 ) + β ln (c 2 ) + λ[i (1.0 + r) c c 2 ] b. The derivative of the function with respect to c 1 (when set to zero) is 1 c 1 λ (1.0 + r) = 0 c. The derivative of the function with respect to c 2 (when set to zero) is β c 2 λ = 0 d. Use the answer to part c to set λ = β/c 2. Then plug this into the equation from part b: 0 = 1 c 1 β c 2 (1.0 + r) 1 c 1 = β c 2 (1.0 + r) c 2 c 1 = β (1.0 + r) 16

17 Question 2. Björn of Norway lives two periods, period t (today) and period t+1 (next year). Björn is assumed to have a utility function from consumption this year C t and consumption next year C t+1 of C 1 σ t 1 σ + βc1 σ t+1 1 σ Björn has no assets as of the start of period t (i.e. A t = 0) but is given $100 of income in period t and again in period t + 1 by the Norweigan government (from all of its oil money). Björn s problem is to choose C t, C t+1 and A t+1 to maximize utility. Write down the budget constraints for years t and t + 1 and then use the Lagrangian technique to derive the expression linking Björn s optimal consumption at time t and at time t + 1 with the gross after-tax rate of interest on assets at t + 1, ˆRt+1 (i.e. ˆRt+1 is defined exactly as in the book). Your result should show that the ratio of marginal utilities is optimally equal to the ratio of prices, where the ratio of the price of consumption at t to the price of consumption at t + 1 is ˆR t+1. Budget constraints for years t and t + 1 are Lagrangian is: Period t $100 C t A t+1 = 0 Period t+1 A t+1 ˆRt+1 + $100 C t+1 A t+2 = 0 Ct 1 σ max C t,c t+1,a t+1 1 σ + βc1 σ [ ] t+1 1 σ + λ t [$100 C t A t+1 ] + λ t+1 A t+1 ˆRt+1 + $100 C t+1 A t+2 where λ t is the Lagrange multiplier on the period t budget constraint and λ t+1 is the Lagrange multiplier on the period t + 1 budget constraint. The first order conditions that characterize the optimal solution are: C t : = λ t C t+1 : βct+1 σ = λ t+1 A t+1 : λ t = λ t+1 ˆRt+1 C σ t Inserting the third equation into the first yields: C t : C t+1 : C σ t = λ t+1 ˆRt+1 βc σ t+1 = λ t+1 Dividing yields C σ t βc σ t+1 = ˆR t+1 The left-hand side is the ratio of marginal utilities; the right-hand side is the ratio of prices. 17

18 Question 3. Suppose the ratio of annual rents to house prices is 5 percent in Further, suppose you have observations for a nominal house price index ( HPI ) and a consumer-price index ( CPI ) for 2000 and 2001 as follows: Year HPI CPI What is the total annual inflation-adjusted return to housing between 2000 and 2001? The total inflation-adjusted return to housing is equal to the dividend yield plus growth in house prices after adjusting for inflation. This can be computed as div yield + HPI 2001/CPI 2001 HPI 2000 /CPI / / = or 12.3 percent. A different answer that is not quite right but is close is or 12.5 percent. div yield + HPI 2001 CPI 2001 HPI 2000 CPI =

19 4 Chapter 4 Extra Questions and s Question 1. In the United States, one hour of labor can make 25 units of manufactures or 15 units of services. In England, one hour of labor can make 12 units of manufactures or 10 units of services. Labor is the only costly input into production for both manufactures and services. a. Assume the United States and England do not trade. The price of one manufactured good is $10.00 in the United States and 5.00 in England. What do you think the price of one service is in the United States (in $) and in England (in ). b. Assume the United States and England trade. What does the United States export to England and what does England export to the United States. For both parts a. and b. show work and explain your answer. a. In the United States, the value of one hour of labor making manufacturing goods is $10.00 * 25 = $ This means that the value of one hour of labor making services in the United States must also be $ Since 15 units of services can be produced in one hour in the United States, the price of one service must be $250.00/15 = $ Likewise, in England it must be the case that the price of a service solves 5.00 * 12 /10 = b. In the United States, the price of one manufactured good, in units of services is $10.00/$16.67 = (This means that 0.60 services must be given up in order to produce one more manufactured good; this could also have been solved as 15/25). In England, the price of one manufactured good, in units of services, is 5.00/ 6.00 = (This means that 0.83 units of services must be given up to produce one more manufactured good; this could also have been solved as 10/12). The United States is the low cost producer of manufactures so, it exports manufactures to England. England exports services to the United States. Question 2. Consider Wayne and Garth. In one hour, Wayne can make either 5 guitars or 6 amps. In one hour, Garth can make either 3 guitars or 2 amps. a. What is the price of 1 guitar in units of amps for Waye? Show work. b. What is the price of 1 guitar in units of amps for Garth? Show work. 19

20 c. Using the answers to parts a and b, explain the patterns of trade: What does Wayne export to Garth, what does Garth export to Wayne, and why. a. It takes Wayne 12 minutes to make 1 guitar and 10 minutes to make one amp. In the time it takes Wayne to make 1 guitar, he could have made 12/10 = 1.2 amps. b. It takes Garth 20 minutes to make 1 guitar and 30 minutes to make one amp. In the time it takes Garth to make 1 guitar, he could have made 20/30 = amps. c. Garth exports guitars because he is the relatively low cost producer: He only foregoes amps to make 1 guitar, whereas Wayne foregoes 1.2 amps to make 1 guitar. Wayne exports amps to Garth. Question 3. Prior to trade, U.S. residents face the following budget constraint between consumption today C t and future consumption C t+1 : $100 = (1.03) C t + C t+1. And Japanese residents face the following budget constraint between C t and future consumption C t+1 : = (1.04) C t + C t+1. Assume Japan and the U.S. start to trade. Which country runs a trade deficit and why? Explain your answer. The price of one unit of consumption today relative to consumption in the future is 1.03 in the United States and 1.04 in Japan. This means that the United States is the relatively low-cost producer of consumption today and Japan is the relatively low cost producer of future consumption. The pattern of trade is that the United States exports C t to Japan and Japan exports C t+1 to the United States. Thus, Japan runs a trade deficit, since it is accepting C t today in exchange for assets, where the assets entitle the Japanese to future consumption from the U.S. (C t+1 ). Question 4. The yearly interest rate on assets is 8 percent. Given yearly income of $100, write down the budget constraint linking consumption today C t and consumption next year C t+1. The budget constraint you write down should look like: x 1 = x 2 C t + x 3 C t+1 20

21 where you fill in numerical values for the x 1, x 2 and x 3. C t+1 = (1.08) ($100 C t ) implying $108 = 1.08 C t + C t+1. It is acceptable to divide the equation by So either: x 1 = $108, x 2 = 1.08, x 3 = 1 or x 1 = $100, x 2 = 1.00, x 3 = (1.08) 1 = Question 5. What is a trade deficit? Why do trade deficits occur? Do not talk about exchange rate manipulation - stick to what was directly discussed in the book and notes. A trade deficit occurs when, on-net, goods and services are exchanged for assets. A trade deficit occurs when one country has a comparative advantage in producing future consumption (relative to current consumption). Prior to trade, this country will have a higher interest rate. (Alternative explanation: Question 6. On January 1, 2009 $1 U.S. can buy English The one-year real risk-free interest rate in both countries is 3.5%. The expected inflation rate in 2009 is 2.5% in the U.S. and and the inflation rate in England is expected to be 4.0%. Assuming that the Fisher equation, purchasing power parity, and covered interest parity hold, what is the forward exchange rate in $ / for January 1, 2010 when contracted on January 1, 2009? (NOTE: Take all calculations out to four decimal points). We use the Fisher equation to get the nominal interest rate in both countries: United States (1 + i $ ) = (1.035) (1.025) = Now apply the uncovered interest parity formula: England (1 + i ) = (1.035) (1.040) = (1 + i $ ) S $/ = (1 + i ) F $/ (1.0609) ($1.00/ 0.50) = (1.0764) x 21

22 This gives a solution to the forward rate of x = $1.9712/ Question 7. Consider the US and Japan. The inflation rate in the US is 2.5 percent per year and the inflation rate in Japan is 1 percent year. The real return to assets in both the U.S. and Japan is 5 percent per year. a. What is the annual nominal interest rate in the United States and Japan. b. According to covered interest parity, a forward contract would call for an appreciation or depreciation of the dollar relative to the Yen? c. If 1 dollar buys 100 at the start of the year, then what is the exchange rate ( /$) that you can lock-in using a forward rate at the end of the year? d. Given your answer to (c), by what percentage did the dollar appreciate / depreciate relative to the Yen. a. U.S. 1 + i US = (1.05)(1.025) = i US = 7.625% Japan 1 + i J = (1.05)(1.01) = i J = 6.05% b. The forward contract would call for the dollar to depreciate relative to the Yen. (The inflation rate in the US is greater than the inflation rate in Japan.) c. The calculation works as follows: ( )(1/100) = x x = The forward rate is equal to $/, or more commonly stated as (1/0.0105) = /$. d. The rate of depreciation of the dollar solves: ( )/100 = 1.46% 22

23 5 Chapter 6 Extra Questions and s Question 1. Consider a Taylor-rule of the form r ff t = [100 (ln(gdp t ) ln(gdp t ))] π t, where r ff t is the current (nominal) fed funds rate, GDPt is trend real GDP and π t is the inflation rate of core consumer prices. Suppose the nominal fed funds rate is now 5.25 percent and the rate of core consumer price inflation is 2.7 percent. a. What is the Fed s implied estimate of [100 (ln (GDP t ) ln (GDP t ))]. b. Show that [100 (ln (GDP t ) ln (GDPt ))] is (approximately) the percentage deviation of real GDP from trend real GDP. a. Here, we just solve for [100 (ln (GDP t ) ln (GDP t ))]. Call this quantity as the variable x 5.25 = x = 0.65 x x = b. As mentioned in the book, ln (GDP t ) ln (GDPt ) can be manipulated as follows: ( ) GDPt = ln = ln GDPt ) (1 + GDP t GDP t GDP t GDP t GDPt GDPt times the above is the percent deviation of GDP from its trend 23