ECON 5118 Macroeconomic Theory

ECON 5118 Macroeconomic Theory Assignments April 23, 2013 Chapter 1 1 Explain why the monetarists think that using monetary policy to fine tune the economy may not be a good idea 2 What are the key assumptions in a dynamic general equilibrium model? Discuss each of the assumptions in the context of the traditional Keynesian model 3 Choose one of the following questions a) Write a concise history of macroeconomics in the 20th century Divide your discussions into three periods: pre-1940, 1940 1980, and 1980 to the end of the century b) Some critiques charge that academic macroeconomic theories are elegant mathematical models which have little relevance to the real world Provide a counter argument with three key developments in macroeconomic theory that have influenced public policy c) Does the job of an macroeconomist resemble that of an scientist or of an engineer? Explain in details d) Some critiques suggest that macroeconomic theorists fall into to one of two rivalrous groups, the salter water economists and the fresh water economists The first group is often called New Keynesians and the second Real Business Cycle Theorists Do you agree with this division? Explain your argument in details Chapter 2 1 Consider a simple economy with two goods, output y t and capital k t, with one aggregate household and one aggregate competitive firm The production function is given by y t F k t ) in each period t Output can be consumed this period or invested for the next period, that is, y t c t + i t Therefore capital available in period t + 1 satisfies k t+1 k t+1 k t i t δk t, where δ is the rate of depreciation of capital a) Derive the dynamic resource constraint b) Derive the consumption function in the steady state c) What is the necessary condition for optimal consumption in the steady state? 2 Consider the economy in question 1 Suppose the household s preferences can be represented by a stationary and separable utility function β s log c t+s 1) where β 1/1 + θ) is a discount factor a) Set up the intertemporal optimization problem b) Find the Euler equation Explain the economic meaning of the Euler equation c) Show that in the steady state, F k ) δ + θ 3 Repeat question 2 with the constant relative risk aversion utility function β s c1 σ t+s, σ > 0 2) 1 σ 4 The relative risk aversion of a household is defined as cu c)/u c) Find the relative risk aversion for the utility function in 2) in each period 5 Suppose that the production function of a centralized economy is given by F k t ) ak α t, 0 < α < 1 Utility of the aggregate house is the same as in 1) An central planning committee has estimated the values of the parameters given in Table 1 Assume that

Table 1: Estimated Parameters of An Economy Parameter Estimated Value α in F 035 Social discount rate, θ 005 Depreciation rate, δ 006 Output/capital ratio at steady-state, y /k 010 Source: Robert E Hall 2010) Why Does the Economy Fall to Pieces after a Financial Crisis? Journal of Economic Perspectives, 244), 3 20 the dynamic of the economy is given by ct+1 c 1 + F U k t+1 k 1+θ)U F U ct U c 1 1 + θ k t k, or x t+1 Ax t where x t c t c, k t k ) T and the derivatives in the matrix A are valued at the steady-state capital k and consumption c You can normalize the scale of the economy such that y 1 a) Determine the stability of the economy b) Does it matter if y is rescaled to, say, 100? 6 Consider a closed centralized economy as described in section 24 of the textbook The economy is at steady-state equilibrium at t 0 Suppose that production technology remains the same in period 1 but the depreciation rate of capital has decreased due to a permanent improvement in maintenance technology That is, δ 0 > δ 1 Describe the dynamic and the new steady-state equilibrium of the economy 7 Suppose that the household s utility function in each period t is Uc t, l t ) c1 σ t 1 σ + log l t, where l t is leisure time The production function is F k t, n t ) Ak α t n 1 α t, where n t is labour input and n t + l t 1 a) Derive the Euler equation for intertemporal consumption b) Derive the relationship between labour supply and consumption given the capital stock c) Can you get a reduced form solution for labour supply, consumption, and capital in the steady state? Why or why not? 8 Suppose that in question 1 the production function is F k t ) Ak α t, 0 < α < 1 Investment, however, requires installation cost of 1 2 φi t/k t per unit, where φ > 0 Household utility in each period t is given by Uc t ) log c t a) Set up the resource constraint and the optimization problem b) Find Tobin s q in each period Is q t > 1? c) Find the steady state value of q 9 Use the first-order conditions to derive equation 235) in the textbook 10 Derive equation 237) in the textbook 11 Derive equation 238) in the textbook Chapter 3 1 Consider the following production functions: 1 F t K t, N t ) A t α k K t + α n N t ), { Kt F t K t, N t ) A t min, N } t, α k α n F t K t, N t ) A t α k K ρ t + α n N ρ t ) 1/ρ, where, in each case, A t, α k, α n > 0, and 0 ρ < 1 a) Do all functions exhibit constant returns to scale? b) Express each function in per capita form c) One of the key result of the Solow-Swan model is that the larger the capital stock per person, the lower the growth rate Does this result still hold for each of the above production function? 2 Suppose that a strictly concave production function Y t F t K t, N t ) exhibits constant returns to scale a) Show that in per capita terms, the function can be expressed as y t f t k t ) b) Show that the capital elasticity of output is always less than one, that is, k t y t dy t dk t < 1 Hint: Recall that a function f is strictly concave if and only if for every x, y, fx) < fy) + f y)x y) 1 See Geoffrey A Jehle and Philip J Reny 2011) Advanced Microeconomic Theory, Third Edition, Essex: Pearson Education Limited, p 129 131 2

3 Let γ x, γ y, and γ z be the growth rates of X t, Y t, and Z t respectively Show that, for α > 0, a) if Y t αx t, then γ y γ x, b) if Z t X t Y t, then γ z γ x + γ y, c) if Z t X t /Y t, then γ z γ x γ y, d) if Y t X α t, then γ y αγ x e) Use the results above to show that the growth rate of capital per person is k t+1 k t K t+1 K t N t+1 N t γ n 4 Suppose that the aggregate production function in period t of a closed economy is given by Y t F K t, A t N t ), where A t 1 µ) t and F is strictly concave and exhibits constant returns to scale The population N t grows at a constant rate n Define effective labour as N # t A t N t a) Show that output per effective labour, y # t Y t /N # t, can be expressed as where k # t K t /N # t y # t fk # t ), b) Let household utility be UC t ) log C t in each period and the discount factor is β 1/1 θ) Derive the Euler equation for intertemporal consumption c) Find the steady-state solution for y #, k # and c # d) Does the economy achieve balance growth in the steady state? 5 Consider an economy that is on the optimal balanced growth path Suppose that technological progress is at a constant rate µ Now suppose that the rate of population growth drops to zero What happens to the balanced-growth-path values of capital per worker, output per worker, and consumption per worker? 6 In the endogenous model with human capital, use the first-order conditions to show that the growth rate of consumption is log ct+1 c t ) 1 σ Aα α 1 α) 1 α) δ θ 7 Define an economy s incremental capital-output ratio ICOR) as ψ t I t, Y t+1 where I t and Y t are investment and output respectively in period t The ICOR is often used as a measure of efficiency of investment Comment on the effectiveness of this measure 8 Solving the Solow-Swan model) Consider an economy with a Cobb-Douglas production function: Y t 1 + µ) t K α t N 1 α t Define effective labour as N # t 1 + µ) t/1 α) N t The capital accumulation equation can be written as 1 + η)k # t+1 sk# t ) α + 1 δ)k # t, where s is the exogenous saving rate and η n + µ/1 α) a) Derive an expression for the growth rate of capital per effective labour γ b) Since k # t is adjusted for the growth rates of population n, technology µ, and depreciation δ, in the steady state γ 0 Find the steady-state value k # c) Does k # conform with the results of the Solow- Swan model with respect to exogenous changes in s, δ, n, and µ? d) From the steady-state results from the optimal growth model, set the parameter σ and θ to zero Show that the results of the two models converge by showing that the values of k # are the same Chapter 4 1 Suppose a consumer wants to maximize the following function with respect to consumption c t and c t+1 : V t Uc t ) + βuc t+1 ) 3) subject to the intertemporal budget constraint c t + c t+1 1 + r x t + x t+1 1 + r, 4) where U is the instantaneous utility function, β is a discount factor, x t and x t+1 are incomes in periods t and t + 1 respectively, and r is the interest rate a) By writing 3) as fx, θ) Uc t ) + βuc t+1 ) V t with x c t+1 and θ c t, apply the implicit function theorem to find the marginal rate of time preference, dc t+1 /dc t 3

b) Find the slope dc t+1 /dc t of the intertemporal budget constraint 4) as well c) Combine your results to get the Euler equation: βu c t+1 ) U 1 + r) 1 c t ) 2 Suppose that the household s utility function in each period t is Uc t, l t ) log c t + η log l t, η > 0, where l t is leisure time, n t is labour supply and n t + l t 1 The budget constraint is a t+1 + c t w t n t + x t + 1 + r t )a t, where a t is financial asset, w t is the wage rate, and x t is an exogenous income in period t a) Substitute the labour constraint n t + l t 1 into the utility function and set up the optimization problem b) Derive the Euler equation for intertemporal consumption c) Derive the labour supply function in terms of consumption and the wage rate 3 The representative household s problem is max c t+s,a t+s+1 subject to the budget constraints β s Uc t+s ), 41) a t+s+1 + c t+s x t+s + r t+s a t+s, s 0, 1,, where a t, x t, and r t are exogenous asset, income, and interest rate respectively in period t a) Identify the state variables) and the control variables) b) Set up the Bellman equation c) Find the necessary conditions for optimization d) Derive the Euler equation for intertemporal consumption 4 Some households think that since the amount of land is fixed and limited, the prices of houses and apartments will have very high growth rates due to economic development a) Do you think this is a reasonable prediction? Why or why not? b) If a household which own a house expects that the price of the house will increase at a rate of 10% per year indefinitely, what would be the effect on personal consumption? Use a formal economic model to justify your answer c) Suppose that housing prices indeed increase at a pace of 10% per year for a period of time Now an interest rate shock causes the housing price to decrease by 40% in one year What is the effect on personal consumption? Why is demand for housing is more volatile than other goods and services such as food and education? 5 The average income of farmers is less than the average income of urban workers, but fluctuates more from year to year How does the permanent income theory predict that estimated consumption functions for farmers and urban workers differ? 6 Suppose that 0 < ρ < 1 Show that sρ1 ρ) s 1 1/ρ s1 7 Let the production function of a firm be F k t, n t ) Ak α t n 1 α t, A > 0, 0 < α < 1, where k t and n t are capital and labour inputs respectively The firm maximizes the present value of current and future profit using the exogenous real interest rate r as the discount rate The market real wage rate is w t The debt level of the firm in period t is b t a) Set up the optimization problem facing the firm s manager What are the control variables? b) Write down the first-order conditions c) Find the labour demand function in each period given the level of capital stock in that period d) Find the investment function in each period given the level of capital stock in that period e) Can you find the optimal debt level? why not? Why or f) What effects has an increase in the interest rate on labour demand and investment? g) What effects has an increase in A on labour demand and investment? 8 Use the results in questions 2 and 7 to find the equilibrium wage rate Using the equilibrium conditions in the bond market, the goods market, and the capital market, can you find an explicit solution of the economy for c t, n t, w t, and k t? 4

9 Use equation 434) in the textbook to prove equation 436) 10 Consider a stock with price p t and pays dividends d t in period t Assume that consumers are risk-neutral and have a discount rate of θ r, thus they maximize c t+s/1 + r) s If consumers sell the stock, it always happens after the company has paid the dividends a) Show that in equilibrium p t d t+1 + p t+1 1 + r b) Assume that there is no bubble in the stock market, that is, lim s p t+s /1+r) s 0 Solve the above difference equation for p t Interpret the result c) Suppose that the dividends are constant for every period, that is, d t d What will happen to the price of the stock if interest rate goes up? 11 According to the US Survey of Consumer Finances released in June 2012, the median family wealth and income, in constant dollar, declined from 2007 to 2010: Year Net Asset Income 2007 126,400 49,600 2010 77,300 45,800 a) Explain why both wealth and income decline for the median family b) What would the effect on the median household consumption according to the permanent income hypothesis? c) The survey also reports that consumer spending has remained surprisingly resilient Explain See the New York Times article by Binyamin Appelbaum for more details 12 Count Dracula lives alone in a castle in the Carpathian Mountains near Transylvania Being a vampire, he rejuvenates his power by sucking the blood of nearby villagers His instantaneous utility in each period t is represented by the function Ub t ), where b t is measured in the unit of blood per person, and U satisfies the Inada conditions The population of the villagers, h t, is assumed to grow at a constant rate n Although Dracula is immortal, he is being pursued by a group of vampire hunters There is a probability of ρ 0, 1) in each period that the vampire hunters will annihilate him a) What is the life expectancy of Count Dracula at each period? b) What is the dynamic resource constraint faced by Dracula? c) Assuming that Dracula is an expected utility maximizer, set up the maximization problem and derive the Bellman equation d) Derive the Euler equation for preying on his victims Is there a steady-state solution? Chapter 5 Bonding Financing Dynamics Period GBC t 1 : g t 1 + Rb t T t 1 t : g t 1 + g t + Rb t T t 1 + b t+1 t + 1 : g t 1 + g t + Rb t + R b t+1 T t 1 + b t+2 t + n 1 : g t 1 + g t + Rb t + R n 1 s1 b t+s T t 1 + b t+n 1 The table above shows the dynamics of bonding financing of a permanent increase of g t in period t 2 By using mathematical induction, show that new bond issued government deficit) in period t + n 1 is given by b t+n 1 + R) n 1 g t 5) 2 In section 541, we assume that R < π + γ so that the difference equation b t+1 y t+1 1 + R b t + 1 + π)1 + γ) y t 1 d t 512) 1 + π)1 + γ) y t is stable Assume also that the deficit/output ratio is constant, that is, show that b t+n lim n y t+n d t+n y t+n d t y t, n 1, 2,, 1 1 + π)1 + γ) 1 + R) 1 π + γ R 3 In the case that R > π + γ, show that b t 1 y t 1 + R d t y t d t y t < 514) ) s 1 + π)1 + γ) dt+s 1 + R 2 There are typos in the textbook y t+s ) 518) 5

4 Consider the lump-sum taxation model where the central planer maximizes household utility β s Uc t+s, g t+s ) subject to the resource constraint F k t ) c t + k t+1 1 δ)k t + g t and the government budget constraint g t T t a) Identify the choice variables and the state variable b) Set up the Bellman equation c) Derive the necessary conditions from the Bellman equation d) Find the Euler equation, the marginal rate of substitution of household consumption to government services, and the steady-state condition 5 Suppose the household budget constraint in real terms is given by 1 + τ c )c t + k t+1 + b t+1 1 τ w )w t n t + 1 + 1 τ k )r k t k t + 1 + r b t )b t where b t is bond investment, k t is capital stock, w t n t is labour income, rt k and rt b are rates of return on capital and bonds, τ c, τ w, and τ k are tax rates on consumption, wages, and capital respectively The household utility function is Uc t, l t ) c1 σ t l 1 σ ) 1 σ + η t, 1 σ where l t is leisure Time constraint is given by n t + l t 1 a) Set up the utility maximization problem and find the Euler equation for intertemporal consumption b) Find the marginal rate of substitution between leisure and consumption c) What is the effect of a higher income tax rate τ w on labour supply? d) What is the effect of a higher consumption tax rate τ c on consumption? e) Find the relations between r k t, r b t, and social discount rate θ in the long run 6 Suppose that the household utility and tax rates are the same as those described in the last question a) Find the implementability condition b) Set up the government optimization problem c) Find the marginal rate of substitution between leisure and consumption Compare the result with your answer in the previous question Chapter 8 1 Combine the two constraints in the cash-in-advance model into one Identify the state and control variables, set up the Bellman equation, and derive the Euler equation 2 Assume that r R π > 0 and in the steady state c t+s c t and m t+s m t for s 1, 2, With the approximation 1 + R 1 + r for low inflation rate, show that the consumption function is c t r 1 + r in the cash-in-advance model x t+s 1 + r) s + rb t πm t 3 For each of the following utility functions, a) Uc t, m t ) log c t + η log m t, b) Uc t, m t ) c 1 η t m η t, c) Uc t, m t ) c ρ t + ηm ρ t ) 1/ρ, 0 ρ < 1, derive the Euler equation and the demand function for money Does an increase in interest rate reduce demand for money in each case? 4 For the original utility function on page 193 of the book, Uc t, m t ) c1 σ t 1 σ + η m 1 σ ) t, 1 σ what is the effect on money demand if η increases? Repeat the question for the utility functions in question 3 5 In the transaction cost model of money demand, solve for all the partial derivatives in c/ x c/ b c/ R m/ x m/ b m/ R by assuming that T mc 0 and determine their signs 6 a) Use equation 818) to show that m c 1 + T c π + T m b) Use equation 819) to show that m/ c > 0 6

7 Suppose 0 < x < 1 Show that sx s x 1 x) 2 Use the result to show that equation 827) becomes 1 1 + α ) s α m t + µs) m t + αµ 1 + α 8 Given the household budget constraint c t + 1 + π t+1 )k t+1 + b t+1 + m t+1 ) 1 τ t )w t n t + 1 + R k t )k t + 1 + R b t)b t + m t and the government budget constraint g t +1+R b t)b t +m t τ t w t n t +1+π t+1 )b t+1 +m t+1 ) Derive the dynamic resource constraint of the economy 9 For each of the following utility functions, a) Uc t, l t, m t ) log c t + η log m t + zl t ), b) Uc t, l t, m t ) c 1 η t m η t + zl t ), c) Uc t, l t, m t ) c ρ t + ηm ρ t ) 1/ρ + zl t ), 0 ρ < 1, determined whether the Friedman rule of money supply is optimal or not 10 Let the household s utility function Uc t, l t, m t ) be homothetic and separable in l t and m t Show that the Friedman rule is optimal Chapter 9 1 Consider a firm with a CES production function N ) 1/ρ Q α i X ρ i, 0 ρ < 1, i1 N α i 1 i1 and faces a downward sloping demand function The N input suppliers are monopolies so that input prices W i are increasing function of input quantities X i, that is, W i W X i ), W X i ) > 0 What will be the price transmission mechanism from W i to the final good P for the case of a) ρ 1, b) ρ 0, c) ρ 2 Assume a slightly more general CES production function N ) 1/ρ y α i x ρ i, 0 ρ < 1, i1 N α i 1, i1 where y is the final output and x i is the ith intermediate input Derive all the key results in Section 933 3 Prove equation 923) in the textbook 4 In the modified Calvo model on page 234, show that π t ρ1 γ)p t p t 1 ) + γe t π t+1 + 1 γ)1 ρ)π t 1 1 + γ 2ργ 5 Let the production function of a firm in period t be Y t A t N φ t, 0 < φ < 1, where Y t is output, A t is technology, and N t is labour input Using lower case letters as the log values of variables, show that the log of marginal product of labour is see equation 936) in textbook) mp t log φ + a t φ 1 φ φ y t 6 Walsh, 2003, 217 219) An aggregate firm produces the final good y with a CES technology and N inputs, N ) 1/ρ y x ρ i, 0 ρ < 1, i1 where x i is intermediate good i a) Show that the demand function for input i is ) φ P x d i y, where P is the price of the final output, p i is the price of the intermediate input i, and φ 1/1 ρ) b) Assuming a competitive market for the final good, show that P N i1 p i p 1 φ i ) 1/1 φ) c) Each intermediate good is produced by a single firm with a production function x i K α i L 1 α i, 0 < α < 1, 7

where K i and L i denote capital and labour inputs of firm i with nominal market prices P r and P w so that r and w are the real prices relative to the final good price P ) Show that profit maximizing price of firm i given the final good price and demand is p i P C ρ, where C cr, w) is the unit cost in producing input i d) Assuming that all intermediate firms are identical, show that the total labour demand function is N ρ1 α)y L L i w i1 7 Romer, 2006, 343 344) Consider an economy consisting of many identity firms In any period only a fraction ρ of firms can set new prices, with the firms chosen at random The objective of the firms is min p # t 1 ρ) s E t p # t p t+s) 2, where p t is the profit maximizing price in period t a) Show that the optimal solution is p # t ρ 1 ρ) s E t p t+s b) Express p # t in term of p t and E t p # t+1 Subtract p t from both sides to find an expression for the relative price charged by firms that set prices in t, p # t p t, in terms of p t p t, E t p # t+1 p t+1, and E t π t+1 c) Show that the average log) price in period t is p t ρ 1 ρ) s p # t s d) Express p t in terms of p # t and p t 1 Use the result to express the inflation rate π t in terms of p # t p t e) Use the result in part 4) to substitute for p # t p t and E t p # t+1 p t+1 in your answers for parts 2), and solve for the inflation rate π t in terms of E t π t+1 and p t p t Suppose that p t p t φy t, where y t is the output and φ > 0, express π t in terms of E t π t+1 and y t 8 Suppose that a small economy has no capital market and the government does not finance its spending by bonds Therefore the households rely on holding money to save a) What is the household budget constraint? b) Suppose the government announces in period t that it will change the unit of its currency Households in period t+1 can exchange 100 units of the old currency for one unit of the new one Wage rate, however, will remain the same using the new currency What is the impact of the policy on the economy, in particular the price level, after the announcement? Review Questions 1 Refer to the article We re Spent by David Leonhardt a) Do consumer behaviours over the years in the US support the theory of household intertemporal consumption? b) What about rational expectations? c) Do yo think that household behaviours are better explained under a Keynesian framework? Why or why not? 2 Models in money demand suggest that demand for money m d has a negative relation with the nominal interest rate R a) Draw a diagram of the money market with R on the vertical axis and m on the horizontal axis Assuming that the central bank controls the money supply Draw the supply and demand curves Find the market equilibrium quantity of money and interest rate b) During a recession, the central bank doubles the supply of money but the quantity of money in circulation contracts Explain the situation with your diagram c) Suggest some policies that can potentially solve the problem 3 Recall from the Tobin q model we have, from equations 238) and 240a), q t q βq t+1 q) + βf k t+1 ) F k), φk t+1 q t q + φ)k t Show that the linearized version can be expressed in the form x t+1 Ax t 8

with x t q t q, k t k) T, and A F k) 1 β kf k) φ k φ 1 What do you expect the eigenvalues of A to be? 4 Solve the money in utility problem in Chapter 8 with the technique of dynamic programming 5 In this question we investigate the long-run fiscal stance in the US as of 2013 You can do your own research Paul Krugman s article in the New York Times serves as a starting point a) Krugman claims that the budget doesn t have to be balanced to put us on a fiscally sustainable path Explain b) In view of equation 516) in the textbook, is $460 billion a sustainable deficit for the US in 2013? c) Rising health costs and an aging population will put the budget under growing pressure over the course of the 2020s What would be the best fiscal policy to address this long-run problem? Hints for Selected Assignment and Test Questions Chapter 2 Question 5 a) With the given output capital ratio show that a 01k ) 1 α Then show that F k ) αα 1)01)/k 0002275 With equation 222) you can show that U c )/U c ) c 04 Putting the values in the matrix A and find the two eigenvalues Is one of the absolute values less than 1? b) You can show that the rescaling does not change the result Question 9 The first-order conditions are β s U c t+s λ t+s 0, s 0, 6) λ t+s 1 + φi ) t+s + µ t+s 0, s 0, 7) k t+s λ t+s F k t+s ) + φ ) 2 it+s 2 k t+s µ t+s 1 + 1 δ)µ t+s 0, s 1 8) Substitute 6) into 7) and putting s 0 gives µ t U c t ) 1 + φ i ) t 9) k t Equation 7), with s 1, also implies that or, rearranging, becomes it+1 k t+1 q t+1 1 + φ i t k t, 10) ) 2 1 φ 2 q t+1 1) 2 11) Equation 235) can be obtained by substituting 6), 9), 10), and 11) into 8) 9

Question 10 In the long run, i δk Equation 233) implies that It follows that Now 235) implies that i 1 q 1)k φ q 1 + φδ 1 236) F k) 1 + θ)q 1 δ)q 1 q 1)2 2φ θ + δ + φδ θ + δ ) 2 Question 11 θ + δ Using the first-order Taylor approximation at q and 236), Thus q t+1 1) 2 q 1) 2 + 2q 1)q t+1 q) φ 2 δ 2 + 2φδq t+1 1 + φδ) 2φδq t+1 2φδ φ 2 δ 2 1 δ)q t+1 + 1 2φ q t+1 1) 2 q t+1 δ φδ2 2 Putting this into 235) and setting c t+1 c t c gives q t β From 237) and 236), which implies that q t+1 + F k t+1 ) δ φδ2 238a) 2 F k) θ + δ + φδθ + φδ2 2 θq + δ + φδ2 2, δ + φδ2 2 F k) θq Putting this into 238a) gives q t β q t+1 + 1 β β q which implies that + β F k t+1 ) F k), q t q βq t+1 q) + β F k t+1 ) F k) 238) Chapter 3 Question 1 Consider the case that F t K t, N t ) A t α k K ρ t + α n N ρ t ) 1/ρ, 0 ρ < 1 a) It is straight forward to show that F t is linearly homogeneous b) f t k t ) A t α k k ρ t + α n ) 1/ρ c) The growth rate of capital per person γ is decreasing in k t if dγ sy t dk t kt 2 1 k t dy t < 0, y t dk t which requires the capital elasticity of output to be less that one, ie, k t dy t < 1 y t dk t This condition holds if f t k t ) is straightly concave You should be able to show that and f f tk t ) α k A t k ρ 1 t α k k ρ t + α n ) 1 ρ 1, t k t ) α k A t ρ 1)k ρ 2 t α k k ρ t + α n ) 1 ρ 1 1 1 1 + α n /α k k ρ t ) The second derivative f t k t ) is negative since ρ < 1 Question 4 a) By definition y # t 1 N # t F K t, N # t ) F k # t, 1) fk # t ) b) Assuming that A t 1 + µ) t and N t 1 + n) t N 0, it can be shown that N # t 1 + η) t N 0, where 1 + η 1 + µ)1 + n) The dynamic resource constraint is fk # t ) c # t + 1 + η)k # t+1 1 δ)k# t Instantaneous utility becomes setting N 0 1) log C t log1+η) t c # t log c # t +t log1+η) log c # t +tη The optimization problem is max c # t+s,k# t+s+1 β s log c # t+s + ηt + s) subject to fk # t ) c # t + 1 + η)k # t+1 1 δ)k# t 10

The problem can be solved by the Lagrangian method or by the Bellman equation The Euler equation is Question 6 β c# t 1 c # f k # t ) + 1 δ 1 + η t Write the Euler equation as ) σ β ct+1 c t 1 + αa kt+1 h t+1 ) 1 α) δ Take log on both sides, apply the log approximation, and use the fact that Question 7 k t+1 α h t+1 1 α The following comments may be helpful: 3 Pivot Capital Management points to China s incremental capital-output ratio ICOR), which is calculated as annual investment divided by the annual increase in GDP, as evidence of the collapsing efficiency of investment Pivot argues that in 2009 China s ICOR was more than double its average in the 1980s and 1990s, implying that it required much more investment to generate an additional unit of output However, it is misleading to look at the ICOR for a single year With slower GDP growth, because of a collapse in global demand, the ICOR rose sharply everywhere The return to investment in terms of growth over a longer period is more informative Measuring this way, BCA Research finds no significant increase in China s ICOR over the past three decades Moreover, if we put aside the effect of depreciation, ψ t is the reciprocal of the marginal product of capital, which is decreasing in K This relationship gives rise to the downward sloping curve in chart 4 of the above quoted article As the chart indicates, China is still in an early stage of economic development Question 8 It follows from the capital accumulation equation that γ k# t+1 k # t 1 sk # t ) α 1 δ + η 1 + η 1 + η 3 Not just another fake, Economist, January 14, 2010 In the steady state γ 0, which implies that the steadystate value of k # t becomes ) 1/1 α) s k # 12) η + δ From the optimal growth model, the optimal saving rate and the steady-state effective capital are and s αη + δ) ση + δ + θ k # α ση + δ + θ ) 1/1 α) respectively Setting σ and θ to zero, we have s k # αη + δ), 13) δ α ) 1/1 α) 14) δ Substituting the saving rate in 13) into 12), the effective capital becomes the same as in 14) This means that the Solow-Swan model is a special case of the optimal growth model when the social discount rate θ becomes zero Chapter 4 Question 6 1 Let I n n s1 sρ1 ρ)s 1 2 Find 1 ρ)i n 3 Derive I n 1 ρ)i n ρi n 4 Find lim n ρi n Question 10 a) The budget constraint of the household is p t a t+1 + c t x t + d t a t, where x t is the exogenous income and a t is the number of shares owned, both in period t The price of a t+1 is p t because it is bought before the dividend is paid in period t + 1 The Bellman equation is v t a t ) max c t { c t + 1 1 + r v t+1 The necessary conditions are 1 1 1 + r λ t+1 1 + r xt + d t a t + p t a t c t p t )} λ t+1 p t 0, 15) p t + d t p t λ t, 16) 11

where λ t v ta t ) Equation 15) implies that λ t+1 p t 1 + r) and λ t p t 1 1 + r) Substituting these into equation 16) gives the result b) The first-order difference equation in p t is The solution is p t+1 1 + r)p t d t+1 p t s1 d t+s 1 + r) s, 17) which means the price of the stock in period t is the sum of the present values of future dividends Notice that no market bubble means that the transversality condition is satisfied c) It is obvious from equation 17) that an increase in interest rate will cause the price of the stock to go down Chapter 5 Question 1 For n 1, and from the table provided, the GBCs at periods t and t 1 imply that b t+1 g t Now assume that equation 5) holds for periods up to t + n 2 From the GBC in period t + n 1, where b t+n g t {1 + R) + R1 + R) + 1 + R) 2 + + 1 + R) n 2 } g t {1 + R) + Rσ}, 18) σ 1 + R) + 1 + R) 2 + + 1 + R) n 2 Multiplying both sides by 1 + R and subtracting to get σ 1 R 1 + R)n 1 1 + R) Substitute σ into 18) and the result follows Question 4 The Bellman equation is vk t ) max c t,g t {Uc t, g t ) + βvf k t ) c t g t + 1 δ)k t )} The necessary conditions for dynamic optimization are v c t U c,t βλ t+1 0, v g t U g,t βλ t+1 0, v k t βλ t+1 F k t ) + 1 δ λ t, where λ t v k t ) The Euler equation is βu c,t U c,t 1 F k t ) + 1 δ 1, and the marginal rate of substitution between private and government consumptions is MRS U c,t U g,t 1 In the steady state F k t ) δ + θ Question 5 b) The MRS between leisure and consumption is ηc σ t l σ t 1 τ w )w t 1 + τ c 19) c) Using l t 1 n t, equation 19) becomes η1 + τ c 1/σ ) n t 1 1 τ w c t )w t Therefore n t is decreasing in τ w Chapter 8 Question 1 Since c t m t for all t, by choosing m t+1 in period t, c t+1 becomes a state variable in period t + 1 Therefore the control variable is m t+1 and the state variables are c t and b t The utility maximization problem becomes max β t Uc t ), m t+1 subject to the transition equation ct+1 m t+1 b t+1 x t + 1 + R t )b t /1 + π t+1 ) m t+1 The Bellman equation is { vc t, b t ) max Uc t ) + βvm t+1, m t+1 } x t + 1 + R t )b t /1 + π t+1 ) m t+1 ) Let vc t, b t ) λ t λ c,t λ b,t T, the necessary conditions are βλ c,t+1 λ b,t+1 ) 0, 20) and U c t ) 0 +β 0 0 0 1 + R t )/1 + π t+1 ) λc,t+1 λ b,t+1 λc,t λ b,t 21) 12

From 20) we have λ c,t+1 λ b,t+1 so that λ c,t λ b,t 22) The first line of 21) implies that λ c,t U c t ) so that λ c,t+1 U c t+1 ) 23) By 22), the second line of 21) can be written as β1 + R t ) 1 + π t+1 λ c,t+1 λ c,t Substitute 23) into the above, we get the Euler equation Question 2 βu c t+1 ) U c t ) 1 + R t 1 1 + π t+1 If r R π > 0, the budget constraint implies that b t + m t 1 1 + R ) s 1 + π c t+s x t+s + Rm t+s) 1 + R 24) Putting c t+s c t and m t+s m t for s 1, 2,, into 24) gives b t +m t 1 + r c t 1 + R r 1 1 + R x t+s 1 + r) s + R 1 + R 1 + r m t, r where have used 1/1 + r) 1 + π)/1 + R) and results from geometric series Now assume that 1 + R 1 + r for low inflation rate, we have b t + m t c t r 1 1 + r x t+s 1 + r) s + R r m t Using r R π and rearranging yields the result Question 3 b) The Euler equation is Money demand is ) η ct m t+1 1 + R t+1 β 1 c t+1 m t 1 + π t+1 m t η c t, 1 η R t which implies that a higher interest rate reduces money demand Question 7 Let Then I sx s x + 2x 2 + 3x 3 + Proceed to show that xi x 2 + 2x 3 + 3x 4 + I xi x 1 x, and solve for I In 822), result for the m t term is straight forward For the µs term, let x α/1+α) and apply the above result to get αµ Question 10 The following argument may be helpful 1 If preferences are homothetic, U can be expressed as a linearly homogeneous function 2 Apply the Euler theorem part 2) to U That is, if we let x c, l, m), then 2 Ux 0 3 Show that the last row of the above matrix multiplication means that U cm,t c t + U lm,t l t + U mm,t m t 0 4 Now show that the first-order condition of the government s problem with respect to m t becomes V m,t 1 + µ)u m,t µu lm,t 0 5 Since U is separable in l t and m t, U lm,t 0 6 Conclude that U m,t 0 Chapter 9 Question 2 The first-order conditions for the final-good producer s profit maximization problem are ) 1/φ y α i P p i 0, i 1,, N, x i where φ 1/1 ρ) is the elasticity of substitution The conditional demand function for input i is therefore ) φ αi P x i y 915a) p i 13

The zero profit condition implies that P N i1 α φ i p1 φ i ) 1/1 φ), 25) which corresponds to equation 3) in the class note For the intermediate-goods suppliers, profit in terms of labour input is Π α i P y 1/φ A i n i ) 1 1/φ) W n i The labour demand function derived from the first-order condition is φ n i A φ 1 φ 1 α i P i y φ W The supply function for intermediate good i is φ x i A i n i A φ φ 1 α i P i y φ W Using 915a) above, p i α iw ρa i, which corresponds to equation 4) in the class note Summing up the labour demand of all firms, we get y vn, where N ) 1 φ 1 αi P v A i1 i Putting A i 1 for all i, it can be shown that p i v 1 N < 1 From equation 25), P is also linearly homogeneous in all the p i It follows that if all the input factors have the same inflation rate π t, then the final good has the same inflation rate as well Question 3 We have inflation rate in period t given by Therefore so that π t ρ1 γ) π t+1 ρ1 γ) E t π t+1 ρ1 γ) γ s E t p t+s p t 1 26) γ s E t+1 p t+s+1 p t, γ s E t p t+s+1 p t 27) Now we can rewrite equation 26) as π t ρ1 γ)p t p t 1 ) + ρ1 γ) γ s E t p t+s p t 1 28) s1 The summation above can be rewritten as γ s E t p t+s p t 1 s1 γ s+1 E t p t+s+1 p t 1 γ γ s E t p t+s+1 p t + γ γ s p t p t 1 γ γ s E t p t+s+1 p t + γ γ s π t Substitute this result into 28) and make use of 27), we have π t ρ1 γ)p t p t 1 ) + γe t π t+1 + ργπ t Solving for π t gives equation 923) in the textbook Question 6 Parts a) and b) are the same as class discussion c) Since the production function for x i is linearly homogeneous, the cost function is separable in input prices and output, that is, total cost is cp r, P w)x i P cr, w)x i P Cx i The firm s profit maximization problem is therefore ) φ P max p i x i P Cx i max p i P C) y p i p i The price p i can be solved from the first-order condition d) For cost minimization, the intermediate firms set the unit cost times the marginal product of labour equal to the wage rate, that is, P C x i L i P w Using the result from part c), this means P w 1 α)xi ρ, p i L i which gives the labour demand L i ρ1 α)p ix i P w ρ1 α) pi ) 1 φ y w P Next, find P in part b) for the case of N identical firms Substituting into L i and adding yields the result p i 14

Question 7 a) The FOC is 1 ρ) s E t p # t p t+s) 0, which implies, with some calculation, the formula for p # t b) The result in part a) can be written recursively as p # t ρp t + 1 ρ)e t p # t+1 Subtracting p t from both sides and rearranging gives p # t p t ρp t p t )+1 ρ)e t p # t+1 p t+1+1 ρ)e t π t+1 29) c) Average price in period t is p t ρp # t + 1 ρ)p t 1 30) Solve this difference equation backward to get the result d) Equation 30) can be written as p t p t 1 ρp # t p t + p t p t 1 ) Putting p t p t 1 π t gives π t e) Combining 29) and 31) gives π t ρ 1 ρ p# t p t ) 31) ρ2 1 ρ p t p t ) + 21 ρ)e t π t+1 Therefore inflation in period t is a linear combination of deviation of price from its optimal level and the expected inflation in period t + 1 Question 8 Compare the permanent income of the households before and after the announcement What will a household with rational expectations do in periods t and t + 1? See the New York Times article by Choe Sang-Hun for more details Review Questions Question 3 Using the first-order Taylor approximation of F k t+1 ) about the steady-state value k, equation 238) becomes q t+1 q 1 β q t q) F k)k t+1 k) 32) Similarly, the linearized version of equation 240a) is k t+1 k k φ q t q) + k t k) 33) Substitute k t+1 k in 33) to 32), we have 1 q t+1 q β kf k) q t q) F k)k t k) 34) φ Equations 34) and 33) can be written as a twodimensional linear dynamical system as required One of the eigenvalues of A is expected to be greater than 1 and the other less than 1 in absolute value Question 4 In the household budget constraint, there are three control variables, c t, b t+1, and m t+1 The household has to choose two so that the third one can be determined by the constraint Here we pick c t and m t+1 as the control variables and leave b t+1 as a state variable In period t, however, m t is already chosen in the last period so it is also a state variable To avoid confusion, we let z t m t in its role as a state variable The transition equation is therefore z t+1 m t+1 The utility maximization problem becomes max β t Uc t, z t ), c t,m t+1 subject to the transition equation zt+1 m t+1 b t+1 x t + 1 + R t )b t + z t c t /1 + π t+1 ) m t+1 The Bellman equation is { vz t, b t ) max Uc t, z t ) + βvm t+1, c t,m t+1 } x t + 1 + R t )b t + z t c t /1 + π t+1 ) m t+1 ) Let vz t, b t ) λ t λ z,t λ b,t T, the necessary conditions are Uc,t 0 1/1 + πt+1 ) λz,t+1 0 + β, 35) 0 1 1 0 and Uz,t 0 + β λ b,t+1 0 1/1 + πt+1 ) λz,t+1 λz,t 0 1 + R t )/1 + π t+1 ) λ b,t+1 Solving for λ t+1 in equation 35), we get λz,t+1 1 + π t+1)u c,t 1 λ b,t+1 β 1 Substitute λ t and λ t+1 into equation 36), we have Uz,t + U c,t 1 + π t)u c,t 1 1 1 + R t )U c,t β 1 The second line gives the Euler equation βu c,t U c,t 1 1 + R t 1 + π t 1, and the first line implies that U m,t U c,t R t λ b,t 36) 15

Tests 2012W, Test 2, Question 3 Since c t 1 is given in period t, it is effectively a state variable So let y t c t 1 The problem becomes max c t β t Uc t, y t ), t1 subject to the transition equation yt+1 c t w t+1 w t c t The Bellman equation is Let vy t, w t ) max c t { Uct, y t ) + βvc t, w t c t ) } vy t, w t ) The necessary conditions are and λy,t λ w,t U c,t + βλ y,t+1 λ w,t+1 ) 0, 37) Uy,t + β 0 0 0 0 1 Putting 38) into 37) to get so that λy,t+1 λ w,t+1 U c,t + βu y,t+1 λ w,t 0, λy,t λ w,t 38) λ w,t U c,t + βu y,t+1 39) Push forward one period, we have λ w,t+1 U c,t+1 + βu y,t+2 40) Substitute 39) and 40) into the second line of 38) and replace y t with c t 1, we obtain the Euler equation β Uc t+1, c t ) c t+1 + β 2 Uc t+1, c t ) c t Uc t, c t 1 ) c t + β Uc t, c t 1 ) c t 1 c 2013 The Pigman Inc All Rights Reserved 16