ECON 5118 Macroeconomic Theory Winter 013 Test 1 February 1, 013 Answer ALL Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages for formal answers and the left-side pages for your rough work Remember to put your name on the front page 1 Suppose that f : R! R is defined as f(x) = log(1 + x) Also, let x, y, and z be the growth rates of X t,y t, and Z t respectively in period t (a) Find the first-order Taylor approximation of f at the point x = 0 (b) Show that x = log X t log X t 1 (c) Show that if Z t = X t Y t,then z = x + y Suppose that the household s utility function in each period t is U(c t,l t )=c 1 t l t, 0 < < 1, where c t is aggregate consumption and l t is leisure time The production function is F (k t,n t )=Akt n 1 t, 0 < < 1, 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 3 Consider the first-order di erence equation x t+1 = ax t + b t, t =0, 1,, (a) Suppose that {x t } is bounded and Find the backward solution for x t 1 <a<1 (b) Suppose that b t = c (a constant) for every t Find the fixed point(s) of the system 4 Consider the following comments by Paul Krugman: As I see it, the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth Until the Great Depression, most economists clung to a vision of capitalism as a perfect or nearly perfect system That vision wasn t sustainable in the face of mass unemployment, but as memories of the Depression faded, economists fell back in love with the old, idealized vision of an economy in which rational individuals interact in perfect markets, this time gussied up with fancy equations Write a short essay discussing Krugman s opinion 5 In the Tobin q model two necessary conditions for utility maximization are and i t = 1 (q t 1)k t, > 0, F 0 (k t+1 )= U 0 (c t ) U 0 (c t+1 ) q t (1 )q t+1 1 (q t+1 1) where =1/(1 + ) is the social discount rate, c t is consumption, i t is investment, k t is aggregate capital stock, and q t is the Tobin q in period t The capital accumulation equation is k t+1 = i t +(1 )k t, where is the depreciation rate of capital Show that in the steady state the marginal product of capital is greater than +
ECON 5118 Macroeconomic Theory Winter 013 Test March 1, 013 Answer ALL Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages for formal answers and the left-side pages for your rough work Remember to put your name on the front page You may find the information in the Appendix useful 1 Consider an economy with a Cobb-Douglas production function: Y t =(1+µ) t K t N 1 t Define e ective labour as N # t = (1 + µ) t/(1 ) N t The capital accumulation equation can be written as (1 + )k # t+1 = s(k# t ) +(1 )k # t, where s is the exogenous saving rate and = n + µ/(1 ) (a) Derive an expression for the growth rate of capital per e ective labour, (b) Find the steady-state value k # (c) What is the impact of an exogenous increase of the saving rate s on k #? Figure 1 on the next page depicts the growth history of four countries In view of the empirical evidence, write a short essay on the assumptions, key results, and implications of the following growth models: (a) the Solow-Swan model, (b) the optimal growth model, (c) the endogenous growth model 3 According to the US Survey of Consumer Finances released in June 01, the median family wealth and income, in constant dollar, declined from 007 to 010: Year Net Asset Income 007 16,400 49,600 010 77,300 45,800 (a) Explain why both wealth and income decline for the median family (b) What would the e ect on the median household consumption according to the permanent income hypothesis? (c) The survey also reports that consumer spending has remained surprisingly resilient Explain 4 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 U(b t ), where b t is measured in the unit of blood per person, and U is increasing and concave The population of the villagers, h t, grows at a constant rate n Although Dracula is immortal, he is being pursued by a group of vampire hunters There is a probability of (0, 1) in each period that the vampire hunters will annihilate him (a) What is the dynamic resource constraint faced by Dracula? (b) Assuming that Dracula is an expected utility maximizer, set up the maximization problem and derive the Bellman equation (c) Derive the Euler equation for preying on his victims Is there a steady-state solution? 5 Let the production function of a firm be F (k t,n t )=Akt n 1 t, A > 0, 0 < <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
Figure 1: Growth History (a) Set up the optimization problem facing the firm s manager What are the control variables? (b) Write down the Lagrangian and 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 Appendix Geometric Series For Also, 1 <x<1, ax s = a 1 x 1 (1 + r) s = 1+r, r 1 (1 + r) s = 1 r Logarithmic Approximation For small values of x, log(1 + x) ' x Let f(x) be any di erentiable function The log-linear approximation of f(x) around a point x is where ˆx = log x log x f(x) ' f(x )+x f 0 (x )ˆx First-Order Di erence Equation Suppose that {x t } is a sequence of real variables which satisfies the first-order di erence equation x t+1 = ax t + b, t =0, 1,,, where a and b are constant and For the non-constant coe lim x t+n = b n!1 1 a cient equation 1 <a<1 Then x t+1 = ax t + b t, t =0, 1,,, if a 1 or a apple 1, then x t = 1 a On the other hand, if x t = a s = 1 <a<1, then a s 1 b t s a s+1 For x around 1, log x ' x 1
ECON 5118 Macroeconomic Theory Winter 013 Test 3 March, 013 Answer All Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages for formal answers and the left-side pages for your rough work Remember to put your name on the front page You may find the information in the Appendix useful 1 Table 1 on the next page shows the dynamics of bonding financing of a permanent increase of g t in period t 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 Figure 1 shows the fiscal stance of selected OECD countries in 010 The second column shows the OECD s forecast for each country s net debt-to-gdp ratio in 010 The third column measures the gap between bond yields on debt of average maturity for each country and the OECD s forecasts for growth in 010 and 011 Compare Ireland and Japan in the following questions: (a) Which country s fiscal position is unsustainable in 010? Explain (b) What should the country or countries do to address the problem? (c) The fourth column lists the average time to maturity of outstanding government debt How does this a ect the financial risk of the countries? 3 Suppose that a linear autonomous dynamical system in R is x t+1 = f(x t ) The matrix representation of f with respect to the standard basis is Figure 1: Data for Question A = apple 1 1 (a) What is the fixed point of the system?
Table 1: 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+ t + n 1: g t 1 + g t + Rb t + R P n 1 = T t 1 + b t+n (b) Explain in details that if the fixed point is a sink, a source, or a saddle point 4 Consider a money demand model with household utility function given by log c t, and budget constraint (1+ t+1 )b t+1 b t +(1+ t+1 )m t+1 m t +c t = x t +R t b t Suppose that c t = m t for all t (a) Identify the state and control variables and set up the utility maximization problem (b) Derive the Bellman equation (c) Derive the necessary conditions (d) Derive the money demand function from the Euler equation 5 In the transaction cost approach to money demand, the necessary conditions for household utility optimization implies that in the steady state, where c + m + T (c, m) x b =0, (1) T m (c, m)+r =0, () T (0,m)=0, T c 0,T cc 0,T m apple 0,T mm 0,T mc apple 0 (a) Use equation (1) to show that @m @c = 1+T c + T m (b) Use equation () to show that @m/@c 0 Provide an economic interpretation for the above results Appendix Geometric Series For 1 <x<1, ax s = a 1 x Also, 1 (1 + r) s = 1+r, r 1 (1 + r) s = 1 r Logarithmic Approximation For small values of x, For x around 1, log(1 + x) ' x log x ' x 1 Let f(x) be any di erentiable function The log-linear approximation of f(x) around a point x is where ˆx = log x log x f(x) ' f(x )+x f 0 (x )ˆx First-Order Di erence Equation Suppose that {x t } is a sequence of real variables which satisfies the first-order di erence equation x t+1 = ax t + b, t =0, 1,,, where a and b are constant and For the non-constant coe lim x t+n = b n!1 1 a cient equation 1 <a<1 Then x t+1 = ax t + b t, t =0, 1,,, if a 1 or a apple 1, then x t = 1 a a s = On the other hand, if 1 <a<1, then x t = a s 1 b t s a s+1
ECON 5118 Macroeconomic Theory Winter 013 Final Examination April, 013 Answer All Questions Time Allowed: 3 hours Attention: Please write your answers on the answer book provided Use the right-side pages for formal answers and the left-side pages for your rough work All mathematical formulations must be explained clearly in English Read the questions carefully and do not include materials that are not required Remember to put your name on the front page You may find the information in the Appendix useful 1 In the centralized model with capital installation costs, the key results can be summarized by two dynamic equations q t q = (q t+1 q)+ [F 0 (k t+1 ) F 0 (k)], k t+1 = (q t q + )k t, where is the social discount factor, is a positive constant relating to the installation costs, q t is the Tobin q and k t is the aggregate capital stock in period t, with q and k being their respective steady-state values (a) Show that the linearized version can be expressed in the form x t+1 = Ax t with x t =(q t q, k t k) T, and find A (b) What is the fixed point of the system? (c) What do you expect the eigenvalues of A to be? Explain Consider an economy with the following production function: Y t =(1+µ) t Kt Nt 1, 0 < <1 (a) Explain whether the function exhibits constant returns to scale (b) Express the 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 Explain if this result holds for the above production function 3 According to the US Survey of Consumer Finances released in June 01, the median family wealth and income, in constant dollar, declined from 007 to 010: Year Net Asset Income 007 16,400 49,600 010 77,300 45,800 (a) Explain why both wealth and income decline for the median family (b) What would the e ect on the median household consumption according to the permanent income hypothesis? (c) The survey also reports that consumer spending has remained surprisingly resilient Explain 4 Let x t+1 = f(x t ) be an autonomous dynamical system on a convex and compact set in R n Show that Fix(f) is a closed set if f is continuous 5 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 6 Suppose that there are many identical firms in the market Each firm has a probability of that it can adjust the price of its product to the optimal level p t in any period t When a firm have the opportunity to change its price it chooses a p # t to minimize where 1 [ (1 )] s E t hp # t pt+si, is a discount factor (a) Let = (1 ) Find p # t and show that it is a linear combination of present and expected future optimal prices (b) Show that the general price level in the whole economy is 8 Let a two-state Markov chain be apple apple apple (t+1)1 p11 p = 1 t1, (t+1) p 1 p t where p ij = Prob(x t+1 = e j x t = e i ), i,j =1,, and t =[ t1 t ] T is the probability distribution of the two states in each period t (a) Find the eigenvalues of the system (b) What are the implications of the results in part (a) 9 Write a short critique of the dynamic stochastic general equilibrium model in macroeconomics p t = (1 ) s E t p t+s +(1 )p t 1 (c) Show that the inflation rate is t = 1 1 [ (1 )(p t p t 1 )+ E t t+1 ] 7 An aggregate firm produces the final good y with a CES technology y =! 1/ NX i x i, 0 6= <1, i=1 NX i =1, i=1 where y is the final output Each of the N intermediate inputs x i with price p i is produced by a monopoly using only labour input Households consume the single final good with price P and supply labour n with wage rate W (a) Set up the aggregate firm s profit maximization problem and derive the necessary condition (b) Derive the conditional demand function for input i in terms of the elasticity of substitution = 1/(1 ) and aggregate output y (c) Assuming a competitive market and therefore zero profit, show that P = NX i p 1 i i=1! 1/(1 ) (d) What is the e ect on P if inflation for all the N inputs are of the same rate Explain
Appendix For the non-constant coe cient equation Geometric Series x t+1 = ax t + b t, t =0, 1,,, For 1 <x<1, if a 1 or a apple 1, then ax s = a 1 x x t = 1 a a s = a s+1 Also, On the other hand, if 1 <a<1, then 1 (1 + r) s = 1+r, r 1 (1 + r) s = 1 r x t = a s 1 b t s Logarithmic Approximation For small values of x, For x around 1, log(1 + x) ' x log x ' x 1 Let f(x) be any di erentiable function The log-linear approximation of f(x) around a point x is where ˆx = log x log x f(x) ' f(x )+x f 0 (x )ˆx First-Order Di erence Equation Suppose that {x t } is a sequence of real variables which satisfies the first-order di erence equation x t+1 = ax t + b, t =0, 1,,, where a and b are constant and 1 <a<1 Then lim x t+n = b n!1 1 a 3