Lecture 4A The Decentralized Economy I

Lecture 4A The Decentralized Economy I From Marx to Smith Economics 5118 Macroeconomic Theory Kam Yu Winter 2013

Outline 1 Introduction 2 Consumption The Consumption Decision The Intertemporal Budget Constraint Interpreting the Euler Equation Consumption Function Permanent and Temporary Shocks 3 Life-Cycle Theory Implications Model of Perpetual Youth 4 Durable Goods 5 References Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 2 / 36

The Deciders Introduction Instead of a social planner making all decisions, a more realistic model involves households and firms making their own decisions with different objectives. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 3 / 36

Consumption The Consumption Decision The Intertemporal Utility Maximization Problem The representative household s problem is max c t+s,a t+s+1 subject to the budget constraints β s U(c t+s ), (4.1) s=0 a t+s+1 + c t+s = x t+s + r t+s a t+s, s = 0, 1,..., (4.2) where a t, x t, and r t are exogenous asset, income, and interest rate respectively in period t. The choice variables are c t and a t+1. Once c t is chosen, a t+1 is determined by the budget constraint. Therefore the household is effectively choosing the consumption path {c t, c t+1, c t+2,...}. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 4 / 36

Optimization Consumption The Consumption Decision The Lagrangian is { } L t = β s U(c t+s ) + λ t+s [x t+s + (1 + r t+s )a t+s c t+s a t+s+1 ] s=0 The first-order conditions are L t c t+s = β s U (c t+s ) λ t+s = 0, s 0, L t a t+s = λ t+s (1 + r t+s ) λ t+s 1 = 0, s 1, and the budget constraint (4.2). The Euler equation is (4.3) βu (c t+1 ) U (1 + r t+1 ) = 1. (4.4) (c t ) Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 5 / 36

Consumption The Intertemporal Budget Constraint Two-Period Budget Consider the budget constraints in periods t and t + 1: a t+1 + c t = x t + (1 + r t )a t, a t+2 + c t+1 = x t+1 + (1 + r t+1 )a t+1. The two equations can be combined to eliminate a t+1 to give a t+2 1 + r t+1 + c t+1 1 + r t+1 + c t = x t+1 1 + r t+1 + x t + (1 + r t )a t. (4.5) The left-hand side is the present value of total expenditure in consumption and investment. The right-hand side is the present value of total incomes. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 6 / 36

Consumption Extension to n 1 Periods The Intertemporal Budget Constraint By successive substitution, the present value of total expenditure is W t = n 1 a t+n n 1 s=1 (1 + r t+s) + c t+s s s=1 u=1 (1 + r t+u) + c t. (4.7) Similarly, the present value of total income is W t = n 1 s=1 x t+s s u=1 (1 + r t+u) + x t + (1 + r t )a t. (4.8) The infinite horizon budget constraint, assuming that interest rate is constant at r, is obtained as n : W t = s=0 c t+s (1 + r) s = x t+s (1 + r) s + (1 + r)a t. (4.9) s=0 Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 7 / 36

Consumption The Intertemporal Budget Constraint Permanent Income W t is the present value of life-time income of the household and is often called permanent income. Equation (4.9) can be used a the alternative single budget constraint in the optimization problem. One more necessary condition called the transversality condition: lim s βs a t+s U (c t+s ) = 0. (4.10) This ensures that the terminal value of the asset has no utility. In particular, the household cannot finance extra consumption indefinitely by borrowing (a t < 0). Therefore it is sometimes called no-ponzi-scheme. See Kamihigashi (2006) for details. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 8 / 36

Consumption Intertemporal Substitution Interpreting the Euler Equation The Euler equation has the usual meaning that it prescribes the 4. The Decentralized Econom intertemporal substitution of consumption between to consecutive periods. c t + 1 c* t + 1 V t = U(c t ) +βu(c t + 1 ) c t * c 1 + r t + 1 t Figure 4.1. Two-period solution. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 9 / 36

Consumption Optimal Consumption Path Consumption Function Using a first-order Taylor approximation of U (c t+1 ) about c t, we have U (c t+1 ) U (c t ) U (c t ) + U (c t )(c t+1 c t ) U (c t ) = 1 + U (c t ) U (c t ) c t+1 = 1 σ c t+1 c t, (4.15) where σ = cu /U is the coefficient of relative risk aversion (see Jehle and Reny, 2011, p. 123). Putting (4.15) into the Euler equation (4.4) gives c t+1 = 1 [ ] 1 1 = r t+1 θ c t σ β(1 + r t+1 ) σ(1 + r t+1 ) If r = θ, consumption growth is zero. (4.16) Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 10 / 36

Consumption Function Consumption Consumption Function From (4.9) with constant interest rate and the steady-state consumption c t, we have W t = = s=0 s=0 c t+s (1 + r) s = 1 + r r c t x t+s (1 + r) s + (1 + r)a t Therefore the consumption function is c t = r 1 + r W t = r s=0 x t+s (1 + r) s+1 + ra t. (4.17) This means that in steady state consumption in each period is a fixed portion of total wealth. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 11 / 36

Empirical Evidence Consumption Consumption Function Slacalek: What Drives Personal Consumption? Figure 1: Consumption Growth and Wealth Growth 1994 2002 Consumption Growth.05.1.15.2.25.3 AUT JAP GER CAN ITA DEN USA BEL FIN SWE!1 0 1 2 3 4 Wealth Growth x W/C Note: Consumption growth and rescaled wealth growth between 1994Q4 and 2002Q4; wealth growth is rescaled by multiplying with the wealth consumption ratio of 1994Q4. Slope of the regression line, MPC LR w = 0.032, t-statistic: 2.36, p-value: 0.018. FRA NED UK SPA 1 Introduction Source: Jiri Slacalek (2009) What Drives Personal Consumption? The Role of Housing and Financial Wealth, The B.E. Journal of Macroeconomics, Figure9(1), 1 plots Article consumption 37. growth in major industrial countries against wealth Kam Yu (LU) growth multiplied with Lecture the 4A wealth consumption The Decentralized Economy ratio. 1 It I suggests that larger Winter 2013 12 / 36

Income Shocks Consumption Permanent and Temporary Shocks Suppose income x t is constant in every period. Then (4.17) becomes c t = x t + ra t. (4.18) A permanent shock in income will therefore have the same effect on consumption, which is the Keynesian idea of marginal propensity to consume (MPC) equals to 1. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 13 / 36

Consumption Permanent and Temporary Shocks Income Shocks Suppose income x t is constant in every period. Then (4.17) becomes c t = x t + ra t. (4.18) A permanent shock in income will therefore have the same effect on consumption, which is the Keynesian idea of marginal propensity to consume (MPC) equals to 1. To see the effect of a temporary shock, rewrite (4.17) as c t = r 1 + r x r t + (1 + r) 2 x t+1 + + ra t. A temporary income shock in period t, therefore only have a MPC equals to r/(1 + r). Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 13 / 36

Consumption Permanent and Temporary Shocks Income Shocks Suppose income x t is constant in every period. Then (4.17) becomes c t = x t + ra t. (4.18) A permanent shock in income will therefore have the same effect on consumption, which is the Keynesian idea of marginal propensity to consume (MPC) equals to 1. To see the effect of a temporary shock, rewrite (4.17) as c t = r 1 + r x r t + (1 + r) 2 x t+1 + + ra t. A temporary income shock in period t, therefore only have a MPC equals to r/(1 + r). Therefore the effects of a permanent tax cut and a temporary tax cut are very different. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 13 / 36

Interest Rate Shocks Consumption Permanent and Temporary Shocks 1 Permanent shocks Recall equation (4.18): c t = x t + ra t. Consumption therefore changes depending on the household has asset (a t > 0) or debt (a t < 0). MPC = 1. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 14 / 36

Consumption Permanent and Temporary Shocks Interest Rate Shocks 1 Permanent shocks Recall equation (4.18): c t = x t + ra t. Consumption therefore changes depending on the household has asset (a t > 0) or debt (a t < 0). MPC = 1. 2 Temporary shocks Recall equation (4.8): W t = n 1 s=1 x t+s s u=1 (1 + r t+u) + x t + (1 + r t )a t. The effect of a temporary change in period t again depends on whether a t is positive or negative. The effect on consumption, however, is small due to intertemporary substitution. MPC = r/(1 + r). Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 14 / 36

Consumption Permanent and Temporary Shocks Interest Rate Shocks 1 Permanent shocks Recall equation (4.18): c t = x t + ra t. Consumption therefore changes depending on the household has asset (a t > 0) or debt (a t < 0). MPC = 1. 2 Temporary shocks Recall equation (4.8): W t = n 1 s=1 x t+s s u=1 (1 + r t+u) + x t + (1 + r t )a t. The effect of a temporary change in period t again depends on whether a t is positive or negative. The effect on consumption, however, is small due to intertemporary substitution. MPC = r/(1 + r). 3 In practice, consumption depends of expectations of future asset returns. See Shiller (2009). Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 14 / 36

Consumption Permanent and Temporary Shocks Anticipated and Unanticipated Income Shocks Confusion about (4.16) and (4.17): c t+1 c t = r t+1 θ σ(1 + r t+1 ), c t = r s=0 x t+s (1 + r) s+1 + ra t. The first equation does not contain information on income. If r t+1 = θ, then c t+1 = c t. But if income change is anticipated, the information is embedded in c t. That is, c t+1 = c t + e t+1, where e t+1 is a random shock (E t [e t+1 ] = 0). In this case consumption is called a martingale process (E t [c t+1 ] = c t ). Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 15 / 36

Saving Consumption Permanent and Temporary Shocks For constant interest rate, saving is Substitute ra t c t from (4.17), s t = x t + ra t c t. s t = x t r = r 1 + r = s=1 s=0 x t+s (1 + r) s+1 s=1 x t+s x t (1 + r) s x t+s (1 + r) s+1 This means that saving is used to offset anticipated future change in income. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 16 / 36

Life-Cycle Theory Implications Happily Ever After? Our models so far do not address the problem of consumer heterogeneity. Young households need to borrow, middle-age households want to save, and mature households just spend. Our consumption function smooth expenditure throughout the life cycle. Assumptions: 1 Life is infinite and predictable 2 No borrowing constraint What are the empirical evidences? Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 17 / 36

Life-Cycle Theory 3000 Implications 2000 Income and Consumption in the 1000 U.S. 8000 7000 6000 5000 4000 3000 2000 1000 0 1950 1960 1970 1980 1990 2000 4. The Figure Decentralized 4.4. U.S. total Economy real consumption and real disposable income 194 RDPI RCONS 0 1950 1960 1970 1980 1990 2000 RDCONS 0 1950 1960 1970 1980 1990 2000.4. U.S. total real consumption and real disposable Figure income 4.5. U.S. 1947 2003. total, nondurable, and durable real consumption 1947 7000 6000 5000 4000 3000 2000 1000 RCONS RNDCONS RDPI = real disposable income, RCONS = real consumption, 7000 Consequently, their consumption would be limited to their curren RNDCONS = nondurables, RDCONS = durables Their 6000 RCONS consumption would therefore tend to fluctuate with their incom than be smoothed over time as life-cycle theory predicts. What does 5000 evidence show? Figures 4.4 and 4.5 give a rough guide. Figure 4.4 4000 posable (after-tax) income against total consumption for the United the period 1947 2003. Figure 4.5 shows total, real nondurable, an 3000 Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 18 / 36

A modification to life-cycle theory that explicitly recognizes the household s finite Kam Yu tenure (LU) on life is the Lecture Blanchard 4A The and Decentralized Fischer (1989) Economyand I Yaari (1965) theory Winter 2013 19 / 36 Life-Cycle Theory Durables and Nondurables 4.4. Life-Cycle Theory Implications 67 Table 4.1. Standard deviations in growth rates. Disposable income 1.12 Total consumption 0.74 Nondurables and services 1.22 Nondurables 0.60 Durables 17.13 % standard deviation of nondurable consumption is only 3.5% of that of durable expenditures. We conclude that there is evidence of consumption smoothing, but only of nondurable expenditures. Neither service nor durable expenditures appear to be smoothed relative to income. We note, however, that unlike nondurable and service expenditures, durable expenditures are not a flow variable but a stock; it is the services from the durable stock that are a flow. Strictly speaking, therefore, the theory derived above does not apply to durables. We therefore reexamine the determination of durable consumption below. 4.4.2 Model of Perpetual Youth

Life-Cycle Theory Model of Perpetual Youth Who Wants to Live Forever? The probability of death in each period is ρ (independent of age). The probability of staying alive is 1 ρ. The probability of having a life of s periods long is f (s) = ρ(1 ρ) s 1, s = 1, 2,... Life expectancy is E[s] = E[s] if ρ 0. sρ(1 ρ) s 1 = ρ 1. s=1 Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 20 / 36

The Power of Posterity Life-Cycle Theory Model of Perpetual Youth Under uncertainty, a household in period t plans ahead with the probability of being alive in period t + s equals to (1 ρ) s. Life time expected utility is therefore [ ] E β s U(c t+s ) = β s (1 ρ) s U(c t+s ) s=0 where β = β(1 ρ). = s=0 β s U(c t+s ), s=0 Our models still work, with a different interpretation of β. See Brooks (2009) for an alternative explanation. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 21 / 36

Durable Goods Durable Consumption Durable goods are similar to capital, providing consumption services over time with depreciation (δ). Let D t be stock of durables, d t new investment in durables, and c t consumption of nondurables. Durable accumulation equation: The household budget constraint is D t+1 = d t + (1 δ)d t. (4.21) a t+1 + c t + p D t d t = x t + (1 + r t )a t, where p D t is the price of durables relative to nondurables. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 22 / 36

Durable Goods Utility Maximization Problem The household s problem is max β s U(c t+s, D t+s ) s=0 subject to a t+s+1 + c t+s + pt+s[d D t+s+1 (1 δ)d t+s ] = x t+s + (1 + r t+s )a t+s. (4.22) Notes: U(c, D) is assumed to be differentiable, increasing, and concave. The control variables are c t+s, D t+s+1, and a t+s+1. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 23 / 36

First-Order Conditions Durable Goods The Lagrangian is L t = s=0 { β s U(c t+s, D t+s ) + λ t+s [ xt+s + (1 + r t+s )a t+s c t+s p D t+s[d t+s+1 (1 δ)d t+s ] a t+s+1 ] }. First-Order Conditions: L t c t+s = β s U c,t+s λ t+s = 0, s 0, L t D t+s = β s U D,t+s + λ t+s p D t+s(1 δ) λ t+s 1 p D t+s 1 = 0, s 1, L t a t+s = λ t+s (1 + r t+s ) λ t+s 1 = 0, s 1. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 24 / 36

Durable Goods Euler Equation The first and the third FOCs give the Euler equation βu c (c t+1, D t+1 ) (1 + r t+1 ) = 1, U c (c t, D t ) which relates the intertemporal marginal rate of time preference in nondurable consumption to β and r. Notice that it depends on the levels of the durable stock. Examples are ice cream and refrigerators, gasoline and cars, Internet services and computers, data plans and smart phones, cable services and HDTVs, fire insurance and houses, gym memberships and degrees etc. (These are all complements, can you think of some substitutes?) Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 25 / 36

Durable Goods Substitution Between Durable and Nondurable Combining all three FOCs gives U D (c t+1, D t+1 ) = U c(c t, D t ) β Equation (4.23) can be written as [ pt D 1 δ ] pt+1 D. (4.23) 1 + r t+1 βu D (c t+1, D t+1 ) U c (c t, D t ) = p D t 1 δ 1 + r t+1 p D t+1. The left hand side defines the marginal rate of substitution between durable in period t + 1 and nondurable in period t. The right hand side is the relative price. The first term is the purchasing price of durable. The second term is resale value in the next period net of depreciation and interest cost. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 26 / 36

Parametric Analysis Durable Goods Let U(c t, D t ) = c α t D 1 α t. The Euler equation is Equation (4.23) becomes ( ct+1 D t+1 ) α = ( ) ct+1 /D (1 α) t+1 β (1 + r t+1) = 1. (4.24) c t /D t ( ) α α 1 [ ct pt D 1 δ ] pt+1 D. (4.25) β(1 α) D t 1 + r t+1 Therefore an increase in interest rate r t+1 reduces the purchases of durables D t+1 relative to nondurables c t+1. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 27 / 36

Durable Goods Steady-State Solution In the steady state, Hence (4.25) becomes c t+1 = c, D t+1 = D, p D t+1 = p D t = p D, c p D D = In terms of investment of durables, c p D d = r t+1 = θ. α (θ + δ). 1 α α 1 α θ + δ. δ Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 28 / 36

Short-run Dynamics Durable Goods In the short-run the stock of durables is fixed. Multiply equation (4.21) by p D t /c t, we get pt D d t = pd t D t+1 (1 δ) pd t D t c t c t c t = c ( ) ( ) t+1 ct /D t p D t D t (1 δ) pd t D t c t c t+1 /D t+1 c t c t [ ( ) ] ct+1 ct /D t p D = 1 + δ t D t c t c t+1 /D t+1 c t From the Euler equation (4.24) we have c t /D t c t+1 /D t+1 = [β(1 + r t+1 )] 1/(1 α) = ( ) 1 + 1/(1 α) rt+1. 1 + θ Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 29 / 36

Short-run Dynamics Durable Goods Combining the last two equations gives [ pt D ( ) d t c t+1 1 + 1/(1 α) rt+1 pt = 1 + δ] D D t. c t c t 1 + θ c t Apply the log approximation x 1 log x to the above, we finally get p D t d t c t [ ct+1 1 ] p D c t 1 α (r t+1 θ) + δ t D t. c t Since p D t d t /c t 0 and p D t D t /c t > 0, the expression inside the square bracket is nonnegative. For example, if r t+1 rises high above the steady-state value of θ, nondurable consumption must increase in period t + 1 to a level such that c t+1 c t 1 1 α (r t+1 θ) δ. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 30 / 36

Durable Goods Empirical Evidence Under the permanent income hypothesis consumption is a martingale. Using the above framework, Mankiw (1982) shows that durable consumption follows an ARMA(1, 1) process. He tests the model with U.S. post-war data using an additively separably utility function in c and D, the model was rejected. In particular, the estimated depreciation rate δ is 1.038. Other economists suggest that consumers do not constantly adjust their stock of durable goods. For example, we buy houses, cars, computers, etc. only occasionally. Purchases of these items follow what is called an (S, s) rule. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 31 / 36

(S, s) Policy Durable Goods Source: Caplin and Leahy (2010) Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 32 / 36

Durable Goods Mechanics of the (S, s) Model A household has an optimal level of durable goods within an interval (s, S). But it does not buy new goods until the stock falls below the lower bound s. This is due to the presence of a fixed cost when buying new durable goods. Some writers argue that the aggregate demand may smooth out the individual household cycles. But the effect of a large aggregate income shock depends on the distribution of existing stock and may create complicated dynamic pattern such as coupled oscillations. The (S, s) model has been applied to other problems such as inventory control, money demand, capital investment, marketing, household finance, and monetary policy analysis. See Caplin and Leahy (2010) for a survey. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 33 / 36

Durable Goods Evidence from the Great Recession Source: Hall (2010) Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 34 / 36

Car Sales Durable Goods Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 35 / 36

References References Brook, David (2009) The Power of Posterity, The New York Times, July 27, 2009. Caplin, Andrew and John Leahy (2010) Economic Theory and the World of Practice: A Celebration of the (S, s) Model, Journal of Economic Perspectives, 24(1), 183 202. Hall, Robert E. (2010) Why Does the Economy Fall to Pieces after a Financial Crisis? Journal of Economic Perspectives, 24(4), 3 20. Jehle, Geoffrey A. and Philip J. Reny (2011) Advanced Microeconomic Theory, Third edition, Essex: Pearson Education Limited. Kamihigashi, Takashi (2006) Transversality Conditions and Dynamica Economic Behavior, in The New Palgrave Dictionary of Economics, 2nd Edition. Mankiw, N. Gregory (1982) Hall s Consumption Hypothesis and Durable Goods, Journal of Monetary Economics, 10, 417 425. Shiller, Robert J. (2009) Unlearned Lessons from the Housing Bubble, The Economists Voice, 6(7), Article 6. Kam Yu (LU) Lecture 4A The Decentralized Economy I Winter 2013 36 / 36