Permanent Income Hypothesis Intro to the Ramsey Model
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1 Consumption and Savings Permanent Income Hypothesis Intro to the Ramsey Model Lecture 10 Topics in Macroeconomics November 6, 2007 Lecture 10 1/18 Topics in Macroeconomics
2 Consumption and Savings Outline 2 Last Week and Yesterday Household s consumption and savings decision Determinants of household s savings (preferences, interest rate) Effect of capital gains taxes on savings behavior Today Multi period model and the permanent income hypothesis Next Week We will introduce the household problem into the growth model (Ramsey model) Lecture 10 2/18 Topics in Macroeconomics
3 Permanent income/life-cycle hypothesis: special case Permanent income/life-cycle hypothesis: in general Long horizons 3 In general, we can write V({c t }) = T t=1 ( 1 ) t 1u(ct ) 1 + ρ and the period t budget constraint (assuming r is constant) c t + a t+1 = y t + (1 + r)a t The intertemporal budget constraint T t=1 ( 1 ) t 1ct = a r T t=1 ( 1 ) t 1yt 1 + r Lecture 10 3/18 Topics in Macroeconomics
4 Permanent income/life-cycle hypothesis: special case Permanent income/life-cycle hypothesis: in general Household s maximization problem 4 Given {y t } T t=1 and r max {c t } T t=1 s.t. T t=1 T t=1 ( 1 ) t 1u(ct ) 1 + ρ ( 1 ) t 1ct = a r T t=1 ( 1 ) t 1yt 1 + r Same Euler Equation as in 2 period model (for every t = 1,..., T 1): u (c t ) = 1 + r 1 + ρ u (c t+1 ) Lecture 10 4/18 Topics in Macroeconomics
5 Permanent income/life-cycle hypothesis: special case Permanent income/life-cycle hypothesis: in general Solution 5 Consider the simplest case and assume ρ = r = 0. Then for all t = 1,..T 1, the Euler equation is: u (c t ) = u (c t+1 ) c 1 = c 2 =... = c T c Using this in the intertemporal budget constraint gives: c t = 1 T ( a 1 + T ) y τ c, all t τ=1 Lecture 10 5/18 Topics in Macroeconomics
6 Permanent income/life-cycle hypothesis: special case Permanent income/life-cycle hypothesis: in general The Permanent income/life-cycle hypothesis 6 Consider the simplest case and assume ρ = r = 0. Transitory income shock: income changes for 1 period, t Transitory shocks unimportant for consumption y t c = y t T Transitory shocks important for savings s t = y t c t = y t 1 T ( a 1 + T ) y τ τ=1 y t s t = y t y t T = (T 1) y t T Lecture 10 6/18 Topics in Macroeconomics
7 Permanent income/life-cycle hypothesis: special case Permanent income/life-cycle hypothesis: in general The Permanent income/life-cycle hypothesis 7 Consider the simplest case and assume ρ = r = 0. Permanent income shock: y t changes for ALL periods, t = 1,..., T Permanent shocks important for consumption y t + y c = y Permanent shocks unimportant for savings s t = y t + y t (c t + c) unchanged Lecture 10 7/18 Topics in Macroeconomics
8 Permanent income/life-cycle hypothesis: special case Permanent income/life-cycle hypothesis: in general Transitory versus permanent income disturbances 8 In general,... Transitory changes in disposable income have a minor impact on current consumption (consumption is smoother than income) Permanent changes in disposable income have a large effect on consumption Intertemporal substitution drives the effects of temporary changes in income Important distinction for real business cycles (temporary shocks) versus growth (permanent changes) Lecture 10 8/18 Topics in Macroeconomics
9 The Ramsey Model 9 Neoclassical model of the firm (Topics 1 & 2) Consumption-savings choice for consumers (Topic 3, Certainty) Solow model + incentives to save (recall example with taxes) Lecture 10 9/18 Topics in Macroeconomics
10 10 Markets and ownership Agents Firms produce goods, hire labor and rent capital Households own labor and assets (capital), receive wages and rental payments, consume and save Markets Inputs: competitive wage rates, w, and rental rate, R Assets: free borrowing and lending at interest rate, r Output: competitive market for consumption good Lecture 10 10/18 Topics in Macroeconomics
11 11 Firms / Representative Firm Seeks to maximize profits Profit = F(K, L) RK wl The FOCs for this problem deliver F K = R F L = w In per unit of labor terms, let f(k) F(k, 1) f (k) = R f(k) kf (k) = w Recall Euler s Theorem: factor payments exhaust output Lecture 10 11/18 Topics in Macroeconomics
12 12 Households / Representative household Preferences U 0 = Budget constraint β t u(c t ) t=0 c t + a t+1 = w t + (1 + r)a t, for all t = 0, 1, 2,... a 0 given Note: labor supplied inelastically, l t = 1 Lecture 10 12/18 Topics in Macroeconomics
13 13 Households / Representative household Intertemporal version of budget constraint t=0 s=0 t ( r s ) c t = a 0 + t=0 s=0 t ( r s We rule out that debt explodes (no Ponzi games) a t+1 B for some B big, but finite ) w t More compactly, PDV(c) = a(0) + PDV(w) Lecture 10 13/18 Topics in Macroeconomics
14 14 Household s problem max (a t+1,c t ) t=0 s.t. β t u(c t ) t=0 c t + a t+1 = w t + (1 + r)a t, for all t = 0, 1, 2,... a t+1 = B for some B big, but finite a 0 given Lecture 10 14/18 Topics in Macroeconomics
15 15 Euler equation In general, u (c t ) = β(1 + r t+1 )u (c t+1 ) From here on, CES utility, u(c) = c1 σ 1 σ, Euler eqn. becomes, ( ) σ ct+1 = β(1 + r t+1) c t Lecture 10 15/18 Topics in Macroeconomics
16 16 Transversality condition HH do not want to end up with positive values of assets lim t βt u (c t )a t 0 HH cannot think they can borrow at the end of their life lim t βt u (c t )a t 0 Hence, lim t βt u (c t )a t = 0 Lecture 10 16/18 Topics in Macroeconomics
17 Definition of 17 A competitive equilibrium is defined by sequences of quantities of consumption, {c t }, capital, {k t }, and output, {y t }, and sequences of prices, {w t } and {r t }, such that Firms maximize profits Households maximize U 0 subject to their constraints Goods, labour and asset markets clear Lecture 10 17/18 Topics in Macroeconomics
18 Characterizing Quantities* 18 k t+1 + c t = f(k t ) + (1 δ)k t c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/θ lim t βt u (c t )k t = 0 k 0 > 0 Lecture 10 18/18 Topics in Macroeconomics
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