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Lecture 4 A Dynamic Model Noah Williams University of Wisconsin Madison Economics 312 Spring 2014 Williams Economics 312 Dynamic Model Now start our analysis of dynamic general equilibrium models which we will continue in the rest of the class Today start with optimal allocations solving social planner problem Later consider equilibrium analysis In going from static to dynamic model the main difference is savings and investment Households no longer consume all income each period save some for future consumption or borrow against future income Firm no longer have fixed capital stock on hand each period may choose to invest in order to build up future capital or disinvest to allow future capital to fall Williams Economics 312 Household Preferences Representative household lives infinite number of periods Utility function V0 U c0 U c1 2 U c2 3 U c3 X t U ct t 0 ct is consumption at date t 0 1 discount factor measures household s degree of 1 impatience Define 1 where is discount rate Preferences over c0 c1 satisfy the conditions discussed previously i e monotonicity U 0 0 and convexity U 00 0 Williams Economics 312 More on Preferences V0 X t U ct t 0 Abstract from labor lesiure tradeoff for now Inelastic labor supply work full time h hours yields no utility Consumption smoothing partially offset by discounting Assume all ct are normal more income more consumption at each date t From vantage point of date 0 marginal utility of ct V0 MUct t U 0 ct ct Intertemporal marginal rate of substitution measures willingness to substitute consumption over time MRSct ct 1 Williams MUct U 0 ct MUct 1 U 0 ct 1 Economics 312 Technology Continue to abstract from labor for now Assume h 1 is supplied inelastically Then production is yt F kt 1 F kt where production function is same as before Note since N 1 fixed diminishing marginal returns in k F 0 k 0 F 00 k 0 For technical reasons also assume Inada conditions lim F 0 k k 0 Williams lim F 0 k 0 k Economics 312 Investment Firms can now invest in order to expand future productivity Capital depreciates at rate and investment at t increases kt 1 kt 1 1 kt it We abstract from government spending so the feasibility or goods market clearing condition now includes investment and consumption yt ct it Combining equations gives us the tradeoff between consumption and capital ct F kt kt 1 1 kt Williams Economics 312 Steady States In general kt ct yt it will vary over time But let s look for a steady state where they are constant From the previous expression this implies c F k k 1 k F k k In the steady state consumption equals output minus replacement investment k Williams Economics 312 y c dk F k dK max c k k Output replacement investment and consumption Williams Economics 312 The Golden Rule Allocation We now consider the social planner s problem to determine the optimal allocation We first focus on a simple objective suppose that the planner wanted to maximize utility in the steady state This is known as the Golden Rule allocation as it treats consumption at all dates equally max U c c k subject to c F k k Since U c strictly increasing this is equivalent to max c s t c F k k First order condition determines golden rule capital k F 0 k Williams Economics 312 c c F k dk k k Net output Williams Economics 312 Optimal Allocation While the golden rule gives the maximal amount of steady state consumption in general it is not optimal If households are impatient 1 then they value current consumption more that future consumption So the timing of consumption matters So now let s consider the optimal allocation max ct kt X t U ct t 0 subject to ct F kt kt 1 1 kt t k0 given Williams Economics 312 Characterizing the Optimal Allocation Form the Lagrangian with multipliers t on the constraints L max ct kt X t U ct t F kt kt 1 1 kt ct t 0 First order conditions for any ct and for kt 1 t 0 t U 0 ct t t t 1 F 0 kt 1 1 0 Note that if there were a finite terminal date T we would have kT 1 0 Consume everything in last date For infinite horizon problem need a similar condition known as transversality condition lim T U 0 cT kT 1 0 T Value in utility terms of capital goes to zero at infinity Williams Economics 312 The Euler Equation Combine the two optimality conditions to get U 0 ct U 0 ct 1 F 0 kt 1 1 This is known as an Euler equation and is a key condition for optimality in dynamic models Can also be written MRSct ct 1 U 0 ct F 0 kt 1 1 U 0 ct 1 Here F 0 kt 1 1 is the slope of the production possibility frontier for ct ct 1 To see this note ct 1 F kt 1 kt 2 1 kt 1 F F kt ct 1 kt kt 2 1 F kt ct 1 kt Take derivative with respect to ct F 0 kt 1 1 Williams Economics 312 c t 1 c t 1 U c t b U c t 1 c t c t 1 r t 1 Optimal choice Williams Economics 312 More on The Euler Equation Can also interpret F 0 kt 1 as holding period return rt 1 on capital U 0 ct U 0 ct 1 1 rt 1 Recall that U is concave so U 00 0 or in other words U 0 c is decreasing So if 1 rt 1 1 U 0 ct U 0 ct 1 ct ct 1 1 rt 1 1 U 0 ct U 0 ct 1 ct ct 1 1 rt 1 1 U 0 ct U 0 ct 1 ct ct 1 Behavior of consumption over time depends on rate of time preference relative to interest rate If equal perfect consumption smoothing Williams Economics 312 Optimal Steady State Look for a steady state of the optimal allocation U 0 c U 0 c F 0 k 1 Or recalling that 1 1 F 0 k 1 1 From the previous expression we also have c F k k The optimal steady state is only equal to the golden rule if 0 And since F 00 k 0 we have F 0 k F 0 k k k Williams Economics 312 y F k dK c dk c dk k k k Optimal steady state consumption and capital Williams Economics 312 c c F k dk c k k k Optimal steady state and golden rule Williams Economics 312 An Example Now work out a parametric example using standard functional forms Cobb Douglas production y Ak For preferences set U c c 1 1 1 For 0 Interpret 1 as U c log c These imply the Euler equation 1 ct ct 1 …

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