Technical Appendix. Resolution of the canonical RBC Model. Master EPP, 2010
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1 Technical Appendix Resolution of the canonical RBC Model Master EPP, 2010
2 Questions What are the causes of macroeconomic fluctuations? To what extent optimal intertemporal behavior of households in a walrasssian world can account for fluctuations? or do we need some sources of imperfections: real?; nominal? How can we modelize agents behavior in a stochastic framework ( uncertainty, anticipations, intertemporal choice) How do we test the empirical relevance of the different theory for fluctuations
3 Starting point: Walrassian model To what extent a walrassian model (baseline Ramsey model) without imperfection neither heterogeneity can account for fluctuations? Good starting point: stress what would be the imperfections and sources of heterogeneity required to bridge the gap with the data
4 Two changes to the baseline walrassian Ramsey model: - Shocks to capture uncertainty and unexpected events Natural candidate: productivity shocks for capturing good and bad times (what about government shock, taste shock ) -Intertemporal behavior to get a propagation mechanism: Consumption/Saving trade-off
5 New representation of fluctuations: - Impulse/Propagation framework : Frish (1933) and Slutsky (1937) Fluctuations seen as propagation mechanisms of initial impulse shocks Ex. real business cycles: Fluctuations as optimal answers to real technological shocks
6 Clear caveats Nothing to say about unemployment, money, coordination failures, heterogeneity and inequalities But still a good starting point - Understanding agents behavior under uncertainty: consumption smoothing (labor-leisure trade-off, investment ) - Get used with resolution and methodological techniques to handle stochastic models and check their empirical relevance
7 1. Basic real business cycle model: Productivity shock and Consumption/saving Trade-off 1.1 Agents behavior Set-up Infinitively-lived households with same preferences Each supplies one unit of labor inelastically in a competitive labor market at wage w Aggregate shock to productivity: saving motive to smooth consumption by accumulating capital. Capital is rented out to firm in a competitive financial market at rate r Net supply of bonds is zero (can be ignored) Representative agents: no idiosyncratic income risks (or perfect insurance assumption)
8 Firms Firms have a Cobb-Douglas production function where : - - is an aggregate productivity shock is total employment (=1 since competitive market and inelastic labor supply) Firms rent capital and labor to households. Profits Optimal static decision problem
9 Consumers Intertemporal utility Budget constraint Lagrange Multiplier
10 First order condition: Euler equation!! Here the interest rate is time-varying: depends on the aggregate productivity shock Arbitrage condition: - Decrease consumption by one unit today: loss of utility - Invest to get of consumption tomorrow - Worth in utility terms - At the optimum, indifference between the two
11 Aggregate equilibrium Using the FOC of the firm yielding equality between marginal productivity of capital and the rental rate
12 1.2 Social planner problem Walrassian economy: equivalence between Pareto optimum of a social planner and the competitive equilibrium Social planner problem + I I Lagrangian
13 First order conditions at time t Conditions equivalent to the decentralized problem - Intertemporal dimension of consumption: opportunity cost of consumption depends on expected next period marginal value of capital - Marginal value of capital depends on next period marginal productivity and marginal value of capital one period ahead By substituting FOCs and using we still get the traditional Keynes-Ramsey
14 2. Effect of a productivity shock: first insights What would be the impact and the propagation mechanism of a productivity shock in the economy? Qualitative/ Quantitative answer Two potential opposite effects -Wealth effect: higher current and future output without increasing capital stock. Lead to an increase in consumption -Substitution effect: intertemporal transfer of higher current return on capital to smooth consumption. Lead to an increase in savings
15 Net effect on consumption might depend on key parameters: - Elasticity of substitution of consumption - Transitory or Permanent shock: if transitory, C up less and S,I up more -But potential positive co-movement between C, I, Y (Good news compared to a taste shock ) Quantitative assessment is tough: -Non linear system of stochastic difference equations under rational expectations -In general, no analytical solution, need to rely on numerical method
16 Analytical solution: a nice case Assume full-depreciation Assume a log-utility (Implications?) Cobb-Douglas technology Solution Both consumption and investment increases with a positive productivity shock ( positive co-movement empirically relevant) Response independent of expectations of the productivity shock and whether the shock is transitory or permanent (hunches: log)
17 3. Linearization Solving the model would be easier if the FOC were linear Key tool in the literature: log-linearization of the economy around the steady state (log: elasticity show up)
18 3.1 Steady state Non-stochastic steady state : Z constant and Stationary capital Capital accumulation equation becomes with Cobb-Douglas Stationary consumption Resource constraint yields
19 3.2 Linearization procedure Quick refresher Taylor approximation of a function f around point a First-order approximation Similarly with two variables Let denote the deviation of a variable from its steady state x
20 Ex.: Log-linearization of the resources constraints Terms by terms = Thus Since Eventually,
21 We get three linearized conditions One jump-variable C, two state variable K and Z Blanchard Khan conditions (1980): One root of W outside the circle, two roots inside
22 Solution takes the form With Vector of control variables (C here) Vector of state variables (K and Z here) Impulse response function Slutsky effect: Propagation mechanism of an initial shock Draw a shock in a normal distribution At period 1: Shock of 1 At horizon j
23 Illustration: assume a shock Technological shock Quarters
24 Propagation mechanism
25 4. Dynamic programing
26
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