Technical Appendix. Resolution of the canonical RBC Model. Master EPP, 2011

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1 Technical Appendix Resolution of the canonical RBC Model Master EPP, 2011

2 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)

3 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

4 Consumers Intertemporal utility Budget constraint Lagrange Multiplier

5 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

6 Aggregate equilibrium Using the FOC of the firm yielding equality between marginal productivity of capital and the rental rate

7 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

8 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

9 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

10 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

11 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)

12 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)

13 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

14 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

15 Ex.: Log-linearization of the resources constraints Terms by terms = Thus Since Eventually,

16 We get three linearized conditions One jump-variable C, two state variables K and Z Blanchard Khan conditions (1980): One root of W outside the circle, two roots inside

17 Solution takes the form With Vector of control variables (C here) Vector of state variables (K and Z here) Impulse response function Draw a shock in a normal distribution At period 1: Shock of 1 At horizon j

18 Illustration: assume a shock Technological shock Quarters

19 Propagation mechanism

20 4. Dynamic programing

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