Equilibrium in a Production Economy

Transcription

1 Equilibrium in a Production Economy Prof. Eric Sims University of Notre Dame Fall 2012 Sims (ND) Equilibrium in a Production Economy Fall / 23

2 Production Economy Last time: studied equilibrium in an endowment economy Now: study equilibrium in an economy with production Will produce operational model that can be used to compare to the actual behavior of the economy in the short run Sims (ND) Equilibrium in a Production Economy Fall / 23

3 Equilibrium Definition still the same: set of prices and quantities consistent with (i) agents optimizing, taking prices as given, and (ii) markets clearing Agents: household, firm, government Large number of each kind of agent, but identical: price-taking behavior, can study representative agent problem Time lasts for two periods: present, t, and future, t + 1 Sims (ND) Equilibrium in a Production Economy Fall / 23

4 Firm Produce output using Y t = AF (K t, N t ) Take real wage, w t, as given Different than Solow model, assume that firms own capital stock and make capital accumulation (investment) decisions Would get same results if household owned capital stock as in Solow Model Sims (ND) Equilibrium in a Production Economy Fall / 23

5 Capital Accumulation Same equation as before: K t+1 = I t + (1 δ)k t Terminal condition: K t+2 = 0 I t+1 = (1 δ)k t. Intuition. Sims (ND) Equilibrium in a Production Economy Fall / 23

6 Profits and Firm Value Profit: Π t = Y t w t N t I t Firm value: present value of profit/dividend: V t = Π t r t Π t+1 Firm: picks N t, N t+1, and K t+1 to maximize V t Sims (ND) Equilibrium in a Production Economy Fall / 23

7 Firm First Order Conditions Optimality conditions: w t = A t F N (K t, N t ) w t+1 = A t+1 F N (K t+1, N t+1 ) 1 = 1 (A t+1 F K (K t+1, N t+1 ) + (1 δ)) 1 + r t Intuition: marginal benefit = marginal cost Sims (ND) Equilibrium in a Production Economy Fall / 23

8 Labor Demand First two first order conditions imply labor demand curves Labor demand is static : depends only on current period stuff Decreasing in the real wage Labor demand shifts out if A t goes up Labor demand would shift in if K t were destroyed (natural disaster) Sims (ND) Equilibrium in a Production Economy Fall / 23

9 Investment Demand The last first order condition implicitly defines an investment demand curve Investment a decreasing function of r t Curve shifts out if A t+1 goes up Curve also shifts out if K t goes down exogenously (natural disaster) Investment fundamentally forward-looking Sims (ND) Equilibrium in a Production Economy Fall / 23

10 Household Problem basically the same, but now household chooses amount of labor/leisure Normalize total endowment of time to 1 each period Leisure is 1 N t, where N t is hours worked Household gets utility from leisure via v(1 N t ), with v (1 N t ) > 0 and v (1 N t ) 0 Lifetime utility: U = u(c t ) + v(1 N t ) + β (u(c t+1 ) + v(1 N t+1 )) Sims (ND) Equilibrium in a Production Economy Fall / 23

11 Budget Constraints Basically look same, but have to account for endogenous income now Household income comes from wages, dividend/profit from firm, and pays taxes to government C t + S t = w t N t T t + Π t C t+1 = w t+1 N t+1 T t+1 = Π t+1 + (1 + r t )S t Combine into one: C t + C t r t = w t N t T t + Π t + w t+1n t+1 T t+1 + Π t r t Sims (ND) Equilibrium in a Production Economy Fall / 23

12 Household First Order Conditions Household chooses C t, C t+1, N t, and N t+1 to maximize lifetime utility. Optimality conditions: u (C t ) = β(1 + r t )u (C t+1 ) v (1 N t ) = u (C t )w t v (1 N t+1 ) = u (C t+1 )w t+1 Consumption Euler equation: same as it ever was Two new conditions: implicitly define labor supply curves Sims (ND) Equilibrium in a Production Economy Fall / 23

13 Labor Supply Condition v (1 N t ) = u (C t )w t implicity defines labor supply curve Can analyze in indifference curve-budget line diagram Changes in w t : complicated effect because offsetting income and substitution effects Draw Frisch labor supply curve: how does N t vary with w t, holding u (C t ) (hence C t ) fixed Must be upward sloping. Shifts whenever C t changes Sims (ND) Equilibrium in a Production Economy Fall / 23

14 The Government Same as before. G t and G t+1 chosen exogenously Government s intertemporal budget constraint: C t + C t r t = T t + T t r t Ricardian Equivalence holds: household behaves as though government balances budget every period Sims (ND) Equilibrium in a Production Economy Fall / 23

15 Equilibrium Conditions Labor demand: N d = N(w t, A t, K t ) Labor supply: N s = N(C t, w t ) Consumption: C t = C (Y t G t, Y t+1 G t+1, r t ) Investment: I t = I (r t, A t+1, K t ) Production function: Y t = A t F (K t, N t ) Market-clearing: Y t = C t + I t + G t Sims (ND) Equilibrium in a Production Economy Fall / 23

16 The Y s Curve Set of (r t, Y t ) pairs consistent with production function where labor market clears Basic idea of derivation: Start with an initial r t. Determines a position of N s through C t Try a higher r t. Leads to lower C t, labor supply shifts out. Higher N t higher Y t Hence, Y s slopes up higher r t effectively makes people want to work more, and hence supply more output Sims (ND) Equilibrium in a Production Economy Fall / 23

17 The Y d Curve Set of (r t, Y t ) pairs consistent with agent optimization and Yt d = Y t, where Yt d = C t + I t + G t Basic idea of derivation: Use the expenditure line - 45 degree line diagram. Start with an r t, determines position of expenditure line Increase r t. Causes expenditure line to shift down. Intersects 45 degree line at lower point Hence, Yt d slopes down Sims (ND) Equilibrium in a Production Economy Fall / 23

18 General Equilibrium General equilibrium requires that all markets clear Effectively two markets here: labor (N s = N d ) and goods (Y d = Y ) Labor market-clearing: on Y s curve Goods market-clearing: on Y d curve General equilibrium: on both curves Sims (ND) Equilibrium in a Production Economy Fall / 23

19 Equilibrium: Graphically Sims (ND) Equilibrium in a Production Economy Fall / 23

20 Curve Shifts Effectively four exogenous variables: A t, A t+1, G t, and G t+1 What shifts what: Labor demand: shifts if either A t increases or K t declines suddenly (natural disaster) Labor supply: shifts if anything changes C t other than things which affect current Y t : r t, A t+1, G t, and G t+1 Goods demand: shifts if A t+1, G t, or G t+1 change Sims (ND) Equilibrium in a Production Economy Fall / 23

21 Analyzing Effects of Changes in Exogenous Variables Follow cookbook approach: Start in labor market. See if N t would change for a given r t. Tells you if Y s curve shifts Figure out if Y d curve shifts Combine to find new equilibrium (r t, Y t ) Figure out what happens to components of Y t Work back to labor market to make quantities line up Sims (ND) Equilibrium in a Production Economy Fall / 23

22 Resolving Ambiguities Sometimes the curve shifts will produce ambiguities These can often be resolved by looking at the equations Labor market clearing condition particularly useful: v (1 N t ) = u (C t )A t F N (K t, N t ) Under our standard assumptions, this means that N t and C t must move in opposite direction if neither A t nor K t move Sims (ND) Equilibrium in a Production Economy Fall / 23

23 Qualitative Effects Variable: A t A t+1 G t G t+1 Output +? + + Hours?? + + Consumption +? - - Investment +? - + Real interest rate Real wage +? - - Sims (ND) Equilibrium in a Production Economy Fall / 23