Equilibrium in a Production Economy

Prof. Eric Sims

University of Notre Dame

Fall 2014

Sims (ND) Equilibrium in a Production Economy Fall 2014 1 / 22 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 2014 2 / 22 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 2014 3 / 22 Firm

Produce output using Yt = At F (Kt , Nt )

Take real wage, wt , as given

Time subscript on At : allow it to change period-to-period 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 2014 4 / 22 Capital Accumulation

Same equation as before, with one twist:

Kt+1 = qIt + (1 − δ)Kt q: investment-specific productivity. Measure of how good we are at transforming investment into capital One way to think about financial system health Assume it is the same in t and t + 1, differently than At (1−δ)Kt+1 Terminal condition: Kt+2 = 0 ⇒ It+1 = − q . Intuition.

Sims (ND) Equilibrium in a Production Economy Fall 2014 5 / 22 Proﬁts and Firm Value

Proﬁt: Πt = Yt − wt Nt − It Firm value: present value of proﬁt/dividend:

1 Vt = Πt + Πt+1 1 + rt

Firm: picks Nt , Nt+1, and Kt+1 to maximize Vt

Sims (ND) Equilibrium in a Production Economy Fall 2014 6 / 22 Firm First Order Conditions

Optimality conditions:

wt = At FN (Kt , Nt )

wt+1 = At+1FN (Kt+1, Nt+1) 1 1 = (qAt+1FK (Kt+1, Nt+1) + (1 − δ)) 1 + rt Intuition: marginal beneﬁt = marginal cost

Sims (ND) Equilibrium in a Production Economy Fall 2014 7 / 22 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 At goes up

Labor demand would shift in if Kt were destroyed (natural disaster)

Sims (ND) Equilibrium in a Production Economy Fall 2014 8 / 22 Investment Demand

The last ﬁrst order condition implicitly deﬁnes an investment demand curve

Investment a decreasing function of rt

Curve shifts out if At+1 or q go up

Curve also shifts out if Kt goes down exogenously (natural disaster) Investment fundamentally forward-looking

Sims (ND) Equilibrium in a Production Economy Fall 2014 9 / 22 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 − Nt , where Nt is hours worked

Household gets utility from leisure via v(1 − Nt ), with 0 00 v (1 − Nt ) > 0 and v (1 − Nt ) ≤ 0 Lifetime utility:

U = u(Ct ) + v(1 − Nt ) + β (u(Ct+1) + v(1 − Nt+1))

Sims (ND) Equilibrium in a Production Economy Fall 2014 10 / 22 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

Ct + St = wt Nt − Tt + Πt

Ct+1 = wt+1Nt+1 − Tt+1 + Πt+1 + (1 + rt )St

Combine into one:

Ct+1 wt+1Nt+1 − Tt+1 + Πt+1 Ct + = wt Nt − Tt + Πt + 1 + rt 1 + rt

Sims (ND) Equilibrium in a Production Economy Fall 2014 11 / 22 Household First Order Conditions

Household chooses Ct , Ct+1, Nt , and Nt+1 to maximize lifetime utility. Optimality conditions:

0 0 u (Ct ) = β(1 + rt )u (Ct+1) 0 0 v (1 − Nt ) = u (Ct )wt 0 0 v (1 − Nt+1) = u (Ct+1)wt+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 2014 12 / 22 Labor Supply

0 0 Condition v (1 − Nt ) = u (Ct )wt implicity deﬁnes labor supply curve Can analyze in indiﬀerence curve-budget line diagram

Changes in wt : complicated effect because offsetting income and substitution effects

Assume that substitution eﬀect dominates: Nt increasing in wt

Simple rational: MPC is less than 1, so Ct reacts less than one-for-one to one period change in wt Easy to see with log utility over consumption

Labor supply will shift with anything which aﬀects Ct other than wt s To make life easy, assume that only thing that shifts N is rt : higher s rt , N shifts out

Sims (ND) Equilibrium in a Production Economy Fall 2014 13 / 22 The Government

Same as before. Gt and Gt+1 chosen exogenously Government’s intertemporal budget constraint:

Gt+1 Tt+1 Gt + = Tt + 1 + rt 1 + rt

Ricardian Equivalence holds: household behaves as though government balances budget every period

Sims (ND) Equilibrium in a Production Economy Fall 2014 14 / 22 Equilibrium Conditions

d Labor demand: N = N(wt , At , Kt ) s Labor supply: N = N(wt , rt )

Consumption: Ct = C (Yt − Gt , Yt+1 − Gt+1, rt )

Investment: It = I (rt , q, At+1, Kt )

Production function: Yt = At F (Kt , Nt )

Market-clearing: Yt = Ct + It + Gt

Sims (ND) Equilibrium in a Production Economy Fall 2014 15 / 22 The Y s Curve

Set of (rt , Yt ) pairs consistent with production function where labor market clears Basic idea of derivation: s Start with an initial rt . Determines a position of N Try a higher rt . Leads to labor supply shifting out. Higher Nt → higher Yt s Hence, Y slopes up – higher rt eﬀectively makes people want to work more, and hence supply more output

Sims (ND) Equilibrium in a Production Economy Fall 2014 16 / 22 The Y d Curve

d Set of (rt , Yt ) pairs consistent with agent optimization and Yt = Yt , d where Yt = Ct + It + Gt Basic idea of derivation: Use the expenditure line - 45 degree line diagram. Start with an rt , determines position of expenditure line Increase rt . Causes expenditure line to shift down – both because of Ct and It . Intersects 45 degree line at lower point d Hence, Yt slopes down

Sims (ND) Equilibrium in a Production Economy Fall 2014 17 / 22 General Equilibrium

General equilibrium requires that all markets clear Eﬀectively two markets here: labor (Ns = Nd ) and goods (Y d = Y ) Labor market-clearing: on Y s curve Goods market-clearing: on Y d curve General equilibrium: on both curves d s Real interest rate, rt , links “goods market” (Y − Y ) with “labor market” (Nd − Ns )

Sims (ND) Equilibrium in a Production Economy Fall 2014 18 / 22 Equilibrium: Graphically

d Yt

Yd(r)

Yt rt Ys Wt Ns 0 rt 0 Wt Yd Nd

Yt Nt Y Yt t Yt=AF(Kt,Nt) Yt=Yt

45° 0 N 0 Yt t Nt Yt

Sims (ND) Equilibrium in a Production Economy Fall 2014 19 / 22 Curve Shifts

Effectively five exogenous variables: At , At+1, q, Gt , and Gt+1 What shifts what: Labor demand: shifts if either At increases or Kt declines suddenly (natural disaster) Note caveats about effects of At , q, Gt and their indirect effects on Yt+1! Goods demand: shifts if At+1, q, Gt , or Gt+1 change

Sims (ND) Equilibrium in a Production Economy Fall 2014 20 / 22 Analyzing Eﬀects of Changes in Exogenous Variables

Follow cookbook approach: Start in labor market. See if Nt would change for a given rt . Tells you if Y s curve shifts Figure out if Y d curve shifts Combine to ﬁnd new equilibrium (rt , Yt ) Figure out what happens to components of Yt Work back to labor market to make quantities line up

Sims (ND) Equilibrium in a Production Economy Fall 2014 21 / 22 Qualitative Eﬀects

Variable: ↑ At ↑ At+1 ↑ q ↑ Gt ↑ Gt+1 Output + + + + - Hours ? + + + - Consumption + ? ? - - Investment + ? + - + Real interest rate - + + + - Real wage + - - - +

Sims (ND) Equilibrium in a Production Economy Fall 2014 22 / 22