equilibrium in a production economy

Equilibrium in a Production Economy

Prof. Eric Sims

University of Notre Dame

Fall 2013

Sims (ND) Equilibrium in a Production Economy Fall 2013 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 theactual behavior of the economy in the short run

Sims (ND) Equilibrium in a Production Economy Fall 2013 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-takingbehavior, can study representative agent problem

Time lasts for two periods: present, t, and future, t + 1

Sims (ND) Equilibrium in a Production Economy Fall 2013 3 / 22

Firm

Produce output using Yt = AtF (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 andmake capital accumulation (investment) decisions

Would get same results if household owned capital stock as in SolowModel

Sims (ND) Equilibrium in a Production Economy Fall 2013 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 capitalOne way to think about financial system healthAssume it is the same in t and t + 1, differently than At

Terminal condition: Kt+2 = 0 ⇒ It+1 = − (1−δ)Kt+1

q . Intuition.

Sims (ND) Equilibrium in a Production Economy Fall 2013 5 / 22

Profits and Firm Value

Profit: Πt = Yt − wtNt − It

Firm value: present value of profit/dividend:

Vt = Πt +1

1 + rtΠt+1

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

Sims (ND) Equilibrium in a Production Economy Fall 2013 6 / 22

Firm First Order Conditions

Optimality conditions:

wt = AtFN(Kt ,Nt)

wt+1 = At+1FN(Kt+1,Nt+1)

1 =1

1 + rt(qAt+1FK (Kt+1,Nt+1) + (1− δ))

Intuition: marginal benefit = marginal cost

Sims (ND) Equilibrium in a Production Economy Fall 2013 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 2013 8 / 22

Investment Demand

The last first order condition implicitly defines an investment demandcurve

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 2013 9 / 22

Household

Problem basically the same, but now household chooses amount oflabor/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), withv ′(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 2013 10 / 22

Budget Constraints

Basically look same, but have to account for endogenous income now

Household income comes from wages, dividend/profit from firm, andpays taxes to government

Ct + St = wtNt − Tt + Πt

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

Combine into one:

Ct +Ct+1

1 + rt= wtNt − Tt + Πt +

wt+1Nt+1 − Tt+1 + Πt+1

1 + rt

Sims (ND) Equilibrium in a Production Economy Fall 2013 11 / 22

Household First Order Conditions

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

u′(Ct) = β(1 + rt)u′(Ct+1)

v ′(1−Nt) = u′(Ct)wt

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 2013 12 / 22

Labor Supply

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 andsubstitution effects

Assume that substitution effect dominates: Nt increasing in wt

Simple rational: MPC is less than 1, so Ct reacts less thanone-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

To make life easy, assume that only thing that shifts Ns is rt : higherrt , N

s shifts out

Sims (ND) Equilibrium in a Production Economy Fall 2013 13 / 22

The Government

Same as before. Gt and Gt+1 chosen exogenously

Government’s intertemporal budget constraint:

Gt +Gt+1

1 + rt= Tt +

Tt+1

1 + rt

Ricardian Equivalence holds: household behaves as thoughgovernment balances budget every period

Sims (ND) Equilibrium in a Production Economy Fall 2013 14 / 22

Equilibrium Conditions

Labor demand: Nd = N(wt ,At ,Kt)

Labor supply: Ns = N(wt , rt)

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

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

Production function: Yt = AtF (Kt ,Nt)

Market-clearing: Yt = Ct + It + Gt

Sims (ND) Equilibrium in a Production Economy Fall 2013 15 / 22

The Y s Curve

Set of (rt ,Yt) pairs consistent with production function where labormarket clears

Basic idea of derivation:

Start with an initial rt . Determines a position of Ns

Try a higher rt . Leads to labor supply shifting out. Higher Nt →higher Yt

Hence, Y s slopes up – higher rt effectively makes people want to workmore, and hence supply more output

Sims (ND) Equilibrium in a Production Economy Fall 2013 16 / 22

The Y d Curve

Set of (rt ,Yt) pairs consistent with agent optimization and Y dt = Yt ,

where Y dt = Ct + It + Gt

Basic idea of derivation:

Use the expenditure line - 45 degree line diagram. Start with an rt ,determines position of expenditure lineIncrease rt . Causes expenditure line to shift down – both because of Ct

and It . Intersects 45 degree line at lower pointHence, Y d

t slopes down

Sims (ND) Equilibrium in a Production Economy Fall 2013 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

Real interest rate, rt , links “goods market” (Y d − Y s) with “labormarket” (Nd −Ns)

Sims (ND) Equilibrium in a Production Economy Fall 2013 18 / 22

Equilibrium: Graphically

Wt

Wt0

Yt

Nt0 Nt

Nt Yt

Yt0

45°

Yt

Yt

Yt

rt0

rt

Ytd

Yd(r)

Ys

Yd

Yt=Yt Yt=AF(Kt,Nt)

Nd

Ns

Sims (ND) Equilibrium in a Production Economy Fall 2013 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 onYt+1!

Goods demand: shifts if At+1, q, Gt , or Gt+1 change

Sims (ND) Equilibrium in a Production Economy Fall 2013 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 youif Y s curve shiftsFigure out if Y d curve shiftsCombine 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 2013 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 2013 22 / 22