notes iii - labor supply, firm optimization, production...

Notes III - Labor Supply, Firm Optimization,Production, and Equilibrium

Julio GarınIntermediate Macroeconomics

Fall 2019

General Equilibrium

I Level of analysis:1. Decision theory.

I How agents make decisions, given prices.

2. Partial equilibrium.I How does a price clears one market, taking other prices as

given.

3. General equilibrium.I How do all prices work to clear all markets simultaneously.

I Several endogenous variables have been treated as exogenous.I Let’s start changing that.

Intermediate Macroeconomics Labor Supply, Firm Optimization, Production, and Equilibrium 2 / 49

Labor Supply


To Start: Understanding Consumption/Leisure Decisions

I What are the main mechanism driving labor/leisure decisions?

I What are the income and substitutions effects?


Outline of Topics and Relevant Material

I Introducing a desire for leisure.

I Defining the set of possibilities: budget constraint.

I Household’s problem.I Analyzing the household’s problem in a static setting.

I Substitution and income effect and the Frisch labor supply.

1. Chapter 12 GLS: Production, Labor Demand, Investment,and Labor SupplyI Section 12.2.


Household

I Problem the same as before, except...I We are endogenizing labor/leisure decisions.

I We normalize the total endowment of time to 1 in eachperiod.I This implies that leisure, l , is 1−N, where N is hours worked.

I We will assume that the consumption and leisure componentsof utility are separable.

I Households get utility from leisure via u(c , 1−N).I ul > 0 and ul ,l < 0.

I Lifetime utility given by:

U = u(C , 1−N + βu(C ′, 1−N ′

).


Household: Budget Constraint

I Now we have to account for endogenous income.I Wages.I Dividends/profits from firms.I Households also pay taxes to the government.

C + S = wN − T + ΠC ′ = w ′N ′ − T ′ + Π′ + S(1 + r).

I Intertemporal budget constraint:

C +C ′

1 + r= wN − T + Π +

w ′N ′ − T ′ + Π′

1 + r.


Household Problem

maxC ,C ′,N,N ′

U = u(C , 1−N) + βu(C ′, 1−N ′)

Subject to:

C +C ′

1 + r= wN − T + Π +

w ′N ′ − T ′ + Π′

1 + r.

Or...

maxC ′,N,N ′

U =u

(wN − T + Π +

w ′N ′ − T ′ + Π′

1 + r− C ′

1 + r, 1−N

)+ βu(C ′, 1−N ′).

I Before solving this, let’s some intuition from a static model.


Optimal Decisions in a Static, One-Period Problem

maxC ,N

u(C , 1−N)

Subject to:C = wN + Π− T

Or...maxN

u(wN + Π− T , 1−N)

I FOC:uL = wuC

I Relative price = MRS.I It is also the slope of the budget constraint.I This implicitly defines a labor supply curve.

I The optimal choices should satisfy the FOC and the BC.

I Remember (Π is D in GLS).


Analyzing the Optimal Decisions in a Static Problem

I Can analyze this in the indifference curve-budget line diagram.

I An ↑ Π or ↓ T have a pure income effect.I Individual responds by consuming more goods and leisure.

I The question of greatest interest is how leisure behaves wrt tochanges in wage.

I Offsetting effects:I Income effect lower labor supply.I Substitution effect raises labor supply.

I Draw Frisch labor supply curve: How does N vary with w ,holding uC constant.I Must be upward sloping and shifts whenever C changes.

I What happens if ↑ r?I What happens if ↓ G or ↓ G ′?


Household Problem

maxC ′,N,N ′

U =u

(wN + Π− T +

w ′N ′ − T ′ + Π′

1 + r− C ′

1 + r, 1−N

)+ βu

(C ′, 1−N ′

)I Optimality conditions:

C : uC = β(1 + r)uC ′

N : uL = uCw

N ′ : uL′ = UC ′w′.

I Identical to the conditions from the static one-period problem.

I In each period one has to decide optimal split betweenconsumption goods and leisure —essentially a static decision.

I Labor supply is obtained as before.


Firm:Production, Investment, and

Labor Demand


What we will be covering?

I How can we represent the production process?

I We will formalize the firm’s decisions.

I How is investment affected by changes in productivity?



I The production function.

I Firm’s problem.

I Labor demand.

I Investment Demand.

1. Chapter 12 GLS: Production, Labor Demand, Investment,and Labor SupplyI Section 12.1.


Production Function

I There is an aggregate production function:

Yt =

I Where At represents technology and it’s a measure ofproductivity

I Properties of F (·):I Increasing in both factors of production.

I FK (·) > 0.I FN (·) > 0.

I Diminishing returns to each factor:I FKK (·) < 0.I FNN (·) < 0.

I Constant returns to scale in K and N:I F (λKt , λNt ) = λF (Kt ,Nt ).

I What would be its graphical representation?


Example: Cobb-Douglas

Yt = AtKαt N

1−αt with 0 < α < 1

I Let’s check the properties:I Increasing in K and N.I Diminishing marginal products.I Constant returns.


Firm

I Uses an increasing and concave c.r.s. production function thatrequires capital and labor:

Y = AF (K ,N)

I Take real wage, w , as given.

I Owns the capital stock and make investment decisions.I Capital goods differ from consumption goods.

I Resources can be transferred across time.I When households save, saving is lent to firms, that then buy

capital goods.I More items can be produced in the future.

I We will assume that capital can be converted back intoconsumption.I Only a two-period model and we don’t want capital to be

wasted.


About Inputs

I Capital:I Capital must be produced.I It depreciates at a constant rate, δ.I Denominated in units of goods.

I Labor:I Supplied by households.


Capital Accumulation, Firm Profit, and Firm Value

I Capital accumulation:

K ′ = qI + (1− δ)K

I Terminal condition: K ′′ = 0 ⇒ I ′ = −(1− δ)K ′.

I Profits (or Dividends):

Π = Y − wN − I

I Firm value: present value of lifetime profits/dividends:

V = Π +1

1 + rΠ′

I Saving at interest rate r is another way of transferringresources into the future.

I Firm chooses investment, capital, and labor to maximize V .


Firm’s Problem

maxN,N ′,I ,K ′

V = zF (K ,N)−wN− I +1

1 + r

[A′F (K ′,N ′)− w ′N ′ +

(1− δ)

q′K ′

]Subject to

K ′ = I + (1− δ)K

Which can also be expressed as,

maxN,N ′,K ′

zF (K ,N)−wN− K ′

q+

(1− δ)

qK +

1

1 + r

[A′F (K ′,N ′)−w ′N ′ +

(1− δ)

q′K ′

]

I FOCs.

I N: ∂V/∂N = 0I N ′: ∂V/∂N ′ = 0I K ′: ∂V/∂K ′ = 0


First Order Conditions

I Optimality Conditions.

∂V

∂N= 0⇔ AFN(K ,N) = w

∂V

∂N ′= 0⇔ A′FN ′(K

′,N ′) = w ′

∂V

∂K ′= 0⇔ 1 =

1

1 + r

[A′FK ′(K

′,N ′) + (1− δ)q

q′

]I Intuition?

I Marginal benefit = Marginal cost.

I Note that firm could:I Save and earn the market rate.I Invest and earn the MPK minus depreciation.

I In the Solow model, the firm rental rate covered both interestand depreciation costs.


Labor Demand

I The first two first order conditions imply labor demand curves.

Nd (w ,A,K )

I Labor demand (those equations) are “static”.I Depend only on current period variables.

I Labor demand decreasing function of real wage.I Labor demand shifts out if ↑ A.

I Technological improvement.

I Labor demand would shift if in if ↓ K .I Natural disaster.


Investment DemandI We can deduce that

K ′ = Kd (r ,A′, q)

I The last FONC implies an investment demand curve.

I (r ,A′, q,K )

I Investment is a decreasing function of r .

I Curve shifts out if ↑ A.

I Investment demand would shift out in if ↓ K .I Investment is fundamentally forward-looking.

I It depends importantly on expected future productivity.

I We express the demand as a function of the variables the firmtakes as given.I We don’t express it as a function of N ′.

I What about labor supply?


Back to the Basics:Equilibrium in an Endowment

Economy


What we will be covering?

I Defining a competitive equilibrium.

I Endogenizing the interest rate.

I Thinking about supply and demand ‘shocks.’



I The representative agent (RA).

I Competitive Equilibrium.

I Y d and Y s curves.

I Adjustment to shocks.

I What is the real interest measuring?

I Adding Government.

1. Chapter 11 GLS: Equilibrium in an Endowment EconomyI Section 11.3.


Environment

I Economy populated by many identical agents.I Normalize to one:

I The representative agent.

I Agent lives for two periods and solves standardconsumption-saving problem.I Takes income as given.

I Endowment economy.

I For now, we are not going to pay attention to the“supply-side” of the economy.


Equilibrium

Definition (Competitive Equilibrium)

Set of prices and allocations such that:

1. All agents in the economy are behaving optimally, takingprices as given.

2. Prices are such that markets clear.

I What are the key ingredients here?I Prices adjust so quantity demanded equals quantity supplied

when agents behave according the optimal decision rules.

I We focused on 1., now were are going to focus more on 2.I Supply still exogenous for now.


Demand Side

I We are moving away from individual’s decision.I More towards understanding aggregates.

I Denoted by capital letters.

I Optimization problem:I Before.I Now?

I From the household problem we obtained the consumptionfunction.

I Demand side: total desired expenditures equals totalconsumption.

Y d = C (Y ,Y ′, r)

I Y is in both sides!

I Supply side: Y s = Y .I Exogenously supply aggregate endowment.


Graphical Derivation of the Y d CurveI We can deal with the complication mentioned before.

I A graph with a 45 degree line.I Y d against Y .I What is the slope?

I MPC.I Less than 1.

The “aggregate demand curve”, or what we will also sometimessimply call the “Y d curve” plots out how the quantity of goodsdemanded in aggregate varies with the real interest rate.

I How do changes in interest rates affect Y d?I Current consumption is decreasing in real interest rates.

I Substitution effect dominates.

I How does that look in the (Y d , r) space?

I Remember, supply of current goods, Y s , doesn’t change withr .I There is no production, so it is exogenous.


Comparative Statics

I Increase in current income: “supply shock”.I ↓ r .

I Increase in future income: “demand shock”.I ↑ r .

I Intuition:I Market-clearing condition.I Real interest rates has to “undo” desired changes in

consumption.I Conditional on the real interest rate, people want to smooth.

I Interest rate has to move to “prevent” smoothing.


Algebraic Example

I Assuming logarithmic utility we have household consumption:

C =1

1 + β

(Y +

Y ′

1 + r

)I What about for aggregates?

I In equilibriumY = Y s = Y d = C

Therefore, 1 + r =1

β

Y ′

YI This is just an Euler equation.

I The economy as a whole doesn’t save.I Interest rate is then a measure of how plentiful future is

relative to present.I What if Y = Y ′?


Adding Government

I Aggregate government spending is G and G ′.

I Can tax,T and T ′.

I Can borrow, SG .

I What would be the government intertemporal budgetconstraint?

I What would be the household intertemporal budgetconstraint?I What if we combined both?I Taxes drop out!

I Ricardian equivalence.I Makes no difference whether current spending financed with

taxes or debt.


Consumption Function and Aggregate DemandI Timing of taxes does not matter, but spending does.

Household consumption function is:

C = C (Y − G ,Y ′ − G ′, r)

I With logarithmic utility:

C (Y − G ,Y ′ − G ′, r) =1

1 + β

(Y − G ′ +

Y ′ − G ′

1 + r

)I To obtain the total demand for current goods:

I Note that in equilibrium government borrowing must be equalto household saving.

I Combine the definition of government savings/borrowing withthe household budget constraint.

I Aggregate demand relationship is given by

Y d = C (Y − G ,Y ′ − G ′, r) + G


Effects of Changes in Government Spending

I We have that

Y d = C (Y − G ,Y ′ − G ′, r) + G

I Increase in current government expenditure, ↑ G .I Y d ↗, r ↑, c ↓.I What are households doing with the remaining income?I Consider two scenarios:

1. Increase in government expenditures financed entirely by taxes.2. Increase in government expenditures financed entirely by

borrowing.

I Increase in future government expenditure, ↑ G ′.I Y d ↙, r ↓, c unchanged.

I Intuition:I Remember what r was measuring.


Equilibrium in a ProductionEconomy



I The representative agent (RA).

I Competitive Equilibrium.

I Y d and Y s curves.

I Adjustment to shocks.

I What is the real interest measuring?

I Adding Government.

1. Chapter 12 GLS: Production, Labor Demand, Investment,and Labor Supply.

2. Chapter 17 GLS: The Neoclassical Model.

3. Chapter 18 GLS: Effects of Shocks in the Neoclassical Model.

4. Chapter 19 GLS: Taking the Neoclassical Model to the Data.


Back to the Production Economy

I So far: study equilibrium in an endowment economy.I Now: we will study equilibrium in an economy with

production.I Why?

I We will construct a model that can be used to compared tothe actual behavior of the economy in the short run.I Why?

I We can study the effects of different policies.


Equilibrium and Framework

I Definition of equilibrium is still the same:I Set of prices and quantities consistent with:

1. Agents optimizing, taking prices as given.2. Markets clearing.

I Agents:

1. Household.2. Firm.3. Government.

I Large number of each kind of agent.I Since they are identical:

I Price taking behavior.I We can use the representative agent problem.

I Only two periods:I Present and future.


Government

I Same as before: G and G ′ chosen exogenously.

I Government’s intertemporal budget constraint:

G +G ′

1 + r= T +

T ′

1 + r

I Ricardian equivalence holds:I Household behaves as though government balances budget

every period.


Remember: Equilibrium Conditions

I Labor demand: Nd = N(w ,A,K ).

I Labor supply: Ns = N(w , θ).

I Consumption: C = C (Y − G ,Y ′ − G ′, r).

I Investment: I = I (r ,A′, q,K ).

I Production function: Y = AF (K ,N).

I Market-clearing: Y = C + I + G


The Y s Curve

The Y s curve represents the set of (r ,Y ) pairs consistent withhousehold and firm optimization on the production side of theeconomy and with labor-market clearing.

I Basic idea behind its derivation:I Start with an initial r .

I This determines a position of Ns through C .

I Try a higher r .I With separability: ↓ C and labor supply shifts out⇒↑ N ⇒↑ Y .

I What about with non-separability?

I Hence, Y s slope, depends on preferences.I With separability: Higher r effectively makes people want to

work more, and hence supply more output.


The Y d Curve

The IS is the set of (r ,Y ) pairs consistent with household andfirm optimization and Y d = Y , where Y d = C + I + G .

I Where did Y d = C (·) + I (·) + G come from?I This is because “total demand”, that comes from combining

the period budget constraints of the different actors in theeconomy, can be expressed as:

Y d = C (·) + I (·) + G .

I Standard accounting identity.I We have the optimal decision rules for each one of these items.

I Basic idea behind its derivation:I Use the expenditure line - 45 degree line diagram. Start with

an r .I Determines position of expenditure line.

I Increase r .I This causes expenditure line to shift down.I The intersection will be a lower point.


General Equilibrium

I General equilibrium requires that all markets clear.I In this case we have effectively two markets:

I Labor market (Ns = Nd ).I Goods market (Y d = Y ).

I Market-clearing:I Labor market-clearing:

I On Y s curve.

I Goods market-clearing:I On Y d curve.

I General Equilibrium:I On both curves.


Defining Equilibrium

Definition (Competitive Equilibrium)

A competitive equilibrium will be composed by a set of allocations—C ,C ′, Ns ′,Ns ,Nd ,Nd ′, I , Π and Π′— and prices —w ,w ′ andr— such that:

1. Given prices, households choose C ,C ′,Ns , and Ns ′ optimally.

2. Given prices, firm chose I ,Nd , and Nd ′ optimally.

3. The prices r ,w , and w ′ are such that markets clear.

4. Profits are given by

Π = AF (K ,N)− wN − qI

and

Π′ = A′F[(1− δ)K + q′I ,N ′

]−w ′N ′+(1− δ)

[(1− δ)K + q′I

]


Graphical Analysis: Curve Shifts

I So we have five exogenous variables: A,A′,G ,G ′, q, and K .I Shifts that may occur:

I Labor demand:I Changes in A and K (more natural to think about a decrease

in K , since K can’t jump up).

I Labor supply:I Anything that changes θ.

I Demand for goods:I Changes in A′,G , and G ′.


Analyzing Effect of Changes in Exogenous Variables

I “Recipe” for thinking about how equilibrium responds to achange in an exogenous variable:

1. Start in the labor market. Holding r and current income fixed,determine whether the change shifts Ns and/or Nd .I Decide whether N would change, for a given r . This will tell

us whether Y s shifts.

2. Take the new value of N, bring it down, take it across andreflect it up.

3. Figure out if Y d shifts.I Y d shifts whenever the quantity demanded would change for

a given r .

4. Combine the shifts in Y d and Y s to find the new equilibriumin the (r ,Y ) dimension.

5. Figure out what happens with components of Y .6. Work back to labor markets to make quantities to line up.

I The adjustment of r along Y s occurs because of the shift ofthe Ns curve that occurs as r changes, so we need to go backto examine the final impact on the labor market (w and N).


Resolving Ambiguities

I Sometimes the curve shifts will produce ambiguities.

I Often, we can resolve them by doing some math.

I In particular, the labor market clearing condition is very useful:

uL = uCAFN(K ,N)

I Under our assumptions, if neither A nor K moved, N and Cmust move in opposite directions.


What’s next?


notes iii - labor supply, firm optimization, production...

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