syllabus: economics 211, second halfweb.stanford.edu/~rehall/econ211winter2011.pdfanswers will try...

Stanford University Department of Economics Winter Quarter, 2011

Professor Robert Hall [email protected] Stanford.edu/~rehall Office hours: 3:15-4:15 Wednesdays in 138 HHMB or at other times by email appointment TA: Krishna Rao [email protected] Office hours 11 am to 1 pm Fridays, in Economics 106, after the section meeting

Syllabus: Economics 211, Second Half

This module of the PhD economics core covers basic building-block subjects in aggregate applied economics: consumption, investment, and the labor market.

There will be two major problem sets. There will be a quiz on the last day of class, Wednesday, March 9. The grade will be based half on the quiz and half on the problem sets.

The class will meet Monday and Wednesday in Economics 140 from 10 to 11:50, with a break. The section will meet on Fridays from 9:00 to 10:50 in 420-050.

Class materials will be available from https://coursework.stanford.edu/.

Course schedule:

Monday Wednesday 9-Feb Consumption

14-Feb Consumption 16-Feb Consumption

21-Feb No class (holiday) 23-Feb Investment

28-Feb Investment 2-Mar Labor market

7-Mar Labor market 9-Mar Quiz in class

Problem sets You are allowed and encouraged to work in groups, but you are required to turn in your own writeup, prepared individually. If anything in the assignment is unclear or you are completely stuck, you can ask Krishna questions by email or during office hours. His answers will try to provide you with the right amount of help to resume your learning process. He won’t give away solutions. He will stop answering questions on a given

assignment 24 hours before it is due. This will encourage you to start in time. Also, there will be no option value to waiting for late tips. Late homework will be graded, but you lose 1/3 of the available points per day that you are late.

Notes and readings Notes will be posted and distributed in advance of each of the three topics. You are strictly responsible for all material in the notes and lectures. Readings are optional.

Consumption David Romer, Advanced Macroeconomics New York: McGraw-Hill, third edition, 2006, Chapter 7: "Consumption"

Hall, R. E. "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence." Journal of Political Economy, 86, 971-987, 1978 (JSTOR).

Caballero, Ricardo, "Earnings Uncertainty and Aggregate Wealth Accumulation" American Economic Review 81-859-871, Sept. 1991 (JSTOR).

Hall, R., "Intertemporal Substitution in Consumption" Journal of Political Economy, 96, 339-357, April 1988. (JSTOR)

Attanasio, Orazio "Consumption" Handbook of Macroeconomics, vol. 1B, Chapter 11, pp. 741-812 (ScienceDirect)

Deaton, Angus, Understanding Consumption, Clarendon Lectures in Economics, Oxford: Clarendon Press, 1992

Ando, A. and F. Modigliani, "The Life Cycle Hypothesis of Saving: Aggregate Implications and Tests." American Economic Review, March 1963, pp. 55-84. (JSTOR)

Abel, Andrew, "Consumption and Investment," Chapter 14 in Benjamin Friedman and Frank Hahn (eds.) Handbook of Monetary Economics, North-Holland, 1990 (ScienceDirect)

Hall, R.E., "Consumption" Chapter 4 in Robert Barro (ed.) Modern Business Cycle Theory, Harvard University Press, 1989.

Flavin, M. "The Adjustment of Consumption to Changing Expectations about Future Income." Journal of Political Economy, 89, 974-1009, 1981. (JSTOR)

Hubbard, Glenn, Jonathan Skinner, and Stephen Zeldes, "The Importance of Precautionary Motives in Explaining Individual and Aggregate Saving" Carnegie-Rochester Series on Public Policy, 40: 59-125, 1994 (ScienceDirect)

Blundell, Richard, Luigi Pistaferri, and Ian Preston, “Consumption Inequality and Partial Insurance” American Economic Review, December 2008, v. 98, No. 5, pp. 1887-1921 (Ebsco)

Investment Romer, Chapter 8: "Investment"

Hall, Robert E. and Dale W. Jorgenson, "Tax Policy and Investment Behavior," American Economic Review, 57, 391-414, 1967 (JSTOR).

Abel, Andrew, "Consumption and Investment," Chapter 14 in Benjamin Friedman and Frank Hahn (eds.) Handbook of Monetary Economics, North-Holland, 1990 (ScienceDirect)

Caballero, Ricardo J. "Aggregate Investment" Handbook of Macroeconomics, vol. 1B, Chapter 12, pp. 813-862 (ScienceDirect)

Ramey ,Valerie A., and Kenneth D. West, "Inventories" Handbook of Macroeconomics, vol. 1B, Chapter 13, pp. 863-926 (ScienceDirect)

Caballero, Ricardo J., Eduardo M.R.A. Engel, and John C. Haltiwanger, "Plant Level Adjustment and Aggregate Investment Dynamics" Brookings Papers on Economic Activity, 2:1995 (JSTOR)

Bernanke, Ben S., "Irreversibility, Uncertainty, and Cyclical Investment," Quarterly Journal of Economics, 98, 85-106, 1983. (JSTOR)

Hall, Robert. “Measuring Factor Adjustment Costs” Quarterly Journal of Economics, vol. 119, pp. 899-927, August 2004. http://www.stanford.edu/~rehall/MFAC-QJE-08-04.pdf

Ramey, Valerie, "Inventories as Factors of Production and Economic Fluctuations" American Economic Review, June 1989, pp. 338-354 (JSTOR)

Dixit, Avinash, and Robert Pindyck, Investment under Uncertainty, 1994

The Labor Market Romer, Chapter 9, "Unemployment"

Christopher Pissarides, Equilibrium Unemployment Theory, second edition, MIT Press, 2000

Shimer, Robert, Labor Markets and Business Cycles, Princeton University Press, 2010

Mortensen, Dale T., and Christopher A. Pissarides, "Job Reallocation, Employment Fluctuations, and Unemployment" Handbook of Macroeconomics, vol. 1B, Chapter 18, pp. 1171-1228 (ScienceDirect)

Shimer, Robert “The Cyclical Behavior of Equilibrium Unemployment and Vacancies” American Economic Review March 2005 http://home.uchicago.edu/~shimer/wp/published/uv.pdf

Hall, Robert. “Employment Fluctuations with Equilibrium Wage Stickiness,” American Economic Review, Vol. 95(1), March 2005, pp. 50-65, http://www.stanford.edu/~rehall/EmpFlucAERMarch2005.pdf

Hall, Robert, “Reconciling Cyclical Movements in the Marginal Value of Time and the Marginal Product of Labor” Journal of Political Economy, April 2009, http://www.stanford.edu/~rehall/ReconcilingJPE2009.pdf

1 Quiz

Answer all questions. Some contain ambiguities. State reasonable assumptions to resolveany ambiguity you find and then answer accordingly. Also note that the quiz tests yourability to recognize concepts by name and turn them into their mathematical expressions.

1. A consumer has a strictly concave utility function u(c) and consumes for just twoperiods, 1 and 2, with no discount for the second period. The consumer has preferencesthat result in precautionary behavior. The consumer receives random wealth in thesecond period of 2−ε with probability 0.5 and 2+ε with probability 0.5. The consumercan borrow at an interest rate of zero to finance consumption in the first period.

(a) Write down the consumer’s first-order condition without using an expectationoperator.

(b) Find the derivative dc1/dε at ε = 0 and explain its value intuitively.

(c) Show that the same derivative is negative for ε > 0 and relate this finding toprecautionary behavior

(d) Does the specific utility function u(c) = −(1 + c)−1 result in precautionary be-havior?

2. A competitive industry contains a large number of firms with identical productionfunctions y = Ak, where A is a fixed parameter. Demand for the product is pε, wherep is the output price. The rental price of capital is z.

(a) What is one firm’s demand for capital, as a function of p and z?

(b) What are the equilibrium values of p and total industry output and capital, Yand K?

(c) Now suppose that the rental price of capital depends on the amount of capital usedin the industry—the supply function for capital is z = Kφ. Find the equilibriumvalues of z, p, Y , and K.

(d) How do z, p, Y , and K change if productivity A rises?

3. An employer derives revenue Z from employing a worker. The worker’s opportunitycost is C (called U − V in class). They pick a wage W before knowing the realizationsof Z and C. These are random variables distributed uniformly and independently onthe square where both variables lie between zero and one.

(a) Given W , what is the probability of an inefficient layoff?

(b) What is the probability of an inefficient quit?

(c) What wage minimizes the probability of an inefficient separation?

(d) As a general matter, does minimizing the probability of an inefficient separationresult in the optimal wage? What about in this special case? (intuitive answeronly, no math)

2 Quiz Solutions

2.1 Problem 1

1. The consumers problem is:

max E [u(c1) + u(y − c1)]

Substituting in the budget constraint directly:

u(c1) + .5 [u(2− ε− c1) + u(2 + ε− c1)]

With resulting FOC:

u′(c1) = .5 [u′(2− ε− c1) + u′(2 + ε− c1)]

2. Differentiate the above expression wrt ε:

u′′(c1)∂c1∂ε

[u′′(2− ε− c1)

(−1− ∂c1

)+ u′′(2 + ε− c1)

(1− ∂c1

)]Rearrange:

∂c1∂ε

[u′′(c1) + u′′(2− ε− c1) + u′′(2 + ε− c1)] = .5 [u′′(2 + ε− c1)− u′′(2 +−ε− c1)]

At ε = 0, the expression on the RHS is 0. Generically, the expression on the LHS inbrackets is not zero so we conclude:

∂c1∂ε

The intuition for this result is that consumers are locally risk neutral. They are notrisk averse to very small amounts of uncertainty.

3. Since u′′′ > 0 (the agent displays precautionary behavior), and ε > 0

u′′(2 + ε− c1)− u′′(2− ε− c1) > 0

And since, generically, u′′ < 0, it must be that ∂c1∂ε

< 0. Increased uncertainty resultsin deferred consumption (increased savings in period 1) because of the precautionarymotive.

4. Yes.u′(c) = (1 + c)−2

u′′(c) = −2(1 + c)−3

u′′′(c) = 6(1 + c)−4 > 0

2.2 Problem 2

1. The price taking firms problem is:

max pAk − zk

Which is ∞ if pA > z, indeterminate if pA = z and 0 if pA < z

2. Thus in equilibrium:

From the demand function we have:

Y = p−ε =( zA

)−εAnd from the production function we have:

A= z−εAε−1

3. We equate capital supply with capital demand to recover the rental price:

Kd = z−εAε−1 = z1φ = Ks

z = Aε−11φ

ε−11φ

+ε−1

−1− 1φ

+ε = A−φ−11+φε

Y = p−ε = Aεφ+ε1+φε

K = z1φ = A

ε−11+φε

4. If A ↑ then variables will increase if the following conditions hold and weakly decreaseif not (remain constant iff expressions = 0):

z :ε− 11φ

+ ε> 0

p :−φ− 1

1 + φε> 0

Y :εφ+ ε

1 + φε> 0

K :ε− 1

1 + φε> 0

2.3 Problem 3

1. Inefficient layoff implies Z < W and Z > C. The probability of that event (for a fixedwage is):

1dcdz =

zdz =1

2. Inefficient quite implies C > W and Z > C. The probability of that event (for a fixedwage is):

1dzdc =

1− Cdc = 1− 1

2− w +

2(1− w)2

3. Minimize the sum of the two probabilities:

2(1− w)2

With FOC:w − (1− w) = 0

(Second order condition holds)

4. In general parties will want to minimize expected loss of value from inefficient sepa-ration. However in this special case (by symmetry) w = 1

2also minimizes expected

Economics 211 slides, Winter 2011

I Consumption

I Investment

I Labor

I Finance

Consumers:

max{c(t)}∞∫0

e−ρτ c(t+τ)1−1/σ

1−1/σdτ

subject to∫e−rτ [w(t+ τ)− c(t+ τ)] dτ + A(t) = 0

Variables:

c(t): consumption

w(t): earnings

A(t): assets (non-human wealth)

Consumers:

max{c(t)}∞∫0

e−ρτ c(t+τ)1−1/σ

1−1/σdτ

subject to∫e−rτ [w(t+ τ)− c(t+ τ)] dτ + A(t) = 0

Variables:

c(t): consumption

w(t): earnings

A(t): assets (non-human wealth)

Parameters:

ρ: Rate of time preference

σ: Intertemporal elasticity of substitution

r: Real interest rate

First-order condition:

∂U∂c(t)

∂U∂c(t+τ)

=Price at time t of c(t)

Price at time t of c(t+ τ)

orc(t)−1/σ

e−ρτc(t+ τ)−1/σ=

e−rτ.

Solve for c(t+ τ):

c(t+ τ) = c(t)eσ(r−ρ)τ .

∂U∂c(t)

∂U∂c(t+τ)

orc(t)−1/σ

e−rτ.

Solve for c(t+ τ):

∂U∂c(t)

∂U∂c(t+τ)

orc(t)−1/σ

e−rτ.

Solve for c(t+ τ):

Consumption Euler equation

c(t) = σ(r − ρ)c(t)

The planned consumption profile rises over time at exponentialrate σr because of the interest reward to saving and declinesat rate σρ because of time preference.

Consumption Euler equation

c(t) = σ(r − ρ)c(t)

The planned consumption profile rises over time at exponentialrate σr because of the interest reward to saving and declinesat rate σρ because of time preference.

Use the budget constraint to

find c(t)

H(t) =

∫ ∞0

e−rτw(t+ τ)dτ,

human wealth.

The budget constraint is∫ ∞0

e−[r−σ(r−ρ)]τc(t)dτ = A(t) +H(t)

c(t) = [r − σ(r − ρ)] [H(t) + A(t)]

= α [H(t) + A(t)]

find c(t)

H(t) =

∫ ∞0

e−rτw(t+ τ)dτ,

human wealth.

c(t) = [r − σ(r − ρ)] [H(t) + A(t)]

= α [H(t) + A(t)]

find c(t)

H(t) =

∫ ∞0

e−rτw(t+ τ)dτ,

human wealth.

c(t) = [r − σ(r − ρ)] [H(t) + A(t)]

= α [H(t) + A(t)]

Llife-cycle consumption model of

Ando and Modigliani

α is the propensity to consume out of wealth.

First special case: Zero intertemporal substitution (σ=0):

c(t) = r [H(t) + A(t)]

Second special case: Unit intertemporal substitution or logutility (σ = 1)

c(t) = ρ [H(t) + A(t)]

Ando and Modigliani

c(t) = r [H(t) + A(t)]

c(t) = ρ [H(t) + A(t)]

Ando and Modigliani

c(t) = r [H(t) + A(t)]

c(t) = ρ [H(t) + A(t)]

AssetsThe consumer’s accumulated savings A(t) are the result of theconsumption decision, not an exogenous variable.

Savings follow the accumulation equation,

A(t) = rA(t) + w(t)− c(t),that is, savings grow by the amount that interest earnings andwage earnings exceed consumption.

Because human wealth can be written

H(t) =

∫ ∞t

e−r(s−t)w(s)ds,

H(t) = rH(t)− w(t)

H(t) =

∫ ∞t

e−r(s−t)w(s)ds,

H(t) = rH(t)− w(t)

H(t) =

∫ ∞t

e−r(s−t)w(s)ds,

H(t) = rH(t)− w(t)

Human wealth, HHuman wealth is like a bank account that never receives anynew deposits, but earns interest at rate r. When earnings arereceived, they deplete H and augment A.

Suppose earnings grow exponentially:

w(t) = w0eγt.

Assume that the growth rate, γ, is less that the interest rate,r. Human wealth is

H(t) = w0eγt

∫ ∞0

e−(r−γ)τdτ

r − γeγt

w(t) = w0eγt.

H(t) = w0eγt

∫ ∞0

e−(r−γ)τdτ

r − γeγt

w(t) = w0eγt.

H(t) = w0eγt

∫ ∞0

e−(r−γ)τdτ

r − γeγt

Consumption function

c(t) = α

(A(t) +

r − γeγt).

The marginal propensity to consume is

∂c(0)

r − γ=r − σ(r − ρ)

r − γ

The MPC will be less than one if

γ < σ(r − ρ).

The consumer will spend more than one dollar out of eachdollar of current earnings if earnings growth is present andeither intertemporal substitution is low or the interest rate isnot much above the rate of time preference.

c(t) = α

(A(t) +

r − γeγt).

∂c(0)

r − γ=r − σ(r − ρ)

r − γ

γ < σ(r − ρ).

c(t) = α

(A(t) +

r − γeγt).

∂c(0)

r − γ=r − σ(r − ρ)

r − γ

γ < σ(r − ρ).

Role of the interest rate

The MPC can be written

(1− σ)r + σρ

r − γ,

which is always a decreasing function of the interest rateexcept in the boundary case ρ = γ = 0.

Stochastic environmentAssume constant interest rate, r. A consumer can consume abit more today, x, and satisfy her budget constraint byconsuming (1 + r)x less next period. The consumer optimizesover this (and other) margins:

{u (ct + x) + Et

1 + ρu (ct+1 − (1 + r)x) + · · ·

The first-order condition is

u′ (ct + x)− Et

[1 + r

1 + ρu′ (ct+1 − (1 + r)x)

When the consumer has chosen optimal consumption, x = 0,so

Et [u′ (ct+1)] =1 + ρ

1 + ru′ (ct)

{u (ct + x) + Et

1 + ρu (ct+1 − (1 + r)x) + · · ·

u′ (ct + x)− Et

[1 + r

1 + ρu′ (ct+1 − (1 + r)x)

Et [u′ (ct+1)] =1 + ρ

1 + ru′ (ct)

{u (ct + x) + Et

1 + ρu (ct+1 − (1 + r)x) + · · ·

u′ (ct + x)− Et

[1 + r

1 + ρu′ (ct+1 − (1 + r)x)

Et [u′ (ct+1)] =1 + ρ

1 + ru′ (ct)

Constant Absolute Risk

Aversion—Exponential UtilityFor simplicity, take ρ = r = 0

u(c) = − 1

θe−θc

Euler equation:Et e

−θct+1 = e−θct

Assume income is a random walk:

yt = yt−1 + εt

εt iid and has mean zero.·

u(c) = − 1

θe−θc

Euler equation:Et e

yt = yt−1 + εt

u(c) = − 1

θe−θc

Euler equation:Et e

yt = yt−1 + εt

Claim: Euler equation is

ct+1 = γ + ct + εt+1

θlog(

E e−θε)

Because the exponential function is convex, Jensen’s inequalityimplies E e−θε ≥ e−θEε = 1, so γ ≥ 0. If ε has any dispersion,γ > 0 and the consumer is strictly precautionary—she planspositive consumption growth when the interest rate equals therate of time preference.

ct+1 = γ + ct + εt+1

θlog(

E e−θε)

ct+1 = γ + ct + εt+1

θlog(

E e−θε)

Proof of the claim

Substitute the conjectured functional form into the generalEuler equation:

[e−θ(γ+ct+εt+1)

]= e−θct .

This implies e−θγ E e−θε = 1 and thus γ = 1θ

E e−θε), as

claimed.·

Proof of the claim

Substitute the conjectured functional form into the generalEuler equation:

[e−θ(γ+ct+εt+1)

]= e−θct .

This implies e−θγ E e−θε = 1 and thus γ = 1θ

E e−θε), as

claimed.·

Asset accumulationThe consumer’s assets or savings accumulate according to

Aτ = yτ − cτ + Aτ−1

Subtract the Euler equation from the process for income to get

yτ − cτ = yτ−1 − cτ−1 − γ

Notice that this implies that

yτ − cτ = − (τ − t) γ + yt − ct

yτ − cτ = yτ−1 − cτ−1 − γ

yτ − cτ = − (τ − t) γ + yt − ct

yτ − cτ = yτ−1 − cτ−1 − γ

yτ − cτ = − (τ − t) γ + yt − ct

Asset accumulation, continuedSubstitute this into the asset accumulation equation andcumulate to the terminal period, T :

AT = −γ∑

(τ − t) + (T − t+ 1) (yt − ct) + At−1.

Asset accumulation is non-stochastic, as consumption adjustsfully to each change in income, because the change ispermanent. Note that the summation is

(T − t) (T − t+ 1)

Now set terminal assets to zero and solve for currentconsumption:

ct = yt +At−1

T − t+ 1− γT − t

AT = −γ∑

(τ − t) + (T − t+ 1) (yt − ct) + At−1.

(T − t) (T − t+ 1)

ct = yt +At−1

T − t+ 1− γT − t

AT = −γ∑

(τ − t) + (T − t+ 1) (yt − ct) + At−1.

(T − t) (T − t+ 1)

ct = yt +At−1

T − t+ 1− γT − t

Stochastic returnsAsset i has stochastic interest rate, ri,t. A consumer canconsume a bit more today, x, and satisfy her budget constraintby consuming (1 + ri,t+1)x less next period.

When the consumer has chosen optimal consumption, x = 0,and

Et (1 + ri,t+1)u′ (ct+1) = (1 + ρ)u′ (ct)

Write this as

Etu′ (ct+1)

(1 + ρ)u′ (ct)(1 + ri,t+1) = 1

Et (1 + ri,t+1)u′ (ct+1) = (1 + ρ)u′ (ct)

Write this as

Etu′ (ct+1)

(1 + ρ)u′ (ct)(1 + ri,t+1) = 1

Et (1 + ri,t+1)u′ (ct+1) = (1 + ρ)u′ (ct)

Write this as

Etu′ (ct+1)

(1 + ρ)u′ (ct)(1 + ri,t+1) = 1

Information aggregation

Et1 + ri,t+1

1 + ρu′(ct+1) = u′(ct)

implies that ct encapsulates all information available at t thathelps predict future consumption, in the sense of theexpectation of future marginal utility multiplied by the returnratio divided by the impatience ratio.

Let R(t, t+ τ) be the realized asset return ratio from t tot+ τ .

EtR(t, t+ τ)

(1 + ρ)τu′(ct+τ ) = u′(ct)

so the consumer projects unchanging consumption over theindefinite future, again in the sense of expected marginal utilitymultiplied by the return ratio divided by the impatience ratio.

Information aggregation

Et1 + ri,t+1

1 + ρu′(ct+1) = u′(ct)

implies that ct encapsulates all information available at t thathelps predict future consumption, in the sense of theexpectation of future marginal utility multiplied by the returnratio divided by the impatience ratio.

Let R(t, t+ τ) be the realized asset return ratio from t tot+ τ .

EtR(t, t+ τ)

(1 + ρ)τu′(ct+τ ) = u′(ct)

so the consumer projects unchanging consumption over theindefinite future, again in the sense of expected marginal utilitymultiplied by the return ratio divided by the impatience ratio.

Consumption surprises—random

1 + ri,t+1

1 + ρu′(ct+1)− u′(ct) = εt+1

Constant Relative Risk

Aversion—Power Utility

u(c) =c1−1/σ

1− 1/σ

ρ = rate of time preference continuously compounded

rt = interest rate continuously compounded

Euler equation:

[e−ρ+rt c

−1/σt+1

−1/σt

u(c) =c1−1/σ

1− 1/σ

Euler equation:

[e−ρ+rt c

−1/σt+1

−1/σt

u(c) =c1−1/σ

1− 1/σ

Euler equation:

[e−ρ+rt c

−1/σt+1

−1/σt

u(c) =c1−1/σ

1− 1/σ

Euler equation:

[e−ρ+rt c

−1/σt+1

−1/σt

Assumptions:

rt is N (rt, γ)

zt+1 = log ct+1 is N (zt+1, vt)

Cov (rt, zt+1) =βt2

Assumptions:

rt is N (rt, γ)

Assumptions:

rt is N (rt, γ)

FactIf y is N (µ, θ) ,

E (ey) = eµ+ 12θ

We needEt

(e−ρ+rt−

zt+1σ

E(−ρ+ rt −

)= −ρ+ rt −

V () = γ +vtσ2− βtσ

E (ey) = eµ+ 12θ

We needEt

(e−ρ+rt−

zt+1σ

E(−ρ+ rt −

)= −ρ+ rt −

E (ey) = eµ+ 12θ

We needEt

(e−ρ+rt−

zt+1σ

E(−ρ+ rt −

)= −ρ+ rt −

Euler equation

(−ρ+ rt −

vtσ2− βtσ

))= exp

(− 1

Take logs

−ρ+ rt −zt+1

Vtσ2− βtσ

)= −zt

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

Euler equation

(−ρ+ rt −

vtσ2− βtσ

))= exp

(− 1

Take logs

−ρ+ rt −zt+1

Vtσ2− βtσ

)= −zt

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

Euler equation

(−ρ+ rt −

vtσ2− βtσ

))= exp

(− 1

Take logs

−ρ+ rt −zt+1

Vtσ2− βtσ

)= −zt

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

The modern econometric Euler

equation

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

)+ εt+1

Driving forces of log

consumption growth

(1) σ (rt − ρ) = deferral from real interest rate

(2) vt2σ

: deferral from precautionary behavior.

A family with its back to the wall could plan consumptiongrowth for both reasons.

consumption growth

(2) vt2σ

consumption growth

(2) vt2σ

Estimation

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

)+ εt+1

rt = rt + ηt+1

OLS: zt+1−zt−σrt = −σρ+σ

vtσ2− βtσ

)+εt+1 +σηt+1

TSLS: zt+1−zt = σrt−σρ+σ

vtσ2− βtσ

)+ εt+1 +σηt+1

Estimation

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

)+ εt+1

rt = rt + ηt+1

vtσ2− βtσ

)+εt+1 +σηt+1

vtσ2− βtσ

)+ εt+1 +σηt+1

Estimation

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

)+ εt+1

rt = rt + ηt+1

vtσ2− βtσ

)+εt+1 +σηt+1

vtσ2− βtσ

)+ εt+1 +σηt+1

Estimation

zt+1 = zt + σ (rt − ρ) +σ

vtσ2− βtσ

)+ εt+1

rt = rt + ηt+1

vtσ2− βtσ

)+εt+1 +σηt+1

vtσ2− βtσ

)+ εt+1 +σηt+1

Finance in one slideLet m be the marginal rate of substitution,

m =u′ (ct+1)

(1 + ρ)u′ (ct)

Then for all assets i,

E m · (1 + ri) = 1

Read John Cochrane, Asset Pricing, to see how this maps intoall the other ways of thinking about finance.

m =u′ (ct+1)

(1 + ρ)u′ (ct)

E m · (1 + ri) = 1

m =u′ (ct+1)

(1 + ρ)u′ (ct)

E m · (1 + ri) = 1

Return depends on asset position

Interest rate earned or paid is r(A).

Consume x more units this period and(1 + r(At − x))(At − x)− (1 + r(At))(At) less next period.

dx|x=0 = −r′(At)At − (1 + r(At))

Et1 + r(At) + r′(At)At

1 + ρu′(ct+1) = u′(ct)

dx|x=0 = −r′(At)At − (1 + r(At))

1 + ρu′(ct+1) = u′(ct)

dx|x=0 = −r′(At)At − (1 + r(At))

1 + ρu′(ct+1) = u′(ct)

Asset constraint—borrowing

Et1 + rt + λt

1 + ρu′(ct+1) = u′(ct)

λt is the Lagrangian multiplier on the constraint.

Quadratic preferences and

certainty equivalenceTwo-period case:

U(c1, c2) = E[−1

2(c− c1)2 − 1

2(c− c2)2

No time preference or asset return: ρ = r = 0.

Budget constraint with random wealth W :

c2 = W − c1

Euler equation (linear marginal utility):

c− c1 − E [c− (W − c1)] = 0

U(c1, c2) = E[−1

2(c− c1)2 − 1

2(c− c2)2

c2 = W − c1

c− c1 − E [c− (W − c1)] = 0

U(c1, c2) = E[−1

2(c− c1)2 − 1

2(c− c2)2

c2 = W − c1

c− c1 − E [c− (W − c1)] = 0

U(c1, c2) = E[−1

2(c− c1)2 − 1

2(c− c2)2

c2 = W − c1

c− c1 − E [c− (W − c1)] = 0

Optimal consumption ruleLet

W = E (W )

c2 = W − 1

Random walk of consumption itself:

E (c2) = c1

W = E (W )

c2 = W − 1

E (c2) = c1

W = E (W )

c2 = W − 1

E (c2) = c1

Certainty equivalence but risk

aversion

2U = −(c− 1

2W )2 − E [c− (W − 1

2W )]2

= −(c− 1

2W )2 − E [c− 1

2W − (W − W )]2

= −2(c− 1

2W )2 + 2(c− 1

2W )[ E (W − W )]− E (W − W )2

aversion

2U = −(c− 1

2W )2 − E [c− (W − 1

2W )]2

= −(c− 1

2W )2 − E [c− 1

2W − (W − W )]2

= −2(c− 1

2W )2 + 2(c− 1

2W )[ E (W − W )]− E (W − W )2

aversion

2U = −(c− 1

2W )2 − E [c− (W − 1

2W )]2

= −(c− 1

2W )2 − E [c− 1

2W − (W − W )]2

= −2(c− 1

2W )2 + 2(c− 1

2W )[ E (W − W )]− E (W − W )2

Expected utility is decreasing in

the variance of wealth V (W )

2U = −2(c− 1

2W )2 − V (W )

Investment

Capital stock

K(t) =

e−δ(t−s)I(s)ds

K(t) = I(t)− δ∫ t

0e−δ(t−s)I(s)ds

= I(t)− δK(t)

Net investment = gross investment – replacement investment

Capital stock

K(t) =

e−δ(t−s)I(s)ds

K(t) = I(t)− δ∫ t

= I(t)− δK(t)

Capital stock

K(t) =

e−δ(t−s)I(s)ds

K(t) = I(t)− δ∫ t

= I(t)− δK(t)

Rental Price of Capital

In continuous time with a variable interest rate, the presentvalue at time t of a dollar to be received at time s is

(−∫ s

r(τ)dτ

The quantity r(t) is called the “force of interest.” It is aninstantaneous interest rate.

Rental Price of Capital

In continuous time with a variable interest rate, the presentvalue at time t of a dollar to be received at time s is

(−∫ s

r(τ)dτ

The quantity r(t) is called the “force of interest.” It is aninstantaneous interest rate.

Rental Price continuedThe zero-profit condition for capital renters is

pK(t) =

∞∫t

e−∫ st r(τ)dτe−δ(s−t)ρ(s)ds

Taking the derivative with respect to t, we get

pK = (r + δ)pK − ρ

ρ = (r + δ − pKpK

pK(t) =

∞∫t

pK = (r + δ)pK − ρ

pK(t) =

∞∫t

pK = (r + δ)pK − ρ

TheoremOnly the instantaneous force of interest at time t, and not

the longer-term interest rate, enters the rental price ofcapital.

The theorem rests on the zero-profit condition. In order forcompetition to enforce the zero-profit condition at each pointin time, there must be immediate entry and exit from thecapital rental business.

To the extent that arbitrage takes time, the zero-profitcondition may fail briefly and the interest rate over the periodneeded for arbitrage will matter.

The term of the relevant interest rate is not set by the lifetimeof the capital but by the time required for arbitrage.

Comments

If the firm that uses the capital has an intertemporallyseparable technology, so that its demand for current factorsdepends only on current factor prices, then investmentdepends only on the interest rates that determine the currentrental price, as discussed above.

If the technology is not time-separable—for example, if thereare adjustment costs—then future rental prices are adeterminant of today’s demand for capital. The future interestrate matters over the span of time when adjustment costsmatter.

Comments

If the firm that uses the capital has an intertemporallyseparable technology, so that its demand for current factorsdepends only on current factor prices, then investmentdepends only on the interest rates that determine the currentrental price, as discussed above.

If the technology is not time-separable—for example, if thereare adjustment costs—then future rental prices are adeterminant of today’s demand for capital. The future interestrate matters over the span of time when adjustment costsmatter.

Theorem

The interest rate relevant for investment is the rate forsecurities whose term matches the time needed to adjust theamount of capital in use in response to changingconditions. The term is not the lifetime of the capital good.

Industry equilibrium

y is output, p is the price of output, and the industry demandfunction is

y = dp−1

where d is a variable that determines the location of thedemand function and which may change over time.

Notice that d = py, so this specification says that industryrevenue is held at the exogenous level d.

Industry equilibrium

y is output, p is the price of output, and the industry demandfunction is

y = dp−1

where d is a variable that determines the location of thedemand function and which may change over time.

Notice that d = py, so this specification says that industryrevenue is held at the exogenous level d.

Industry equilibrium, continuedSuppose further that the technology is Cobb-Douglas:

y = Anαk1−α.

Each firm in the industry is a price taker satisfying thefirst-order condition for capital,

(1− α)Anαk−α =ρ

This can be written

(1− α)y

y = Anαk1−α.

This can be written

(1− α)y

y = Anαk1−α.

This can be written

(1− α)y

Commentary

k = (1− α)py

ρ= (1− α)

The capital stock responds in positive proportion to shifts ofindustry demand, d, and inversely to shifts in the rental priceof capital, ρ.

The response is immediate.

Commentary

k = (1− α)py

ρ= (1− α)

Commentary

k = (1− α)py

ρ= (1− α)

Step increase in demand

If the technology is not time-separable—for example, if there are adjustment costs—then future rental prices are a determinant of today's demand for capital. The future interest rate matters over the span of time when adjustment costs matter. To summarize,

Proposition. The interest rate relevant for investment is the rate for securities whose term matches the time needed to adjust the amount of capital in use in response to changing conditions. The term is not the lifetime of the capital good.

Industry equilibrium Suppose y is output, p is the price of output, and the industry demand function is

1y dp−= where d is a variable that determines the location of the demand function and which may change over time. Notice that d py= , so this specification says that industry revenue is held at the exogenous level d.

Suppose further that the technology is Cobb-Douglas: 1y An kα α−= . Each firm in the industry is a price taker satisfying the first-order condition for capital,

( )1 An kp

α α ρα −− = . This can be written ( )1 yk p

ρα− = or ( ) ( )1 1py dk α αρ ρ

= − = − .

Thus the capital stock responds in positive proportion to shifts of industry demand, d, and inversely to shifts in the rental price of capital, ρ . The response is immediate. The impulse response functions for the capital stock and investment, for a step increase in industry demand, are:

Now consider a similar setup with adjustment costs. As before, let Kp be the price of new capital and now let q be the ratio of the price of installed capital to the price of new capital. Thus the effective price of capital used in production is qp and the rental price of

Adjustment cost

Let pK be the price of new capital and q be the ratio of theprice of installed capital to the price of new capital.

The effective price of capital used in production is qp

The rental price of installed capital is

(r − q

q− pKpK

)qpK .

Here we have assumed no depreciation, δ = 0.

Adjustment cost

(r − q

q− pKpK

)qpK .

Adjustment cost

(r − q

q− pKpK

)qpK .

Adjustment cost

(r − q

q− pKpK

)qpK .

Capital demand

The first-order condition remains the same with this newversion of the rental price.

k = (1− α)d(

r − qq− pK

The endogenous q appears in both levels and time derivativesin the demand for capital.

Capital demand

The first-order condition remains the same with this newversion of the rental price.

k = (1− α)d(

r − qq− pK

The endogenous q appears in both levels and time derivativesin the demand for capital.

Adjustment cost functionLet c (I) be the adjustment cost of investment flow I = k.

Assume convex adjustment cost:

c′(I) ≥ 0, c′(0) = 0, c′′(I) > 0.

The flow of newly installed capital solves

maxIqpKI − c(I)− pKI

so that,c′(I) = pK(q − 1)

c′(I) ≥ 0, c′(0) = 0, c′′(I) > 0.

Optimal path

The variables q and K form a saddle-point system.

The locus of stationary values of q is defined by

k = (1− α)d(

r − pKpK

and the locus of stationary values of k by

Optimal path

k = (1− α)d(

r − pKpK

Optimal path

k = (1− α)d(

r − pKpK

Phase diagram

installed capital is KK

q pr qpq p

= − −

& &. Here we have assumed no depreciation,

0δ = . The first-order condition remains the same with this new version of the rental

price. Thus ( )1K

dkpqr qp

α= −

− −

&&. Now the endogenous q appears in both

levels and time derivatives in the demand for capital.

Let ( )c I be the adjustment cost of investment flow I k= & . Assume convex adjustment cost: ( ) 0, (0) 0, ( ) 0c I c c I′ ′ ′′≥ = > . The flow of newly installed capital solves max ( )K KI

qp I c I p I− − so that, ( ) ( 1)Kc I p q′ = − .

The variables q and K form a saddle-point system. The locus of stationary values of q is

defined by ( )1K

dkpr qpp

α= −

& and the locus of stationary values of k by q = 1.

The phase diagram is:

1 0k =&

Response of k and k

Impulse response functions:

Tobin’s q—Ratio of Market Value to Reproduction Cost of Plant and Equipment

Investment under Uncertainty Reference: Avinash Dixit and Robert S. Pindyck, Investment under Uncertainty, 1994, pp. 40-41. Consider a project that produces 1 unit of output per year. This year, output sells at price P0 , which is known before any decision has to be made. There are four possible decisions and outcomes about the project:

• Decide this year never to build the project. • Decide this year to build the project this year. • Defer the decision this year and decide next year to build the project. • Defer the decision this year and decide next year not to build the project.

Figure 4. Tobin’s q—Ratio of

Market Value to Reproduction

Cost of Plant and Equipment

Impulse response functions:

Tobin’s q—Ratio of Market Value to Reproduction Cost of Plant and Equipment

Investment under Uncertainty Reference: Avinash Dixit and Robert S. Pindyck, Investment under Uncertainty, 1994, pp. 40-41. Consider a project that produces 1 unit of output per year. This year, output sells at price P0 , which is known before any decision has to be made. There are four possible decisions and outcomes about the project:

• Decide this year never to build the project. • Decide this year to build the project this year. • Defer the decision this year and decide next year to build the project. • Defer the decision this year and decide next year not to build the project.

Investment under Uncertainty,

with Irreversibility

Reference: Avinash Dixit and Robert S. Pindyck, Investmentunder Uncertainty, 1994, pp. 40-41

Often called real option theory

Investment under Uncertainty,

with Irreversibility

Reference: Avinash Dixit and Robert S. Pindyck, Investmentunder Uncertainty, 1994, pp. 40-41

Often called real option theory

SetupConsider a project that produces 1 unit of output per year.This year, output sells at price P0, which is known before anydecision has to be made.

There are four possible decisions and outcomes about theproject:

1. Decide this year never to build the project.

2. Decide this year to build the project this year.

3. Defer the decision this year and decide next year to buildthe project.

4. Defer the decision this year and decide next year not tobuild the project.

SetupConsider a project that produces 1 unit of output per year.This year, output sells at price P0, which is known before anydecision has to be made.

There are four possible decisions and outcomes about theproject:

1. Decide this year never to build the project.

2. Decide this year to build the project this year.

3. Defer the decision this year and decide next year to buildthe project.

4. Defer the decision this year and decide next year not tobuild the project.

If the project is built,

it yields a present discounted value

(1 + u)MP0 with probability q

(1− d)MP0 with probability 1− q

M is a capitalization factor with a value like 5 or 10; itembodies discounting as well as capturing the length of thepayoff from the project.

If the project is built,

it yields a present discounted value

(1 + u)MP0 with probability q

(1− d)MP0 with probability 1− q

M is a capitalization factor with a value like 5 or 10; itembodies discounting as well as capturing the length of thepayoff from the project.

The project costs I whether it is built this year or next year.

Assume that this year’s revenue exceeds the interest cost onthe investment:

1 + rI

The project costs I whether it is built this year or next year.

Assume that this year’s revenue exceeds the interest cost onthe investment:

1 + rI

PayoffsLet VN be the value associated with making the decisionabout whether or not to build the project this year:

VN = max [0,−I + P0 + q(1 + u)MP0 + (1− q)(1− d)MP0]

Let VW be the expected value associated with deferring thedecision until next year:

VW = qmax

[0,−I

1 + r+ (1 + u)MP0

(1− q) max

[0,−I

1 + r+ (1− d)MP0

PayoffsLet VN be the value associated with making the decisionabout whether or not to build the project this year:

VN = max [0,−I + P0 + q(1 + u)MP0 + (1− q)(1− d)MP0]

Let VW be the expected value associated with deferring thedecision until next year:

VW = qmax

[0,−I

1 + r+ (1 + u)MP0

(1− q) max

[0,−I

1 + r+ (1− d)MP0

Payoffs

A B DP0C

Four relevant prices are shown on the horizontal axis: A is the cutoff for investing next year if the good outcome occurs:

P Ir u MA0 1 1

=+ +( )( )

B is the cutoff for investing this year:

P Iq u M q d MB0 1 1 1 1

=+ + + − −( ) ( )( )

Assume that P PA B0 0< . (Exercise: interpret this assumption). C is the point where the expected values of the two decision times are equal:

q d MC0

1 1 1=

+ − −( )( )

D is the point where investment will occur with a deferred decision even if the bad outcome occurs:

P Ir d MD0 1 1

=+ −( )( )

Now the optimal decision strategy is apparent: If P P A0 0< , forget the project now. If P P PA C0 0 0≤ < , defer the decision and invest only if the good outcome occurs. If P P C0 0> , invest this year.

Relevant prices

A is the cutoff for investing next year if the good outcomeoccurs:

P0A =I

(1 + r)(1 + u)M

P0B =I

1 + q(1 + u)M + (1− q)(1− d)M

Relevant prices

A is the cutoff for investing next year if the good outcomeoccurs:

P0A =I

(1 + r)(1 + u)M

P0B =I

1 + q(1 + u)M + (1− q)(1− d)M

More relevant prices

Assume that P0A < P0B. C is the point where the expectedvalues of the two decision times are equal:

(1− q

1 + (1− q)(1− d)M

D is the point where investment will occur with a deferreddecision even if the bad outcome occurs:

P0D =I

(1 + r)(1− d)M

More relevant prices

Assume that P0A < P0B. C is the point where the expectedvalues of the two decision times are equal:

(1− q

1 + (1− q)(1− d)M

D is the point where investment will occur with a deferreddecision even if the bad outcome occurs:

P0D =I

(1 + r)(1− d)M

Optimal decision strategy

If P0 < P0A, forget the project now.

If P0A ≤ P0 < P0C , defer the decision and invest only if thegood outcome occurs.

If P0 > P0C , invest this year.

The Bad News Principle

Note that the formula forP0C does not depend on how goodthe good news might be next year; u does not appear in theformula. This establishes

Bernanke’s Theorem: The decision whether or not to investthis year does not depend on how good the good news mightbe, but only on how bad the bad news might be. The valueof waiting comes entirely from avoiding the effect of badnews. All of the value of good news is captured either byinvesting now or investing next year.

The Bad News Principle

Note that the formula forP0C does not depend on how goodthe good news might be next year; u does not appear in theformula. This establishes

Bernanke’s Theorem: The decision whether or not to investthis year does not depend on how good the good news mightbe, but only on how bad the bad news might be. The valueof waiting comes entirely from avoiding the effect of badnews. All of the value of good news is captured either byinvesting now or investing next year.

Effect of increased uncertainty

Consider the family of distributions of subsequent value whosemeans are P0M .

Further restrict the family to those satisfying u = αd—that is,the good news is proportional to the bad news, with aprescribed constant of proportionality, α.

In this family, the parameter d is a measure of dispersion.Higher values of d mean there is more uncertainty about thefuture payoff of the project.

Increased uncertainty, continuedThe restriction on the mean of the subsequent payoff turnsout to restrict the probability of the good outcome:

1 + α

Putting this into the formula for the switchover price gives:

(1 + α− 1

1 + α + α(1− d)M

This is an increasing function of dispersion, d.

Theorem. Deferral of investment is more likely if there ismore uncertainty.

1 + α

(1 + α− 1

1 + α + α(1− d)M

1 + α

(1 + α− 1

1 + α + α(1− d)M

Relation between project

selection and investmentFirm:

min∞∑s=t

Rs,t(Cs(Qs, K0, . . . , Ks) + pk,sKs)

FONC (project selection):∞∑s=t

Rs,tMs,t = pk,t

Ms,t = −∂Cs∂Kt

The marginal investment project just meets the present-valuecriterion.

Relation between project

selection and investmentFirm:

min∞∑s=t

Rs,t(Cs(Qs, K0, . . . , Ks) + pk,sKs)

FONC (project selection):∞∑s=t

Rs,tMs,t = pk,t

Ms,t = −∂Cs∂Kt

The marginal investment project just meets the present-valuecriterion.

The investment equationFONC for next period:

∞∑s=t+1

Rs,tMs,t+1 =pk,t

1 + rt

Key assumption: different ages of capital are perfectsubstitutes with deterioration:

Ms,t =Ms,t+1

1 + δ

So∞∑

Rs,tMs,t =pk,t+1

(1 + δ)(1 + rt)

∞∑s=t+1

Rs,tMs,t+1 =pk,t

1 + rt

Ms,t =Ms,t+1

1 + δ

So∞∑

Rs,tMs,t =pk,t+1

(1 + δ)(1 + rt)

∞∑s=t+1

Rs,tMs,t+1 =pk,t

1 + rt

Ms,t =Ms,t+1

1 + δ

So∞∑

Rs,tMs,t =pk,t+1

(1 + δ)(1 + rt)

Conclusion

Subtract to get

Mt,t = pk,t −pk,t+1

(1 + δ)(1 + rt),

the investment equation involving only the next future period.

Fixed costs of adjustmentDynamic program:

V (K) = π(K) +

[−C(I)− F (I) +

1 + rV (K ′)

]K ′ = (1− δ)K + I

I π(K) is the period profit from capital K.I I is investment, which cannot be negative (irreversible).I F (I) is a fixed cost F if I > 0 with F (0) = 0.I C(I) is the variable cost of investment I.

Fixed costs of adjustmentDynamic program:

V (K) = π(K) +

[−C(I)− F (I) +

1 + rV (K ′)

]K ′ = (1− δ)K + I

I π(K) is the period profit from capital K.I I is investment, which cannot be negative (irreversible).I F (I) is a fixed cost F if I > 0 with F (0) = 0.I C(I) is the variable cost of investment I.

BehaviorLet

V ∗(K) = maxI

[−C(I) +

1 + rV (K ′)

the future value from investing, putting aside the fixed cost.

V 0(K) =1

1 + rV ((1− δ)K),

the future value from zero investment.

Then the condition for positive investment is

V ∗(K) ≥ V 0(K) + F .

BehaviorLet

V ∗(K) = maxI

[−C(I) +

1 + rV (K ′)

V 0(K) =1

1 + rV ((1− δ)K),

V ∗(K) ≥ V 0(K) + F .

BehaviorLet

V ∗(K) = maxI

[−C(I) +

1 + rV (K ′)

V 0(K) =1

1 + rV ((1− δ)K),

V ∗(K) ≥ V 0(K) + F .

Periodic investment

Given N and K, let I satisfy

K = (1− δ)NK + I

Invest I every N periods and zero the rest of the time.

Periodic investment

Given N and K, let I satisfy

K = (1− δ)NK + I

Invest I every N periods and zero the rest of the time.

Dynamic program becomes

V (K) =N−1∑i=0

)iπ((1− δ)iK)

)N (V (K)− F − C([1− (1− δ)N ]K)

Thus we can solve for V explicitly as

V (K) =

∑N−1i=0

)iπ((1− δ)iK)−

)N (F + C()

)NMaximize over N and K, then set I = [1− (1− δ)N ]K.

V (K) =N−1∑i=0

)iπ((1− δ)iK)

)N (V (K)− F − C([1− (1− δ)N ]K)

)Thus we can solve for V explicitly as

V (K) =

∑N−1i=0

)iπ((1− δ)iK)−

)N (F + C()

Maximize over N and K, then set I = [1− (1− δ)N ]K.

V (K) =N−1∑i=0

)iπ((1− δ)iK)

)N (V (K)− F − C([1− (1− δ)N ]K)

)Thus we can solve for V explicitly as

V (K) =

∑N−1i=0

)iπ((1− δ)iK)−

)N (F + C()

)NMaximize over N and K, then set I = [1− (1− δ)N ]K.

Applications—(s, S) rules

Inventory investment: F is cost of restocking. Wait untilinventory falls to (1− δ)NK then buy I = [1− (1− δ)N ]K tobring stock up to K.

Truck or car purchasing: F is cost of selling current vehicleand going to dealer to get a new one. δ is rate of decline ofnet benefit of ownership. Wait until benefit falls to (1− δ)NKthen trade up by I = [1− (1− δ)N ]K.

Applications—(s, S) rules

Inventory investment: F is cost of restocking. Wait untilinventory falls to (1− δ)NK then buy I = [1− (1− δ)N ]K tobring stock up to K.

Truck or car purchasing: F is cost of selling current vehicleand going to dealer to get a new one. δ is rate of decline ofnet benefit of ownership. Wait until benefit falls to (1− δ)NKthen trade up by I = [1− (1− δ)N ]K.

Important non-investment

applications

Tobin’s model of currency demand: F is cost of going to bank(ATM). δ is rate of spending currency. Wait until money inwallet falls to (1− δ)NK then withdraw I = [1− (1− δ)N ]Kto bring wallet up to K.

Menu costs: F is cost of changing price, K. δ is rate ofinflation. Wait until real price falls to (1− δ)NK then raiseprice by I = [1− (1− δ)N ]K to bring to K.

Important non-investment

applications

Tobin’s model of currency demand: F is cost of going to bank(ATM). δ is rate of spending currency. Wait until money inwallet falls to (1− δ)NK then withdraw I = [1− (1− δ)N ]Kto bring wallet up to K.

Menu costs: F is cost of changing price, K. δ is rate ofinflation. Wait until real price falls to (1− δ)NK then raiseprice by I = [1− (1− δ)N ]K to bring to K.

Stochastic case

Dynamic program:

V (K, x) = π(K, x) +

[−C(I)− F (I) + Ex′ mV (K ′, x′)]

K ′ = (1− δ)K + I

The shock must be Markov: the distribution of x′ is a functiononly of x.

Stochastic case

Dynamic program:

V (K, x) = π(K, x) +

[−C(I)− F (I) + Ex′ mV (K ′, x′)]

K ′ = (1− δ)K + I

The shock must be Markov: the distribution of x′ is a functiononly of x.

As beforeLet

V ∗(K, x) = maxI

[−C(I) + Ex′ mV (K ′, x′)] ,

LetV 0(K, x) = Ex′ mV ((1− δ)K, x′),

V ∗(K, x) ≥ V 0(K, x) + F .

V ∗(K, x) = V 0(K, x) + F defines K, the value of K whereinvestment becomes beneficial. K > K is the zone of inaction.

As beforeLet

V ∗(K, x) = maxI

[−C(I) + Ex′ mV (K ′, x′)] ,

LetV 0(K, x) = Ex′ mV ((1− δ)K, x′),

V ∗(K, x) ≥ V 0(K, x) + F .

As beforeLet

V ∗(K, x) = maxI

[−C(I) + Ex′ mV (K ′, x′)] ,

LetV 0(K, x) = Ex′ mV ((1− δ)K, x′),

V ∗(K, x) ≥ V 0(K, x) + F .

As beforeLet

V ∗(K, x) = maxI

[−C(I) + Ex′ mV (K ′, x′)] ,

LetV 0(K, x) = Ex′ mV ((1− δ)K, x′),

V ∗(K, x) ≥ V 0(K, x) + F .

Stochastic evolution of K

After an investment, K is well inside the zone of inaction.

Each period, K ′ = (1− δ)K, so K ′ will eventually drop belowthe zone of inaction and another investment will occur.

A shock x that raises the marginal payoff from capital willraise K and thus raise the zone of inaction and may cause Kto be below the zone, in which case investment will occursooner than without the shock.

Many firms with idiosyncratic

shocks

x = a+ εi.

a affects all firms and εi is idiosyncratic to firm i; both iid.

Ki is distributed across firms within their zones of inaction.This distribution is a state variable of dimension equal to thenumber of firms (infinity in many models).

If a large mass of firms is close to K, a small positiveaggregate shock a will cause a large volume of investment andvice versa.

Hard to generalize about the effects of fixed investment costson the sensitivity of investment to aggregate shocks. Same fornon-investment applications.

shocks

x = a+ εi.

shocks

x = a+ εi.

shocks

x = a+ εi.

Two-sided case

There is another cost F for disinvesting.

Then there will be another critical value of K, K, abovewhich disinvestment will be advantageous.

The zone of inaction is K < K < K.

Two-sided case

Intertemporal labor supply

under uncertaintyLet u(c, h) be period utility when consuming goods c andworking hours h and let β be the worker’s subjective discountratio.

Let ps,t be the Arrow-Debreu time-0 price of a unit ofconsumption at time t and state of the world s and let ws,t bethe price of an hour of work—the wage.

Period zero has only one state and output in that state isnumeraire: p1,0 = 1.

The probability of state s at time t is πs,t with∑

s πs,t = 1.

ChoiceAt time zero, the worker makes a lifetime contingent choice tomaximize expected utility,∑

βtπs,tu(cs,t, hs,t)

subject to ∑t

(ps,tcs,t − ws,ths,t) = 0.

Suppose that the uncertainty facing the worker is purelypersonal, so that a competitive insurance company will berisk-neutral and offer contingent claims on consumption andhours at

ps,t = βtπs,t

ws,t = ps,tws,t.

Here ws,t is the real wage.

ChoiceAt time zero, the worker makes a lifetime contingent choice tomaximize expected utility,∑

βtπs,tu(cs,t, hs,t)

subject to ∑t

(ps,tcs,t − ws,ths,t) = 0.

Suppose that the uncertainty facing the worker is purelypersonal, so that a competitive insurance company will berisk-neutral and offer contingent claims on consumption andhours at

ps,t = βtπs,t

ws,t = ps,tws,t.

Here ws,t is the real wage.

First-order conditionsLet λ be the Lagrangian associated with the budgetconstraint. The first-order conditions for the maximization ofexpected utility are

βtπs,tuc = λβtπs,t

−βtπs,tuh = λβtπs,tws,t

oruc = λ

−uh = ws,tλ.

The first is the Borch-Arrow condition, calling for the equalityof marginal utility of consumption across all states (and timeperiods). The second is analogous—it is called the Frischlabor-supply function in labor economics.

First-order conditionsLet λ be the Lagrangian associated with the budgetconstraint. The first-order conditions for the maximization ofexpected utility are

βtπs,tuc = λβtπs,t

−βtπs,tuh = λβtπs,tws,t

oruc = λ

−uh = ws,tλ.

The first is the Borch-Arrow condition, calling for the equalityof marginal utility of consumption across all states (and timeperiods). The second is analogous—it is called the Frischlabor-supply function in labor economics.

No insuranceNow suppose that workers can’t buy insurance and can’tborrow against future income. They make sequential decisionsand face the constraint cs,t ≤ Wt + ys,t. Here W is theworker’s non-human wealth (savings).

Suppose that the uncertain state follows a Markoff process:

Prob[st+1 = s′|st = s] = φs,s′

Use dynamic programming to find the worker’s optimal choiceof consumption and hours. Let Vs,t(W ) be the worker’sexpected value at the beginning of period t, emerging fromstate s with non-human wealth W .

Bellman equation

Vs,t(W ) = maxc,h

[u(c, h) +

∑s′

φs,s′βVs′,t+1

(W − c+ ws,th

The division by β reflects the interest that the carried-forwardwealth earns: 1 + r = 1/β.

The first-order conditions are:

uc =∑s′

φs,s′V′s′,t+1(Wt+1)

and−uh = ws,t

∑s′

φs,s′V′s′,t+1(Wt+1).

Bellman equation

Vs,t(W ) = maxc,h

[u(c, h) +

∑s′

φs,s′βVs′,t+1

(W − c+ ws,th

The division by β reflects the interest that the carried-forwardwealth earns: 1 + r = 1/β.

The first-order conditions are:

uc =∑s′

φs,s′V′s′,t+1(Wt+1)

and−uh = ws,t

∑s′

φs,s′V′s′,t+1(Wt+1).

Relation to insured case

If the value function is linear in W with a constant coefficientover s, the first-order conditions are the same as in the insuredcase, with λ = V ′.

Insurance matters because of the curvature of V . Thecurvature arises for the reasons we discussed in connectionwith consumption behavior—it is especially pronounced forpeople with powerful precautionary tendencies.

Relation to insured case

If the value function is linear in W with a constant coefficientover s, the first-order conditions are the same as in the insuredcase, with λ = V ′.

Insurance matters because of the curvature of V . Thecurvature arises for the reasons we discussed in connectionwith consumption behavior—it is especially pronounced forpeople with powerful precautionary tendencies.

A utility functionSuppose people order consumption-hours pairs according tothe period utility,

u(c, h) =1

1− 1/δ

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)1−1/δ

The kernel inside the parentheses governs the marginal rate ofsubstitution between consumption and work within a monthand state of the world. Because consumption enters as thelog, these preferences imply that hours of work areindependent of the wage if wage income in the same monthand same state is the only way the consumer financesconsumption. There is a concave transformation of the kernelbefore adding across months or states. δ controls theconcavity—it is the intertemporal and inter-state elasticity ofsubstitution with respect to the kernel.

A utility functionSuppose people order consumption-hours pairs according tothe period utility,

u(c, h) =1

1− 1/δ

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)1−1/δ

The kernel inside the parentheses governs the marginal rate ofsubstitution between consumption and work within a monthand state of the world. Because consumption enters as thelog, these preferences imply that hours of work areindependent of the wage if wage income in the same monthand same state is the only way the consumer financesconsumption. There is a concave transformation of the kernelbefore adding across months or states. δ controls theconcavity—it is the intertemporal and inter-state elasticity ofsubstitution with respect to the kernel.

Separability between c and h

If the parameter δ is infinite—a widely used specification withseparability between c and h—then the Frisch consumptiondemand is

uc(c, h) =1

c= λ or c =

constant over states of the world and time periods. Thisproperty has created the impression that the observed behaviorof consumption is inconsistent with insurance or life-cyclesmoothing. It is known that consumption is lower in states orat times when people are not working. But that finding isconsistent with the impression only in the special case ofseparability.

Frisch complementarityConsumption and non-work as Frisch complements ifconsumption rises when the wage rises (work rises andnon-work falls).

In the specification with δ =∞, consumption and work are atthe boundary.

If δ is finite, consumption and non-work are unambiguouslyFrisch complements. People consume more when wages arehigh because they work more and consume less leisure.

Nothing can be deduced about limited insurance orconsumption smoothing from the reaction of consumption tochanges in work opportunities, without further structure.

Calibration

Calibrate in a stationary, non-stochastic setting whereconsumption c and hours h are 1. The budget constraintimplies that the value of earnings, y, is also one. From thefirst-order condition for balancing c against h in the face of awage, w = 1 (so the budget constraint is c = wh), I infer thatγ = 1 under this normalization, so I will omit γ in whatfollows.

Curvature parameters

The three remaining parameters of preferences are theintercept in the kernel, κ, the overall substitution parameter,δ, and the inverse elasticity of marginal disutility of work, ψ.Calibrate to three basic properties of consumer behavior.

Risk aversion and intertemporal

substitutionThe first property is risk aversion and intertemporalsubstitution in consumption. With additively separablepreferences across states and time periods, the coefficient ofrelative risk aversion (CRRA) and the intertemporal elasticityof substitution are reciprocals of one another. A substantialbody of research suggests that the CRRA is fairly close to 2.Another body of research on the intertemporal elasticity ofsubstitution suggests that 0.5 is a reasonable value.

The CRRA in the preferences above is

CRRA = 1 +1

(κ+ log c− γ

1 + 1/ψh1+1/ψ

The calibration sets this expression equal to 2.

CRRA = 1 +1

(κ+ log c− γ

1 + 1/ψh1+1/ψ

CRRA = 1 +1

(κ+ log c− γ

1 + 1/ψh1+1/ψ

Labor supplyThe second property is the Frisch elasticity of labor supply.Another large body of research finds values for this elasticityaround 0.8. The second condition sets the Frisch elasticity tothis value. In the case of δ =∞, that elasticity is theparameter ψ, but the expression is much more complicatedotherwise.

It is the elasticity of h with respect to w in the two-equationFrisch system:

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)−1/δ

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)−1/δ

= λw.

This elasticity equals 0.8 in the calibration.

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)−1/δ

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)−1/δ

= λw.

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)−1/δ

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)−1/δ

= λw.

Complementarity of consumption

and non-workThe third property is the relation between hours of work andconsumption. A substantial body of work has examined whathappens to consumption when a person stops working, eitherbecause of unemployment following job loss or because ofretirement, which may be the result of job loss. A consensusof this research is that people consume about 20 percent lesswhen their work falls to zero, compared to their consumptionwhen working a normal amount. Incorporate this property inthe calibration by requiring

0.8(κ+ log 0.8)−1/δ = λ

Voluntary work reduction?

Notice that this calibration does not require a stand onwhether people who are not working have chosen thatcondition voluntarily, against other available choices. Theequation in the previous slide holds when the choice is eithervoluntary or involuntary. Some of the research on the effectsof unemployment and retirement on consumption haveinterpreted the decline as the result of frictions in capital andinsurance markets. But the decline may arises from the Frischcomplementarity of non-work and consumption, not fromfailures of insurance and capital markets.

Value of a job

In some models, workers pick among available jobs accordingto their values. The value of a job requiring h hours of workand paying y is, in consumption units, v(h, y) =

1− 1/δ

(κ+ log c− γ

1 + 1/ψh1+1/ψ

)1−1/δ

− c+ y

Job value as a function of hours

and earnings

Hours of workEarnings

CommentsThe previous slide shows the shape of v(h, y). The value of ajob rises linearly with its earnings, y. It falls with increasinglynegative slope with rising hours. The value of a job with zerohours and zero earnings is -0.2. The contour line rising up andto the right from the corner is the boundary between the jobsthat a worker would pick in preference to zero hours and zeroearnings. The contour line above it is the boundary betweenworking zero hours with benefits of y = 0.4 and workingpositive hours with higher earnings. This line would apply ifthe replacement rate for benefits were 40 percent and thebenefits were not integrated with the insurance covering theworker—the benefits are joint revenue to the insurer and theworker, not a transfer from insurer to worker.

Implications

If not working involves no benefits, the minimum acceptablelevel of earnings for a normal amount of work (h = 1) is 0.52.To induce a worker to accept a job with 20 percent more thannormal hours (h = 1.2), earnings need to be at least 0.75. Onthe other hand, if not working involves benefits of 0.4, theminimum acceptable level of earnings for a normal amount ofwork is that much higher, at 0.92. Workers are close to thepoint of indifference between work and non-work. Laborsupply is perfectly elastic in this case. It occurs when benefitsare at 0.48. This figure is not realistic for the United States,where typical replacement rates for unemployment benefits aregenerally less than 20 percent.

Job loss

These findings shed some light on the decline in value theoccurs when a worker loses a job. When working a normalschedule (h = 1) and receiving normal earnings (y = 1),absent any benefits, workers enjoy a value of 0.32consumption units. After a job loss, without benefits(h = y = 0), the value drops considerably to -0.16.

The Employment Relationship

Topics:

I Bilateral efficiency of the relationship

I Mechanism design

I The fixed-wage governance scheme

I The Nash bargain

Basic analysis of efficient

continuation or separation

I Z = Present value of worker’s contribution to revenue(called productivity for short).

I U = Present value of a worker’s remaining career at atime when unemployed

I V = Present value of the U that will prevail at the time anew job ends, as of the time of taking the job

Nature of the employment

opportunity

Employer has unconstrained potential number of jobs.

Z is present value of worker’s marginal revenue product.

Employer does not consider other workers for a particular job,because other workers would also add Z to the present valueof revenue.

opportunity

Worker’s alternative

opportunities

Worker does have other opportunities, but they may requiresome search time.

The asymmetry between employers and workers reflects thefact that labor is the sole primary factor in the economy andearns Ricardian rents.

Thus a worker’s earnings are expected to be close to Z.

opportunities

Assume that there is free entry to job creation, so the firmattributes no value to a vacant job.

Criterion for efficient continuation: Z + V ≥ U

Notice that the wage plays no role in the criterion forefficiency. The wage may play an important role in governanceregimes that try to achieve efficiency, however.

Efficiency

Joint value of continuation, Z+V

Joint value from separation, U

45° line

Worker stays, efficiently

Worker departs, efficiently

Another way to look at

efficiency

Criterion: Z ≥ U − V

Value from working at least as large as worker’s opportunitycost

In general, Z private to employer and U − V private to worker

efficiency

Efficiency again

Job value, Z

Opportunity cost, U-V

45° line

Efficient retention

Efficient layoff

Mechanism design

See Mas-Colell, Whinston, and Green, Chapter 23, especiallypp. 894-897.

The Myerson-Satterthwaite theorem discussed there impliesthat there is no perfect solution to the governance problem ifboth productivity Z and opportunity cost U − V are privateinformation.

Question becomes how to structure the governance of theemployment relationship to achieve at least approximateefficiency.

Mechanism design

Double Vickrey auctionEmployer bids wage WE and worker bids WK .

If WE ≥ WK , employment occurs or continues; workerreceives WE and employer pays WK .

Dominant strategy of employer is to bid WE = Z,productivity. Employer’s bid has no effect on how muchemployer pays and any lower bid would result in a lostopportunity to gain profit by employing the worker.

Dominant strategy of worker is to bid WK = U − V ,opportunity cost. Worker’s bid has no effect on how muchworker earns and any higher bid would result in a lostopportunity to gain a wage that exceeds the opportunity cost.

Why we never see the double

Vickrey auctionThe wage paid falls short of the wage received, so the partiesneed to line up an insurance company to make up thedifference.

The insurance company faces an information problem similarto the one facing the worker and employer—needs to know thedistributions of Z and U − V

Double Vickrey does not satisfy the assumptions ofMyerson-Satterthwaite, which exclude third parties that havean interest in the outcome. M-S mechanism has balancedbudget ex post, not just ex ante.

Split-the-difference auction

Also called the Nash demand auction or theChatterjee-Samuelson auction

Same bidding as in double Vickrey

If WE ≥ WK , employment occurs or continues; wage paid andreceived is .5WE + .5WK

Nash equilibrium of

split-the-difference auction

No dominant strategies

Continuum of equilibria

Any employment that occurs is efficient, but many equilibriafail to generate employment that would be efficient

Nash equilibrium of

Only productivity private

Worker’s opportunity cost, U − V , is predetermined (notnecessarily constant over time, but predictable as of the timewhen the employment relationship is established)

Let the firm make a unilateral choice about whether tocontinue employment.

The firm agrees in advance to a program of wage paymentswhose present value is W .

The firm will choose unilaterally to terminate the workerunless the firm’s benefit from continuing, Z −W , is positive.

Efficiency

The firm will make the efficient decision if the wage equals theopportunity cost, or W = U − V .

The firm considers its profit, Z −W , whereas efficiencyconsiders the surplus, Z + V − U , and the two are equal if thewage, W , causes the firm to internalize the worker’sopportunity cost, U − V .

Note that the firm has a call option on the worker’s services.

Efficiency

Properties of

employer-call-option contract

Best if the worker’s marginal product at the firm is variable,unpredictable, and unverifiable, while the worker’s opportunitycost is known in advance or is contractible.

The contract wage should equal the worker’s opportunity cost.

In principle, the worker has no right to quit, but no incentiveeither.

Employer captures all of the joint surplus.

Properties of

The worker will want to quit whenever U − V > W . Civilizedsocieties grant an almost unlimited right to quit.

In reality, there is some variability in opportunity cost, so quitswill occur.

These quits could be highly inefficient.

At a minimum, need to build in a wage premium to head offquits, but at the expense of incurring some inefficient layoffs.

Reverse case: Only opportunity

cost private

Wage is the known productivity.

Worker has put option—can choose to work at the givenwage, if it exceeds opportunity cost

Worker receives all the surplus.

Taxi drivers. Does not seem to be a common governancemechanism.

cost private

Fixed wage with both

productivity and opportunity

cost private

See Hall and Lazear, “The Excess Sensitivity of Layoffs andQuits to Demand” Journal of Labor Economics 2:233-257,1984.

Tradeoff between inefficient layoffs and inefficient quits resultsin the choice of the fixed wage somewhere in the middlebetween the average value of productivity and the averagevalue of the opportunity cost.

Fixed wage with both

productivity and opportunity

cost private

See Hall and Lazear, “The Excess Sensitivity of Layoffs andQuits to Demand” Journal of Labor Economics 2:233-257,1984.

Tradeoff between inefficient layoffs and inefficient quits resultsin the choice of the fixed wage somewhere in the middlebetween the average value of productivity and the averagevalue of the opportunity cost.

General case

Opportunity cost, U-V

45° line

Contract wage

Con- tract wage

Efficient retention

Inefficient quit

Efficient quit

Efficient layoff-quit

Efficient layoff

Inefficient layoff

Other aspects of the fixed-wage

contract

The firm captures all of the residual value from therelationship, so it has a full incentive to invest inmatch-specific capital.

Workers obtain the advantage of the investments they make toget the job in the first place, as long as termination does notoccur (search effort, moving, making new friends, . . . )

Workers have no marginal incentive to invest once the jobstarts unless the employer compensates them for investment.

contract

Credibility and holdupAt all times when the wage exceeds opportunity cost, theemployer has an incentive to cut the wage. Workers risk theloss of specific investments they made to get the job.

There is a benefit to full commitment to the wage

On the other hand, when a wage turns out to be higher thanboth productivity and opportunity cost, the worker will sufferlayoff inefficiently and the parties would be better off byagreeing on a lower wage, below productivity but aboveopportunity cost. This operates against commitment.

There’s no good overall solution when productivity andopportunity cost are both private.

Offer matching

In some markets (notably the market for economists), wagesare committed to a fixed path with one exception: Employersmatch wage offers from other employers when the wageremains below the worker’s productivity

In principle, offer matching can avoid inefficient quits, but onlywhen the employer can verify the terms of the offer.

Offer matching generates another inefficiency—excess effort toreceive offers just to gain higher pay.

Offer matching

Observable values and Nash

bargainAssume productivity and opportunity cost are observable toboth sides.

Nash bargaining solution to the bargaining problem. Bargainedwage is the equally weighted average of the reservation wagesof the two parties.

Worker’s reservation wage is the value obtained from itsoutside option of remaining unemployed, namely theopportunity cost U − V .

Employer’s reservation wage is the value lost from its outsideoption of not filling the job—recall that filling the job withanother applicant is not an option, because that applicantcould fill another job just as well. Thus the employerreservation wage is productivity Z.

Nash wage bargain, continued

2(Z + U − V )

An advantageous bargain is possible whenever employment isefficient. All separations are efficient.

Nash wage bargain, continued

2(Z + U − V )

An advantageous bargain is possible whenever employment isefficient. All separations are efficient.

Division of the surplus

The surplus from the match is S = Z − (U − V ).

The wage is W = U − V + 12S, so the worker gets half the

surplus. The employer gets the other half.

If the worker can find another job at the same wage fairly sooninstead of taking this job, U − V , alternative earnings if thisjob is not taken will be close to Z, the surplus will be fairlysmall, and W will be close to Z.

The magnitude of the surplus depends on how much searchfriction there is in the labor market, not on the fundamentalproductivity of workers.

Other aspects of the Nash

bargainThere is a 50 percent diminution of match-specificinvestments by either side—this applies to both up-frontinvestments and marginal investments.

Each side has an incentive to reduce the other side’s outsideoption value, to capture more of the joint value.

Fundamentally unrealistic because threats to take outsidevalues instead of making a bargain are not credible.

More realistic approach is alternating-offer bargaining. SeeHall and Milgrom, AER, 2008

Unemployment

Topics:

I Turnover

I Efficiency wage model

I Diamond-Mortensen-Pissarides model

I Sticky-wage model

Turnover

f = Job-finding rate, per period probability of finding a newmatch

s = Separation rate, the probability that match becomesunproductive in a given period

Turnover

f = Job-finding rate, per period probability of finding a newmatch

s = Separation rate, the probability that match becomesunproductive in a given period

Values of U , V , and Zr =Discount rate, risk adjusted

1 + r[(1− s)V + δU ]

1 + r[(1− f)U + f · (W + V )]

Substitute the Nash value for W and rewrite asJob value relation: Z + V = s

r+sU + Z

Unemployment value relation: Z + V = 2r+ffU

1 + r[(1− s)V + δU ]

1 + r[(1− f)U + f · (W + V )]

r+sU + Z

1 + r[(1− s)V + δU ]

1 + r[(1− f)U + f · (W + V )]

r+sU + Z

1 + r[(1− s)V + δU ]

1 + r[(1− f)U + f · (W + V )]

r+sU + Z

Job and unemployment values

45° line

Job value relation

Unemployment valuerelation

Glue: Match-specific capital or

job surplus

The glue is the amount of the match-specific capital, thesurplus or difference between Z + V and U .

Under 50-50 Nash bargaining, the worker owns half the glueand the employer the other half.

The more friction in the market (lower f), the more glue.

job surplus

Efficiency wage theory

Lower f creates more glue.

To make termination costly to workers and to prevent themfrom shirking, f must be low and so unemployment must behigh.

This requires a model where fcan vary. Suppose f = φ(s), anegatively-sloped function of s, the separation rate.

Efficiency wage, continued

Suppose workers will shirk unless match capital is high enoughso that their sacrifice by shirking is less than their gain.

To sustain the employment relationship, the job-finding ratemust be below some critical level, f ∗.

The job destruction rate will be 1 for f > f∗, any values ≤ d ≤ 1 for f = f∗, and s for f < f∗.

Efficiency wage model

Labor-Market Equilibrium

The value of the outside option of the job-seeker whenbargaining over the wage with a prospective employer isU − V .

Workers produce output with a present value Z over thecourse of the job. We will be concerned with the response ofunemployment and other endogenous variables to changes inZ, the driving force of fluctuations.

Add one more element: When a worker is searching ratherthan working, the worker receives unemployment benefits andenjoys not having to work, with a combined flow value of λ.

MatchingMatching of employers and workers results fromnon-contractible pre-match effort by employers—help-wantedadvertising and other recruiting costs. We describe themechanism in terms of vacancies, though this concept need benothing more than a metaphor capturing recruiting effort ofmany kinds. The key variable is θ, the ratio of vacancies tounemployment.

The job-finding rate depends on θ according to

f = φ(θ)

The recruiting rate is

ρ(θ) =φ(θ)

which is assumed to be decreasing.

f = φ(θ)

ρ(θ) =φ(θ)

f = φ(θ)

ρ(θ) =φ(θ)

Recruiting effortThere is free entry on the employer side, so that employerpre-match cost equals the employer’s expected share of thematch surplus in equilibrium. Employers control the resourcesthat govern the rate of job finding. The incentive to deploythe resources is the employer’s net value from a match,Z −W . Recruiting to fill a vacancy costs c per period.

The zero-profit condition is:

ρ(θ)(Z −W ) = c.

Employers create vacancies, drive up thevacancy/unemployment ratio θ, and drive down the recruitingrate ρ(θ) to the point that satisfies the zero-profit condition.

ρ(θ)(Z −W ) = c.

Equilibrium

The model has five endogenous variables, the worker’s value ofbeing unemployed, U , her value of employment after theprospective job, V , the job-finding rate, f , thevacancy/unemployment ratio, θ, and the present value of wagepayments, W . It has five equations.

In a stationary equilibrium, the flow rate of workers intounemployment is (1− u)s and the flow rate out ofunemployment is uf ; equilibrium requires that these ratesmust be equal. So, the unemployment rate is:

Equilibrium

The model has five endogenous variables, the worker’s value ofbeing unemployed, U , her value of employment after theprospective job, V , the job-finding rate, f , thevacancy/unemployment ratio, θ, and the present value of wagepayments, W . It has five equations.

In a stationary equilibrium, the flow rate of workers intounemployment is (1− u)s and the flow rate out ofunemployment is uf ; equilibrium requires that these ratesmust be equal. So, the unemployment rate is:

Flexible wageIn this model with Nash bargaining, the wage is the average ofproductivity Z and the worker’s opportunity cost, U − V . Thewage is highly responsive to changes in productivity because Zand U − V move together—the worker’s opportunity costU − V depends sensitively on the wages of other jobs.Further, if unemployment rises, the wage will fall because theworker’s opportunity cost falls. For both of these reasons, areduction in Z results in correspondingly large changes in Wbut only tiny changes in unemployment.

This flexible-wage property of the standard model is the pointof a paper by Shimer in the AER, March 2005.

Flexible wageIn this model with Nash bargaining, the wage is the average ofproductivity Z and the worker’s opportunity cost, U − V . Thewage is highly responsive to changes in productivity because Zand U − V move together—the worker’s opportunity costU − V depends sensitively on the wages of other jobs.Further, if unemployment rises, the wage will fall because theworker’s opportunity cost falls. For both of these reasons, areduction in Z results in correspondingly large changes in Wbut only tiny changes in unemployment.

This flexible-wage property of the standard model is the pointof a paper by Shimer in the AER, March 2005.

ParametersMeasure time in months and calibrate to a separation rate of 3percent per month and an unemployment rate of 5.5 percent.These imply a job-finding rate of 52 percent per month. Fromthe Job Openings and Labor Turnover Survey, take thevacancy/unemployment ratio, θ to be 2. Normalize Z to 1 atthe calibration point. Take the discount rate to ber = 0.05/12. Take the flow value of unemploymentcompensation and leisure to be λ = 0.4(r + s), 40 percent offlow productivity.

Solve the model for the cost of the employer’s pre-matchrecruiting, c, to fit the job-finding rate. The value isc = 0.036, about a month of wages.

ParametersMeasure time in months and calibrate to a separation rate of 3percent per month and an unemployment rate of 5.5 percent.These imply a job-finding rate of 52 percent per month. Fromthe Job Openings and Labor Turnover Survey, take thevacancy/unemployment ratio, θ to be 2. Normalize Z to 1 atthe calibration point. Take the discount rate to ber = 0.05/12. Take the flow value of unemploymentcompensation and leisure to be λ = 0.4(r + s), 40 percent offlow productivity.

Solve the model for the cost of the employer’s pre-matchrecruiting, c, to fit the job-finding rate. The value isc = 0.036, about a month of wages.

Matching

Take the job-finding function to be

φ(θ) = φ0θ0.5,

so the recruiting rate function is

ρ(θ) = φ0θ−0.5.

Calibrate the efficiency parameter φ0 to the job-finding rateand vacancy/unemployment ratio.

Matching

φ(θ) = φ0θ0.5,

ρ(θ) = φ0θ−0.5.

Matching

φ(θ) = φ0θ0.5,

ρ(θ) = φ0θ−0.5.

Equilibrium values

At the calibrated equilibrium, the wage is W = 0.965 and thejob-seeker’s value while unemployed is U = 7.61.

Equilibrium diagram

The next slide shows the determination of the equilibrium inthe standard model in a diagram with thevacancy/unemployment ratio, θ, on the horizontal axis and thewage, W , on the vertical. The downward-sloping curve depictsvalues where firms earn zero profits from hiring. Theupward-sloping curve describes the equilibrium of the rest ofthe model, including the Nash bargain for the wage. Theequilibrium is stable in the following sense: When thevacancy/unemployment ratio is below the equilibrium, thewage determined in the model leaves hiring profits foremployers. As they expand hiring, they raise thevacancy/unemployment ratio and move the labor markettoward equilibrium.

Determination of the Wage

0.0 0.2 0.4 0.6 0.8 1.0

Vacancy/Unemployment Ratio,

Zero Profit

Bargaining Equilibrium

Response to changes in ZTo see how the bargaining equilibrium curve shifts as Zchanges, take the derivative of W with respect to Z in thesystem of equations underlying the curve, keeping θ constant.The derivative is

r + f + s

2r + f + 2s.

At the calibrated values, the derivative is 0.94. In a recession,W falls directly by the 0.5 coefficient of Z and by another0.44 because the worker’s opportunity cost, U − V , falls. Thederivative of the zero-profit value of W , again holding θconstant, is 1. Hence the two curves in the diagram shiftdownward by about the same amount and conditions in thelabor market, measured by θ, hardly change. This is Shimer’spoint, repeated.

Response to changes in ZTo see how the bargaining equilibrium curve shifts as Zchanges, take the derivative of W with respect to Z in thesystem of equations underlying the curve, keeping θ constant.The derivative is

r + f + s

2r + f + 2s.

At the calibrated values, the derivative is 0.94. In a recession,W falls directly by the 0.5 coefficient of Z and by another0.44 because the worker’s opportunity cost, U − V , falls. Thederivative of the zero-profit value of W , again holding θconstant, is 1. Hence the two curves in the diagram shiftdownward by about the same amount and conditions in thelabor market, measured by θ, hardly change. This is Shimer’spoint, repeated.

· 145

Sticky wageNow consider the same model, except that the wage W isfixed. Suppose that, initially, its fixed value lies in thebargaining set of the wage bargain, [U − V, Z].

Then Z falls. The market has to slacken, with lower θ, to keepemployers at the point of zero profit. Unemployment risessharply.

The sticky-wage model delivers large changes inunemployment from small changes in Z.

Notice that bilateral efficiency is retained as long as Z doesnot fall far enough to be below W , in which case employerswould not hire workers with whom they are matched, eventhough efficiency calls for them to hire.

Finance

Based on John Cochrane, Asset Pricing

Price of an asset with payoff x

pt = Et

[δu′(ct+1)

u′(ct)xt+1

p = E (mx)

Price of an asset with payoff x

pt = Et

[δu′(ct+1)

u′(ct)xt+1

]p = E (mx)

Examples

Stock: pt = Et [mt+1(pt+1 + dt+1)]

Return: 1 = Et [mt+1Rt+1]

Excess return: 0 = Et

[mt+1(R

at+1 −Rb

One-period bond: pt = Et [mt+1]

Examples

[mt+1(R

at+1 −Rb

Examples

[mt+1(R

at+1 −Rb

Examples

[mt+1(R

at+1 −Rb

More examples

Risk-free return: 1 = Rf Et [mt+1]

Option: pt = Et [mt+1 max(St+1 −K, 0)]

More examples

Risk-free return: 1 = Rf Et [mt+1]

Option: pt = Et [mt+1 max(St+1 −K, 0)]

Risk-free rate

With log-normal consumption,

log(Rf ) = δ + γ Et (∆ log ct+1)−γ2

2σ2t (∆ log ct+1)

Risk-free rate

Risk and price

p = E (mx) = E (m) E (x) + cov(m,x)

E (m) =1

p =E (x)

Rf+ cov(m,x)

p = discounted expected return plus risk adjustment

Risk and price

p = E (mx) = E (m) E (x) + cov(m,x)

E (m) =1

p =E (x)

Rf+ cov(m,x)

Risk and price

p = E (mx) = E (m) E (x) + cov(m,x)

E (m) =1

p =E (x)

Rf+ cov(m,x)

Risk and price

p = E (mx) = E (m) E (x) + cov(m,x)

E (m) =1

p =E (x)

Rf+ cov(m,x)

Risk premium over safe rate

1 = E (mRi) = E (m) E (Ri) + cov(m,Ri)

E (m)= E (Ri) +

E (m)cov(m,Ri)

E (Ri) = Rf −Rfcov(m,Ri)

E (m)= E (Ri) +

E (m)cov(m,Ri)

E (m)= E (Ri) +

E (m)cov(m,Ri)

No discount for idiosyncratic risk

x = x+ ε

ε has mean 0 and is uncorrelated with m

p =E (x) + E (ε)

Rf+ cov(m, x) + cov(m, ε)

p =E (x)

Rf+ cov(m, x)

x = x+ ε

p =E (x) + E (ε)

p =E (x)

Rf+ cov(m, x)

x = x+ ε

p =E (x) + E (ε)

p =E (x)

Rf+ cov(m, x)

x = x+ ε

p =E (x) + E (ε)

p =E (x)

Rf+ cov(m, x)

Valuation is linear

px+y = E [m(x+ y)] = E (mx) + E (my) = px + py

No payoff for diversification in a diffusely held firm

No reason for such a firm to hedge any risk, systematic oridiosyncratic

No benefit to acquisition of assets or spinning assets off unlesstheir returns change as a result of change in control

Valuation is linear

E (Ri) = Rf +cov(m,Ri)

V(m)(−RfV(m))

E (Ri) = Rf + βiλ

Expected return is the risk-free rate plus the coefficient of theregression of the return on the stochastic discounter multipliedby a factor that is the same for all assets

λ is the price of risk and βi is asset i’s risk

V(m)(−RfV(m))

E (Ri) = Rf + βiλ

V(m)(−RfV(m))

E (Ri) = Rf + βiλ

V(m)(−RfV(m))

E (Ri) = Rf + βiλ

ApplicationOne-year real returns on one-year Treasury bills and fromholding the S&P 500 stock portfolio for one year (includingdividends), 1959 through 2009.

Form the discounter from consumption of nondurable goodsper person, with γ=2.

Find the value of the utility discount factor from the conditionthat the discounted average real return ratio on bills is one.This turns out to be 1.0064, a problem. The observed realreturn is too small to square with the observed growth ofconsumption per person.

The average value of the discounted return on the stockmarket is 1.014, higher than the CCAPM value of one. Butthe standard error is 0.022, so the discrepancy could easilyarise from sampling error.

· 157

Application, continued

The value of the coefficient of relative risk aversion γ thatresults in a discounted stock-market return ratio of one is11.25, implausibly high.

If the calculation starts with earlier data, and omits the past11 years, the excess return of the stock market over Treasurybills (the equity premium) is much higher and the value of γneed to rationalize is even higher, such as 40.

syllabus: economics 211, second halfweb.stanford.edu/~rehall/econ211winter2011.pdfanswers will try...

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