c©April 26, 2013, Christopher D. Carroll LucasAssetPrice

The Lucas Asset Pricing ModelLucas (1978) considers an economy populated by a large number of identical in-

dividual consumers in which the only assets are a set of identical infinitely-livedtrees. Aggregate output equals the fruit of the trees, and cannot be stored. Thus,ctLt = dtKt where ct is consumption of fruit per person, Lt is the population, dt isthe exogenous output of fruit per tree and Kt is the stock of trees. Normalize the(unchanging) aggregate stock of trees to Kt = 1 ∀ t and the aggregate populationto 1, Lt = 1 ∀ t. Assume that in a given year, each tree produces exactly the sameamount of fruit as every other tree, but the total harvest dt per tree varies from yearto year depending on the weather. Each consumer owns the same number of trees(because consumers are identical).Now consider the market for buying and selling trees. In equilibrium, the price of

trees must be such that, each period, each consumer does not want either to increaseor to decrease his holding of trees (because the aggregate number of trees cannot bechanged). Let Pt denote the price of a tree in period t (in terms of units of fruit),and assume that if the tree is sold, the sale occurs after the existing owner receivesthat period’s fruit. The total resources available for consumption by consumer i inperiod t are the sum of the fruit received from the trees owned, dtkit, plus the potentialproceeds if the consumer were to sell all his stock of trees, Ptk

it. Total resources are

divided into two uses: Current consumption cit and the purchase of trees for nextperiod kit+1 at price Pt. Thus we can write the consumer’s budget constraint as

Uses of resources︷ ︸︸ ︷kit+1Pt + cit =

Total resources︷ ︸︸ ︷dtk

it + Ptk

it (1)

kit+1 = (1 + dt/Pt)kit − cit/Pt. (2)

Finally, assume that consumer i maximizes

v(mit) = max Ei

t

[∞∑n=0

βn log cit+n

](3)

s.t.kit+1 = (1 + dt/Pt)k

it − cit/Pt (4)

mit+1 = (Pt+1 + dt+1)k

it+1. (5)

Rewriting in the form of Bellman’s equation,

v(mit) = max

{cit}u(cit) + β Ei

t

[v(mi

t+1)]

(6)

= max{cit}

u(cit) + β Eit

v

(Pt+1 + dt+1) [(1 + dt/Pt)kit − cit/Pt]︸ ︷︷ ︸

kit+1

The first order condition tells us that

0 = u′(cit) + β Eit

v′(mit+1)

d

dcit

(Pt+1 + dt+1) [(1 + dt/Pt)kit − cit/Pt]︸ ︷︷ ︸

kit+1

u′(cit) = β Eit

[v′(mi

t+1)

(Pt+1 + dt+1

Pt

)]. (7)

The usual Envelope theorem argument tells us that v′(mit+1) = u′(cit+1), so (7)

becomes

u′(cit) = β Eit

[u′(cit+1)

(Pt+1 + dt+1

Pt

)]. (8)

Now using this and the fact that u′(c) = 1/c (because we assumed log utility),1

1

cit= β Ei

t

[1

cit+1

Pt+1 + dt+1

Pt

](9)

Pt = citβ Eit

[Pt+1 + dt+1

cit+1

]. (10)

As mentioned above we assume for simplicity that the aggregate population isnormalized to measure 1 (cf. handout Aggregation). The assumption that all con-sumers are identical means that we can write cit = cjt ∀ i, j ∀ t. Call normalizedaggregate consumption per capita ct. Now recall that aggregate consumption mustequal aggregate production in this economy because we assumed that fruit cannotbe stored. With population Lt = 1 and stock of fruit trees Kt = 1, aggregateconsumption equals aggregate production means

ctLt = dtKt (11)ct = dt. (12)

Thus, substituting ct and ct+1 for cit and cit+1 in (10) and then substituting dt forct we get

Pt = ctβ Et

[Pt+1 + dt+1

ct+1

](13)

= dtβ Et

[Pt+1 + dt+1

dt+1

](14)

1The key results below, including the constancy of the dividend-price ratio, go through even inthe case where relative risk aversion is greater than 1.

2

= dtβ Et [1 + Pt+1/dt+1] (15)Pt

dt= β

(1 + Et

[Pt+1

dt+1

])(16)

because in every period the amount of consumption per capita is exactly equal tothe amount of fruit produced per tree in that year.Now note that an identical equation will hold in period t+ 1 and so on:

Pt+1

dt+1

= β

(1 + Et+1

[Pt+2

dt+2

])(17)

......

Pt+n

dt+n

= β

(1 + Et+n

[Pt+n+1

dt+n+1

])(18)

But the ‘law of iterated expectations’ says that Et[Et+1[Zt+2]] = Et[Zt+2]. Usingthis law, we can repeatedly substitute for P to obtain:

Pt

dt= β

(1 + Et

[Pt+1

dt+1

])(19)

= β

(1 + β(1 + Et

[Pt+2

dt+2

])

)(20)

= β

(1 + β(1 + β(1 + Et

[Pt+3

dt+3

]))

)(21)

= β

(1 + β + β2 + β3 + . . .+ Et

[limn→∞

βn−1[Pt+n

dt+n

]])(22)

=β

1− β+ β Et

{limn→∞

βn−1[Pt+n

dt+n

]}. (23)

If we assume that the price is bounded (it cannot ever go, for example, to a valuesuch that it would cost more than the economy’s entire output to buy a single tree),we can show that the limit term in this equation goes to zero. Thus, the equilibriumprice is:

Pt = dt

(β

1− β

)(24)

= dt

(1

1/β − 1

)(25)

= dt

(1

1 + ϑ− 1

)(26)

=dtϑ. (27)

Note the surprising result that the equilibrium price of trees today does not depend

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on the expected level of fruit output in the future. The reason is that there are twoeffects of an increase in the expected amount of fruit at some future date. The first isthe fact that at a given marginal utility of consumption, the higher expected futurefruit production increases the attractiveness of owning the trees that will producethat future fruit. This tends to raise the current price of a tree. However, sinceconsumption equals current fruit output in this model, higher expected fruit outputin that future period means higher consumption and thus lower marginal utility ofconsumption in that future period. This tends to reduce the attractiveness of owningtrees - the tree is going to pay off more in a time when marginal utility is expected tobe low – and thus tends to lower the current value (and hence price) of a tree. Thesetwo forces are the manifestation of the (pure) income effect and substitution effect inthis model (there is no human wealth, and therefore no human wealth effect).2Note finally that equation (27) can be rewritten as

dtPt

= ϑ. (28)

We are now in position to discuss the determinants of interest rates in this model.To do so, we need to highlight and distinguish between two concepts that are oftenconflated but need to be separated here: The marginal product of capital, and theintertemporal tradeoff between goods today and goods tomorrow.Start with the marginal product. The consumer’s marginal return to the ownership

of an extra unit of trees in period t + 1 is dt+1. In models we have used until now,the price of a unit of capital has implicitly been fixed at Pt = 1 ∀ t. Here, however,the price of a unit of capital varies over time – in fact, the price of capital variesone-for-one with the current return on capital. If we think of ‘the interest rate’ asthe ratio of capital income generated by ownership of a unit of capital to the value ofthat capital, d/P, then the interest rate in this economy is constant at ϑ (cf. (28)).The ‘intertemporal tradeoff’ definition of interest rates, however, gives us a very

different answer. That answer can be understood most directly by examining the con-sumption Euler equation in this model. Notice that, interpreting D as the equivalentof the return factor (R in a riskless model), (8) collapses to the usual Euler equationu′(ct) = β Et[u

′(ct+1)Dt+1] if

Dt+1 =

(Pt+1 + dt+1

Pt

). (29)

Why would this be a sensible definition of the return factor in this context?Because, from the standpoint of an individual, Dt+1 is the return, in terms of nextyear’s resources, to purchasing an additional unit of capital in this period.

2In the case of logarithmic utility, income and substitution effects are of the same size andopposite sign so the two forces exactly offset each other, leaving the current price of a tree unchangedin the face of a rise in expected future fruit output.

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Notice further that we can elaborate this equation by adding and subtracting Pt,obtaining

Rt+1 =

(Pt+1 + Pt − Pt + dt+1

Pt

)(30)

=

(1 +

∆Pt+1 + dt+1

Pt

)(31)

= 1 +

(∆Pt+1 + dt+1

Pt+1

Pt+1

Pt

)(32)

≈ 1 +

(∆Pt+1 + dt+1

Pt+1

)(33)

≈ 1 + ϑ+

(∆Pt+1

Pt+1

)(34)

rt+1 ≈ ϑ+

(∆Pt+1

Pt+1

)(35)

where the approximation is justified by the assumption that prices will not changeby a large amount from one period to the next so that Pt+1/Pt ≈ 1.The distinction between this definition and the other one is therefore essentially due

to ∆Pt+1/Pt+1. What is this? It is the capital gain (or loss) from capital ownership.So the alternative definitions boil down to whether one wants to define the interestrate as reflecting ‘income over value’ or ‘income plus capital gains over value.’ (Theapproximation in the derivation above holds when capital gains are not expected tobe very large, so that Pt+1/Pt will not be too far from 1).Now note one final point: The Euler equation analysis that yielded this definition

of the aggregate interest rate was the result of an optimization problem from thestandpoint of an individual (atomistic) consumer, whose decisions were assumed tohave no effect on market prices or interest rates. But from the standpoint of theaggregate economy for the whole island, there is no ability to increase period t + 1output by reducing period t consumption. So in the aggregate, the intertemporaldefinition of the interest rate yields a conclusion that the aggregate interest factoris zero. This just reflects the fact that if there were an overall ‘social planner’for this economy, capable of setting the aggregate saving rate to any amount lessthan total output, that social planner would set the aggregate level of saving to zero(consumption equal to all fruit), because saving more today (in the aggregate) doesnot yield more output tomorrow (all the fruit will rot; none will result in more trees).

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ReferencesLucas, Robert E. (1978): “Asset Prices in an ExchangeEconomy,” Econometrica, 46, 1429–1445, Available at http://cedec.wustl.edu/azariadis/teaching/e502Sp08/papers/Lucas_ecta78.pdf.

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