$©$ March 14, 2023, Christopher D. Carroll LucasAssetPrice

The Lucas Asset Pricing Model

Lucas-Asset-Pricing-Model is a notebook that solves the model numerically

1 Introduction/Setup

Lucas [1978] considers an economy populated by inﬁnitely many¹ identical individual consumers. The only assets are a set of identical inﬁnitely-lived trees. Aggregate output is the fruit that falls from the trees, and cannot be stored (it would rot!); because $′ u (c) > 0 ∀ c > 0$ , the fruit is all eaten:

ctLt = dtKt

(1)

where $ct$ is consumption of fruit per person, $Lt$ is the population, $Kt$ measures the stock of trees, and $d t$ is the exogenous output of fruit that drops from each tree. In a given year, each tree produces exactly the same amount of fruit as every other tree, but $d t$ varies from year to year depending on the weather.²

An economy like this, in which output arrives without any deliberate actions on the part of residents, is called an ‘endowment’ economy (or, sometimes, an ‘exchange’ economy).³

2 The Market for Trees

If there is a perfect capital market for trees, the price of trees $P t$ must be such that, each period, each (identical) consumer does not want either to increase or to decrease their holding of trees.⁴

If a tree is sold, the sale is assumed to occur after the existing owner receives that period’s fruit ( $Pt$ is the ‘ex-dividend’ price). Total resources available to consumer $i$ in period $t$ are the sum of the fruit received from the trees owned, $dtkit$ , plus the potential proceeds if the consumer were to sell all his stock of trees, $i Ptk t$ . Total resources are divided into two uses: Current consumption $cit$ and the purchase of trees for next period $kit+1$ at price $Pt$ ,

Uses of resources Total resources ◜----◞◟ ---◝ ◜----◞◟ ---◝ kit+1Pt + cit = dtkit + Ptkit i i i kt+1 = (1 + dt∕Pt )kt − c t∕Pt.

(2)

3 The Problem of an Individual Consumer

Consumer $i$ maximizes

[ ] ∑∞ v (mi ) = max 𝔼i βnu (ci ) t t t+n n=0 s.t. i i i kt+1 = (1 + dt ∕Pt)k t − ct∕Pt i i m t+1 = (Pt+1 + dt+1 )kt+1.

Rewriting in the form of Bellman’s equation,

v (mi ) = max u(ci) + β 𝔼i [v(mi )] , t {ci} t t t+1 t

the ﬁrst order condition tells us that

⌊ ( i ) ⌋ ◜------------------mt◞+◟1------------------◝ ′ i i | ′ i -∂-- | ( i i )| | 0 = u (ct) + β 𝔼t ⌈v (m t+1) i ( (Pt+1 + dt+1 ) (1-+-dt∕Pt-)k-t −-ct∕Pt- ) ⌉ ∂c t ◟ ◝i◜ ◞ kt+1

(3)

⌊ ( ) ⌋ ′ i i|| ′ i || Pt+1-+--dt+1-|| || u (c t) = β 𝔼 t⌈v (m t+1 )( P ) ⌉ ◟----◝◜t----◞ ≡Rt+1 i[ ′ i ] = β 𝔼 t Rt+1v (m t+1 )

(4)

where $Rt+1$ is the return factor that measures the resources in period $t + 1$ that are the reward for owning a unit of trees at the end of $t$ .

The Envelope theorem tells us that $′ i ′ i v (m t+1) = u (ct+1 )$ , so (4) becomes

[ ( ) ] ′ i i ′ i Pt+1-+--dt+1- u (ct) = β 𝔼 t u (c t+1 ) P [( ) t ] i u′(cit+1) Pt = β 𝔼 t --′--i--- (Pt+1 + dt+1 ) . u (ct)

(5)

4 Aggregation

Since all consumers are identical, $ci = cj ∀ i,j t t$ , so henceforth we just call consumption per capita $ct$ . Since aggregate consumption must equal aggregate production because fruit cannot be stored, normalizing the population to $L = 1 ∀ t t$ and the stock of trees to $K = 1 ∀ t t$ , equation (1) becomes:

c = d . t t

(6)

Substituting $ct$ and $ct+1$ for $i ct$ and $i ct+1$ in (5) and then substituting $dt$ for $c t$ we get

[( ) ] u′(dt+1)- Pt = β 𝔼t ′ (Pt+1 + dt+1 ) . u (dt)

(7)

We can rewrite this more simply if we deﬁne

( ) u ′(d ) Mt,t+n = βn -----t+n-- u′(dt)

(8)

$Mt,t+n$ is called the ‘stochastic discount factor’ because (a) it is stochastic (thanks to the shocks between $t$ and $t + n$ that determine the value of $dt+n$ ); and (b) it measures the rate at which all agents in this economy in period $t$ will discount a dividend received in a future period, e.g. $t + 1$ :

Pt = 𝔼t [Mt,t+1 (Pt+1 + dt+1 )].

(9)

A corresponding equation will hold in period $t + 1$ (and in period $t + 2$ and beyond):

Pt+1 = 𝔼t+1 [Mt+1,t+2 (Pt+2 + dt+2 )]

(10)

so we can use repeated substitution, e.g. of (10) into (9), to get

Pt = 𝔼t [Mt,t+1dt+1 ] + 𝔼t[Mt,t+1 𝔼t+1 [Mt+1,t+2dt+2 ]] + ....

(11)

The ‘law of iterated expectations’ says that $𝔼t [𝔼t+1 [Pt+2 ]] = 𝔼t [Pt+2 ]$ ; given this, and noting that $Mt,t+2 = Mt,t+1Mt+1,t+2$ , (11) becomes:

Pt = 𝔼t [Mt,t+1dt+1 + Mt,t+2dt+2 + Mt,t+3dt+3 + ...].

(12)

So, the price of the asset is the present discounted value of the stream of future ‘dividends,’ where the (potentially stochastic) factor by which (potentially stochastic) dividends received in $t + n$ are discounted back to $t$ is $Mt,t+n$ .⁵

5 Specializing the Model

This is as far as we can go without making explicit assumptions about the structure of utility. If utility is CRRA, $− 1 1− ρ u (c) = (1 − ρ ) c ,$ substituting $u ′(d ) = d − ρ$ into (7) yields

P = βd ρ 𝔼 [d− ρ (P + d ) ] t t [ t t+1 t+1 ] t+1 Pt Pt+1 1− ρ -ρ-= β 𝔼t --ρ-- + dt+1 dt d t+1

(13)

The particularly special case of logarithmic utility (which Lucas emphasizes) corresponds to $ρ = 1$ , implying $1− ρ d t+1 = 1$ which (again using the law of iterated expectations) allows us to simplify the second version of (13) to

P ( [P ] ) --t = β 1 + 𝔼t --t+1 dt dt+1 ( ( [ ] ) ) = β 1 + β 1 + 𝔼 Pt+2- t dt+2 { [ ] } --β---- n− 1 Pt+n-- = 1 − β + β 𝔼t nl→im∞ β d . t+n

If the price is bounded (it cannot ever go, for example, to a value such that it would cost more than the economy’s entire output to buy a single tree), it is possible to show that the $lim$ term in this equation goes to zero. Using the usual deﬁnition of the time preference factor as $β = 1∕(1 + 𝜗)$ where $𝜗$ is the time preference rate, the equilibrium price is:

( ) --β---- Pt = dt 1 − β ( ) 1 = dt ---------- ( 1∕β − 1 ) 1 = dt ----------- 1 + 𝜗 − 1 d = -t- 𝜗

or, equivalently, the ‘dividend-price ratio’ is always $dt ∕Pt = 𝜗$ .

It may surprise you that the equilibrium price of trees today does not depend on the expected level of fruit output in the future. If the weather was bad this year, but is expected to return to normal next year (and, by deﬁnition, is expected to be equal to normal in subsequent years), you might think that the price today would mostly reﬂect the ‘normal’ value of fruit prodution that the trees produce, not the (temporarily low) value that happens to obtain today.

The above derivation says that intuition is wrong: Today’s price depends only on today’s output.

Nevertheless, the logic (higher future output is a reason for higher current prices) is not wrong; but it is (exactly) counterbalanced by another, and subtler, fact: Since future consumption will equal future fruit output, higher expected fruit output means lower marginal utility of consumption in that future period of (more) abundant fruit (basically, people are starving today, which reduces the attractiveness of cutting their consumption to buy trees that will produce more in a period when they expect not to be starving). These two forces are the manifestation of the (pure) income eﬀect and substitution eﬀect in this model (there is no human wealth, and therefore no human wealth eﬀect). In our assumed special case of logarithmic utility, income and substitution eﬀects are of the same size and opposite sign so the two forces exactly oﬀset.

6 The ‘Rate of Return’ in a Lucas Model

We can decompose the return factor attributable to ownership of a share of capital (cf. (4)) by adding and subtracting $P t$ in the numerator:

( P + P − P + d ) rt+1 = -t+1------t----t-----t+1- Pt ( ) ΔPt+1--- dt+1- = 1 + P + P t t

so the ‘rate of return’ is

ΔPt+1 dt+1 rt+1 = --------+ ----- Pt Pt

(14)

which is a useful decomposition because the two components have natural interpretations: The ﬁrst is a ‘capital gain’ (or loss), and the second can plausibly identiﬁed as ‘the interest rate’ paid by the asset (because it corresponds to income received regardless of whether the asset is liquidated).

In models that do not explicitly discuss asset pricing, the implicit assumption is usually that the price of capital is constant (which might be plausible if capital consists mostly of reproducible items like machines,⁶ rather than Lucas trees). In this case

( ) dt+1- Rt+1 = 1 + Pt

says that the only risk in the rate of return is attributable to unpredictable variation in the size of dividend/interest payments. Indeed, if additional assumptions are made (e.g., perfect capital markets) that yield the conclusion that the interest rate matches the marginal product of capital, then such models generally imply that variation in returns (at least at high frequencies) is very small, because aggregate capital typically is very stable from one period to the next; if the aggregate production function is stable, this implies great stability in the marginal product of capital.

7 Aggregate Returns Versus Individual Returns

One of the subtler entries in Aristotle [350 BC]’s catalog of common human reasoning errors was the ‘fallacy of composition,’ in which the reasoner supposes that if a proposition is true of each element of a whole, then it must be true of the whole.

The Lucas model provides a counterexample. From the standpoint of any individual (atomistic) agent, it is quite true that a decision to save one more unit will yield greater future resources, in the amount $Rt+1$ . But from the standpoint of the society as a whole, if everyone decided to do the same thing (save one more unit), there would be no eﬀect on aggregate resources in period $t + 1$ . Put another way, for any individual agent, it appears that the ‘marginal product of capital’ is $Rt+1$ , but for the society as a whole the marginal product of capital is zero.

The proposition that the return for society as a whole must be the same as the return that is available to individuals is an error because it implicitly assumes that there are no general equilibrium eﬀects of a generalized desire to save more (or, more broadly, there is no interaction between the decisions one person makes and the decisions of another person). The Lucas model provides a counterexample in which, if everyone’s preferences change (e.g., $𝜗$ goes down for everyone), the price of the future asset is aﬀected – indeed, it is aﬀected in a way that is suﬃcient to exactly counteract the increased desire for ownership of future dividends (since there is a ﬁxed supply of assets to be owned, the demand must be reconciled with that preexisting supply).

Aristotle was a smart guy!

8 A Surprise

In the case where dividends are identically individually lognormally distributed, $log dt+1 ∼ 𝒩 (− σ2r∕2, σ2r)$ , the appendix shows that

log P = ρ log d + ρ (ρ − 1)σ2 ∕2 − 𝜗 t t r

(15)

and thus the variances obey

var (log P ) = ρ2 var(log d ).

(16)

Given that $ρ > 1$ , this derivation yields some interesting insights:

(the log of) asset prices will be more volatile than (the log of) dividends
- This would be even more true if $𝜗$ were stochastic, as is often assumed in the asset pricing literature
An increase in risk aversion $ρ$ increases the price $Pt$ (because $2 ρ(ρ − 1)σ r∕2 > 0$ and an increase in $ρ$ increases its size)

The second point is surprising; ChatGPT correctly summarizes the usual received wisdom about risk aversion by saying “In general, an increase in risk aversion can cause a decrease in the overall stock market ...”⁷ The reason for this prediction is intuitive: an increase in risk aversion makes people want the risky asset less, and if they want it less one would think that the price should be lower. But no: an increase in risk aversion increases the price of the risky asset. The resolution of this conﬂict between ChatGPT and Lucas comes from realizing that the usual model (ChatGPT will always summarize ‘the usual model’) is one in which investors have access to a safe asset as well as a risky one, while in the Lucas model presented here the only asset available is risky.

9 Analytical and Numerical Solutions

The appendices derive various results about the solution to the model under diﬀerent assumptions. But, unfortunately, the model has analytical solutions (like, $P = d∕𝜗$ ) or approximate analytical solutions only in special circumstances. The accompanying DemARK notebook shows how to solve the model numerically for a simple case where there is no such analytical solution (the case where dividends follow an AR(1) process), and also shows how the numerical solution compares with the approximate analytical solution in the CRRA utility case.

Appendix: Analytical Solutions in CRRA Utility Case

A When Dividends are IID

When $ρ > 1$ , we can rewrite (13) by multiplying the second term on the right by $Pt+1 ∕Pt+1$ , yielding

( ) [ ( ) ] Pt- Pt+1- Pt+1-dt+1- ρ = β 𝔼t ρ + ρ dt [ d t+1( dt+1 Pt+)1] Pt+1 dt+1 = β 𝔼t --ρ-- 1 + ----- d t+1 Pt+1 [ ( − ρ ρ ) ] Pt+1- dt+1d-t+1dt+1- = β 𝔼t ρ 1 + d t+1 Pt+1 [ P ] = β 𝔼t --t+1- + d1− ρ d ρt+1 t+1

(17)

and we can hypothesize that there is a solution with a constant ratio $δ = d ρ∕P$ . In that case this equation simplifes to

− 1 [ − 1 1− ρ] δ = β 𝔼t δ + d t+1

(18)

Suppose $d t+n$ is identically individually distributed in every future period, so that its expectation as of $t$ is the same for any date $n > 0$ :

d`≡ 𝔼 [d1− ρ]. t t+n

(19)

Now note that (18) can be rewritten as

( [ ]) P P -tρ-= β d`+ 𝔼t --tρ+1 dt dt+1 ( [ ] ) = β `d 1 + β + β 𝔼 Pt+2- t dρ ( t+2 ) | [ [ ]] | `| 2 n− 1 Pt+n-- | = β d|(1 + β + β + ...+ 𝔼t nli→m∞ β d ρ |) ◟-----------◝◜-----t+n--◞ assume goes to zero ( ) β `d = ------- 1 − β ( ) d` = --− 1---- β − 1

(20)

To make further progress, suppose that the iid process for the stochastic component of dividends is a mean-one lognormal: $2 2 log dt+n ∼ 𝒩 (− σ ∕2,σ ) ∀ n$ so that $𝔼t [dt+n ] = 1 ∀ n$ (see $[ELogNormMeanOne ]$ ), in which case $[ELogNormTimes ]$ can be used to show that⁸

ρ(ρ− 1)σ2∕2 `d = e d

(21)

and if we deﬁne the discount factor as $β = 1∕ (1 + 𝜗 )$ then $− 1 β = 1 + 𝜗$ ; substituting into (20),

( ρ(ρ− 1)σ2∕2) Pt- e-------d--- d ρ = 𝜗 t ( ) ρ ρ(ρ− 1)σ2∕2 d-te-------d-- Pt = 𝜗

(22)

So the log is

2 log Pt = ρ log dt + ρ (ρ − 1)σ d∕2 − 𝜗

(23)

as asserted in the main text.

B When Dividends Follow a Random Walk

The polar alternative to IID shocks would be for dividends to follow a random walk: $2 2 log (dt+1∕dt ) ∼ 𝒩 (− σ d∕2, σd )$ . (Recall that this assumption implies that the expected arithmetic growth rate for dividends is zero: $2 2 0 𝔼t[dt+1 ∕dt] = exp (− σd ∕2 + σd∕2 ) = e = 1$ ; later we will consider the case in which dividends have a positive trend growth rate).

Now divide both sides of (13) by $dt$ , and rewrite the object inside the expectations operator by multiplying the ﬁrst term by $dt+1$ and dividing the second term by $dt+1$ , yielding

[ ] Pt = βd ρ𝔼t d − ρ (Pt+1 + dt+1 ) ( ) t t+1 [ ( ) ] Pt- − (1− ρ) 1− ρ Pt+1- = βd t 𝔼t dt+1 + 1 dt [ dt+1 ] ( d )1 − ρ ( P ) = β 𝔼t -t+1- -t+1-+ 1 . dt dt+1

(24)

Note that our assumption here about the distribution of $dt+1∕dt$ is identical to the assumption about $d t+1$ above, so the expectation will be the same $`d$ . Now hypothesize that there will be a solution under which the price-dividend ratio is a constant; call it $r− 1$ :

[ ] r − 1 = β d`(r− 1 + 1 ) ` 1 = β d(1 + r) 1 ----= 1 + r β `d

(25)

but $2 log d` = ρ (ρ − 1)σd ∕2$ , while OverPlus says that $log β ≈ − 𝜗$ and ExpEps says $log (1 + r ) ≈ r$ , so taking the log of both sides of this equation therefore yields

2 𝜗 − ρ (ρ − 1 )σd∕2 ≈ r

(26)

In the case of logarithmic utility ( $ρ = 1$ ), this equation conﬁrms our earlier conclusion in the main text that the arithmetic interest rate must be equal to the time preference rate in order for the economy to be in equilibrium.

But with $ρ > 1$ , the expression subtracted from $𝜗$ must be positive. That is simply saying that if consumers have risk aversion higher than that of logarithmic utility, the equilibrium price-dividend ratio must be such that consumers expect a higher $r$ to make them willing to hold the asset. For example, in the case where $ρ = 2$ , the expression becomes

r ≈ 𝜗 − σ2 , d

(27)

which indicates that the arithmetic rate of return must be lower than the time preference rate by the amount $σ2d$ in order to induce consumers to hold the risky asset. Remembering that, for a given dividend payout, a lower $d ∕P$ must be accomplished by a higher price, this says that the price of the risky asset will be higher in the economy with $ρ > 1$ .

This equation seems puzzing because with a low enough time preference rate, and with $ρ > 1$ and $2 σ d > 0$ , it could possibly imply that $dt∕Pt = r < 0$ . If prices are positive, this must mean that dividends are negative - which was ruled out by assumption in the statement of the model (because aggregate consumption is equal to aggregate dividends, and negative $c t$ would yield undeﬁned utility).

What this reveals is that the model has no solution unless people are suﬃciently impatient. (This is another impatience condition like those articulated in PerfForesightCRRA and TractableBufferStock – in this case, you must be impatient enough to want to hold the risky asset despite its riskiness).

Recall our earlier assumption that $`d = 1$ ; that is, the arithmetic growth rate of dividends is zero. If, instead, dividends had a positive growth factor $γ Γ = e ≈ 1 + γ$ , the consequence is that a further subtraction from $𝜗$ is $(1 − ρ)γ$ :

2 r ≈ 𝜗 + (ρ − 1)γ − ρ(ρ − 1)σ d∕2

(28)

This makes the impatience condition

( ) 𝜗 ≳ (ρ − 1) ρ σ2∕2 − γ . d

(29)

The eﬀect of growth on the required rate of return therefore depends on the sign of $(ρ − 1)$ . If $ρ > 1$ , positive growth increases the required rate of return and makes the impatience condition easier to satisfy. By increasing the required rate of return, for any given level of dividends the price must be lower than in the case where $γ = 0$ . How can an asset with growing dividends be worth less than one with dividends that do not grow? The answer is that—because consumption must be equal to dividends—growing dividends also mean growing consumption, and an agent who knows that he will have a higher consumption in the future will be less willing to pay for further increases in this future level of consumption. Therefore, there are two eﬀects in play:

Growth increases the future dividends, making the asset more valuable for any ﬁxed consumption path.
Growth increases the agent’s future consumption, making him less willing to pay for any stream of dividends.

How $ρ$ compares to $1$ determines which of these two eﬀects dominates.

All of this ﬁnally puts us in position to calculate the price from the dividend, which we can do using an updated version of (22) which incorporates growth:⁹

Pt β --- = -----------2--------- dt e(ρ− 1)(γ− ρσd∕2) − β

(30)

from which we can directly read oﬀ the following propositions:

For any given $dt$ , if $ρ > 1$ and the impatience condition is satisﬁed, the price $Pt$ is higher when:

Dividend growth $γ$ is lower
The time preference rate $𝜗$ is smaller (or $β$ is larger, people are more patient)

References

Aristotle. On Sophistical Refutations. The Wikipedia Foundation, 350 BC. URL https://en.wikipedia.org/wiki/Sophistical_Refutations.

Robert E. Lucas. Asset prices in an exchange economy. Econometrica, 46:1429–1445, December 1978. URL http://www.jstor.org/stable/1913837. Available at http://www.jstor.org/stable/1913837.

Rajnish Mehra and Edward C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics, 15:145–61, 1985. URL http://ideas.repec.org/a/eee/moneco/v15y1985i2p145-161.html.