The Lucas Asset Pricing Model

1Introduction/Setup¶

Lucas (1978) considers an economy populated by infinitely many^[1] identical individual consumers, in which 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 $\uFunc^{\prime}(\cRat)>0~\forall~\cRat$, the fruit is all eaten:

where $\cRat_{t}$ is consumption of fruit per person, $\Pop_{t}$ is the population, $\Kap_{t}$ measures the stock of trees, and $\dvdnd_{t}$ is the exogenous output of fruit that drops from each tree. A crucial assumption is that the stock of trees is exogenous: you cannot consume a little less fruit and have more trees next period. In a given year, each tree produces exactly the same amount of fruit as every other tree, but $\dvdnd_{t}$ varies from year to year depending on the weather. 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).^[2]

2The Market for Trees¶

We now consider a market in which individual consumers can buy and sell trees. This requires some mental gymnastics: with infinitely many identical consumers, we must find a price at which every one of them is simultaneously satisfied. If there is a perfect capital market for trees, the price of trees $\Price_{t}$ must be such that, each period, each (identical) consumer does not want either to increase or to decrease their holding of trees.^[3]

If a tree is sold, the sale is assumed to occur after the existing owner receives that period’s fruit ($\Price_{t}$ 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, $\dvdnd_{t}\kap_{t}^{i}$, plus the potential proceeds if the consumer were to sell their whole stock of trees, $\Price_{t}\kap_{t}^{i}$. Total resources are divided into two uses: Current consumption $c_{t}^{i}$ and the purchase of trees for next period $\kap_{t+1}^{i}$ at price $\Price_{t}$,

3The Problem of an Individual Consumer¶

Consumer $i$ maximizes

subject to

Rewriting in the form of Bellman’s equation,

the first order condition tells us that

Spending one more unit of consumption today buys $1/\Price_{t}$ fewer trees, each of which would have paid off ${\Price}_{t+1}+\dvdnd_{t+1}$ tomorrow, so the derivative inside the expectation is

Substituting this and moving the resulting negative term to the left-hand side,

where $\Risky_{t+1}$ is the return factor: the resources in period $t+1$ that are the reward for investing one unit of consumption in trees at the end of $t$.

The Envelope theorem tells us that $\vFunc^{\prime}({m}_{t+1}^{i})=\uFunc^{\prime}(c_{t+1}^{i})$, so (10) becomes

4Aggregation¶

Since all consumers are identical, $c_{t}^{i} = c_{t}^{j}~\forall~i,j$, so henceforth we just call consumption per capita $\cRat_{t}$. Since aggregate consumption must equal aggregate production because fruit cannot be stored, normalizing the population to $\Pop_{t}=1 ~\forall ~ t$ and stock of trees to $\Kap_{t}=1 ~\forall~t$, equation (1) becomes:

Substituting $\cRat_{t}$ and $\cRat_{t+1}$ for $c_{t}^{i}$ and $c_{t+1}^{i}$ in (11) and then substituting $\dvdnd_{t}$ for $\cRat_{t}$ we get

We can rewrite this more simply if we define

which 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 $\dvdnd_{t+n}$); and (b) it is the factor by which all agents in this economy in period $t$ discount a unit of value received in a future period, e.g. $t+1$:

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

so we can use repeated substitution, e.g. of (17) into (16), and so on to get

The “law of iterated expectations,” together with $\SDF_{t,t+2} = \SDF_{t,t+1}\SDF_{t+1,t+2}$, gives $\Ex_{t}[\SDF_{t,t+1}\Ex_{t+1}[\SDF_{t+1,t+2}\dvdnd_{t+2}]] = \Ex_{t}[\SDF_{t,t+2}\dvdnd_{t+2}]$, and likewise for later terms, so, assuming no bubble, (18) becomes:

So, the price of the asset is the present discounted value of the stream of future “dividends,” where the stochastic factor by which (potentially stochastic) dividends received in $t+n$ are discounted back to $t$ is $\SDF_{t,t+n}$.

5Specializing the Model¶

This is as far as we can go without making explicit assumptions about the structure of utility. If utility is CRRA, $\uFunc(\cRat)=(1-\CRRA)^{-1}\cRat^{1-\CRRA},$ substituting $\uFunc^{\prime}(\dvdnd)=\dvdnd^{-\CRRA}$ into (14) yields

and the particularly special case of logarithmic utility (which Lucas emphasizes) corresponds to $\CRRA=1$, which (again using the law of iterated expectations) allows us to simplify (20) to

If the price-dividend ratio is bounded (there is no bubble), the limit term in this equation goes to zero. Using the usual definition of the time preference factor as $\DiscFac = 1/(1+\timeRate)$, where $\timeRate$ is the time preference rate, the equilibrium price is:

or, equivalently, the “dividend-price ratio” is always $\dvdnd_{t}/\Price_{t} = \timeRate$.^[4]

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 production 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). The first force is the substitution effect: a higher return to buying trees makes saving more attractive. The second is the (pure) income effect: future abundance makes saving less urgent. (There is no human wealth in this model, and therefore no human wealth effect.) In the special case of logarithmic utility considered here, income and substitution effects are of the same size and opposite sign so the two forces exactly offset.

6The Interest Rate and the Rate of Return in a Lucas Model¶

We can decompose the return factor attributable to ownership of a share of capital (cf. (10)) by adding and subtracting $\Price_{t}$ in the numerator:

so the net rate of return is

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 be 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,^[5] rather than Lucas trees). In this case

says that the only risk in the 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 in such models, and, if the aggregate production function is stable, this implies great stability in the marginal product of capital.

7Aggregate Returns Versus Individual Returns¶

One of the subtler entries in Aristotle (n.d.)’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 $\Risky_{t+1}$. 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 $\Risky_{t+1}$, 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, that there is no interaction between the decisions one person makes and the outcomes for another person). In the Lucas model, if everyone’s preferences change (e.g., $\timeRate$ goes down for everyone), the price of trees rises just enough to offset 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!

8Analytical and Numerical Solutions¶

The model has analytical solutions (such as $\Price_{t} = \dvdnd_{t}/\timeRate$) or approximate analytical solutions only in very special circumstances; Analytical Solutions in CRRA Utility Case derives them. When dividends follow an AR(1) process there is no closed form, though When Dividends Follow an AR(1) Process gives an exact series. The accompanying DemARK notebook shows how to solve that case numerically, and also shows how the numerical solution compares with the approximate analytical solution in the CRRA utility case.

9Appendix: Analytical Solutions in CRRA Utility Case¶

9.1When Dividends are IID¶

Suppose $\dvdnd_{t+n}$ is independently and identically distributed in every future period, so that its expectation as of $t$ is the same for any date $n>0$:

Now note that (20) can be rewritten as

To make further progress, suppose that the IID process for dividends is a mean-one lognormal: $\log \dvdnd_{t+n} \sim \mathcal{N}(-\sigma^{2}/2,\sigma^{2})~\forall~n$ so that $\Ex_{t}[\dvdnd_{t+n}]=1~\forall~n$ (see ELogNormMeanOne), in which case ELogNormTimes can be used to show that

and since $\DiscFac^{-1} = 1+\timeRate$, (30) becomes

So the log is

and thus the variances obey

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

1. (the log of) asset prices will be more volatile than (the log of) dividends

2. An increase in risk aversion $\CRRA$ increases the price $\Price_{t}$ at any given $\dvdnd_{t} \geq 1$, and the average price, $\Ex[\Price_{t}] = e^{\CRRA(\CRRA-1)\sigma^{2}}/\timeRate$ (because $\CRRA(\CRRA-1)\sigma^{2}/2 > 0$ and an increase in $\CRRA$ increases its size)

The second point is surprising, so it bears repeating: an increase in risk aversion increases the price of the risky asset. In a sense, this is an implication of the proposition that risk aversion increases the volatility of asset prices (when they are high, they must be very high; when low, very low). But it does not correspond very well to the common narrative in which market analysts often attribute a decline in asset prices to “increased risk aversion.” The result depends, however, on the absence of growth: with growing dividends it can reverse (see When Dividends Follow a Random Walk).

9.2When Dividends Follow a Random Walk¶

The polar alternative to IID shocks would be for dividends to follow a random walk: $\log (\dvdnd_{t+1}/\dvdnd_{t}) \sim \mathcal{N}(-\sigma^{2}/2,\sigma^{2})$.

Now divide both sides of (20) by $\dvdnd_{t}$, and rewrite the object inside the expectations operator by multiplying the first term by $\dvdnd_{t+1}$ and dividing the second term by $\dvdnd_{t+1}$, yielding

Now (1) note that our assumption here about the distribution of $\dvdnd_{t+1}/\dvdnd_{t}$ is identical to the IID case’s assumption about $\dvdnd_{t+1}$, so the expectation will be the same $\edvdnd$; and (2) hypothesize that there will be a solution under which the price-dividend ratio is a constant. Then (36) becomes

so that

and, since $(\DiscFac \edvdnd)^{-1} = (1+\timeRate)e^{-\CRRA(\CRRA-1)\sigma^{2}/2} \approx 1+\timeRate-\CRRA(\CRRA-1)\sigma^{2}/2$,

Compared with (34), the $\CRRA$ multiplying $\log \dvdnd_{t}$ is gone, so the variances of the changes in log prices and log dividends are now the same. Risk also enters differently, in the denominator, where it matters far more: at $\CRRA=3$, $\sigma=0.1$ and $\timeRate=0.04$ the price-dividend ratio is 108, against 25.8 with IID dividends at $\dvdnd_{t}=1$ and $1/\timeRate=25$ with no risk at all. The 108 is the exact ratio, $1/((\DiscFac \edvdnd)^{-1}-1)$; the approximation (39) gives $1/(0.04-0.03)=100$, because its denominator is the small difference between two rates, and the second-order terms it drops change that difference by 7 percent. The price is finite only if $\DiscFac \edvdnd < 1$, approximately $\timeRate > \CRRA(\CRRA-1)\sigma^{2}/2$. The surprising result that the price-dividend ratio increases when risk aversion increases continues to hold.

That result, like the one for IID dividends, depends on the absence of growth. If log dividend growth has mean $\pGro-\sigma^{2}/2$ instead, so that dividends are expected to grow by the factor $\PGro = e^{\pGro}$, the same steps give

so a higher $\CRRA$ raises the ratio only if $(\CRRA-1/2)\sigma^{2} > \pGro$. With CRRA utility, a higher $\CRRA$ is also a lower elasticity of intertemporal substitution: when dividends, and so consumption, are expected to grow, consumers who are less willing to substitute over time want to borrow against the future, which pushes the riskless rate up and prices down. For consumption, whose expected growth and standard deviation are both about 0.02 a year, a higher $\CRRA$ lowers the price for any $\CRRA$ below about 50. In a growing economy, then, the model does deliver the narrative of prices falling because risk aversion rose, but through the riskless rate rather than through risk premia.

9.3When Dividends Follow an AR(1) Process¶

The two cases just examined are the polar ones, and the interesting territory lies between them. Dividends that are IID have no memory at all, while dividends that follow a random walk never forget. Actual dividends fall somewhere in the middle, high this year if they were high last year, though not permanently so.

We impose the persistence on the log of dividends rather than on their level, for two reasons. An AR(1) in levels, $\dvdnd_{t+1} = \tau \dvdnd_{t} + \err_{t+1}$ with normal $\err$, would put positive probability on a negative dividend, which is not a thing a tree can produce. And in logs the value of each future dividend has a closed form, (43), which in levels it does not.

So write $x_{t} \equiv \log \dvdnd_{t}$ and assume

which contains both earlier cases: $\tau=0$ is the IID assumption and $\tau=1$ is the random walk.

Rather than guess at a price-dividend ratio, we price each future dividend separately and add up (the guess-and-verify route of When Dividends Follow a Random Walk needs a constant ratio, and there is not one to find here). The Lucas price is the present discounted value of the dividend stream evaluated at the consumer’s own marginal utility, so with $\cRat=\dvdnd$ and $\uFunc^{\prime}(\cRat)=\cRat^{-\CRRA}$,

Iterating (41) gives $x_{t+n}-x_{t} = (\tau^{n}-1)x_{t} + \sum_{j=1}^{n}\tau^{n-j}\err_{t+j}$, and because the $\err$ are independent and normal, each strip is the expectation of a lognormal variable and can be evaluated with ELogNorm. Writing $S_{n} = \sum_{k=0}^{n-1}\tau^{k}$ and $Q_{n} = \sum_{k=0}^{n-1}\tau^{2k}$,

Substituting (43) into (42) gives the price-dividend ratio for any $\tau$. We know of no closed form for $0<\tau<1$, since $\tau^{n}$ appears in an exponent and resists the geometric sum that rescues the polar cases, though the series itself converges geometrically and evaluating it costs a single line of code.

The two polar cases fall out. Setting $\tau=1$ makes $S_{n}=Q_{n}=n$ and kills the $x_{t}$ term, so every strip is $\edvdnd^{n}$ and the sum collapses to the constant ratio of When Dividends Follow a Random Walk. Setting $\tau=0$ makes $S_{n}=Q_{n}=1$ for every $n\geq 1$, leaving $\dvdnd_{t}^{\CRRA-1}\edvdnd \DiscFac/(1-\DiscFac)$, which is (30) written as a price-dividend ratio.

Except in the random walk limit, the price-dividend ratio depends on $\dvdnd_{t}$, through the $(\tau^{n}-1)x_{t}$ term, as it already does with IID dividends, where it is $\dvdnd_{t}^{\CRRA-1}$ times a constant. Only when $\tau=1$ does the level of dividends carry no information about their growth; only then is the ratio constant, except with logarithmic utility, which makes it constant for every $\tau$. At $\dvdnd_{t}=1$, where the term is zero, raising $\tau$ raises the ratio, steeply as $\tau$ nears one: at $\CRRA=3$, $\sigma=0.1$ and $\timeRate=0.04$ it runs 25.8, 26.4, 29.5, 47.6, 80.8 and 108 for $\tau=0, 0.6, 0.9, 0.99, 0.999$ and 1. Persistence lets the risk of distant dividends accumulate, and with $\CRRA>1$ risk raises a dividend’s value, as with IID dividends; mean reversion is what stops the accumulation and keeps the ratio modest. The random walk is the end of that climb, not a break from it: the ratio is continuous in $\tau$ at one.

Away from $\dvdnd_{t}=1$, persistence has a second effect. A dividend above its mean is expected to fall back, and with $\CRRA>1$ the income effect of that coming scarcity outweighs the substitution effect of the lower return, so the ratio is high; more persistence slows the expected fall, and so weakens the effect. At $\dvdnd_{t}=2$ this second effect wins until $\tau$ is close to one: the ratio falls from 103.0 at $\tau=0$ to 99.4, 89.3 and 73.2 at $\tau=0.6, 0.9$ and 0.99, before rising back toward 108 as $\tau$ approaches one.