The Lucas Tree: Why Asset Prices Are Really About Hunger

The CAPM tells you that an asset's expected return depends on how it co-moves with the stock market. But that raises an obvious follow-up question: why should anyone care about the stock market? The market is not food. It is not shelter. It is a number. Why is co-movement with a number the thing that determines prices?

Robert Lucas answered that question in 1978 with a paper so austere it is almost a parable. He stripped the economy down to a single tree, and in doing so he replaced the market portfolio with something far more fundamental: consumption. The framework he built is the direct ancestor of the stochastic discount factor, which is now the language in which all of modern asset pricing is written.

The problem: finance had no foundations under it

Pre-Lucas asset pricing models started from investors who like high returns and dislike variance. That is a reasonable description of a trader, but it is not a description of a person. People do not eat returns. They eat food. Variance of a portfolio is not intrinsically painful; what is painful is not having enough to live on.

Economics had a well-developed theory of what people actually want, which is to consume things over time and to avoid periods of scarcity. Finance had a separate theory of asset prices, bolted on top, with no bridge between them. Lucas built the bridge.

The key idea, via analogy

Picture an island with a single fruit tree and a population of identical inhabitants. Each year the tree randomly drops some quantity of fruit: a lot in a good year, very little in a bad one. The fruit is perishable, so it cannot be stored. Shares in the tree are traded.

Now here is the crucial observation that makes the whole model work. Everyone on the island must eat exactly the fruit that falls. There is nowhere else for it to go. The islanders can trade shares in the tree with each other all day long, but in aggregate, consumption equals the harvest. Trading cannot conjure more fruit.

Once you see that, pricing becomes a question about hunger rather than about portfolios. What is a share of the tree worth? A share is a claim on future fruit. To decide what to pay for it, an islander asks: how much do I value an extra piece of fruit in the states of the world where that share pays off?

And now everything inverts in an illuminating way. An extra piece of fruit is precious in a famine year and nearly worthless in a glut year. So a claim that delivers fruit in bad harvest years is enormously valuable, because it delivers when you are desperate. A claim that only delivers in bumper years is close to useless, because in a bumper year you already have more fruit than you can eat.

This gives Lucas's pricing rule, and it is the heart of modern asset pricing:

An asset is expensive if it pays off in bad times, and cheap if it pays off in good times. Cheap means high expected return. So the assets with high expected returns are precisely those that let you down when you most need help.

Notice that the asset's own volatility never entered. What matters is not how much the asset bounces, but whether it bounces in sympathy with your hunger. A wildly volatile asset that happened to pay out during famines would be treasured and would earn a low return, despite being volatile.

Stocks earn a premium in this world because they do the opposite. They crash exactly when the economy is bad, when people are unemployed, when consumption is scarce. They kick you when you are down, and therefore they must be sold cheaply enough to compensate.

The technical device that carries all of this is a single object multiplying every payoff: something Lucas expressed through how much the islanders value an extra unit of consumption in each state. Later economists named it the stochastic discount factor, and the entire field, options, bonds, equities, everything, was subsequently rewritten in its terms.

Why it mattered

• It gave finance economic foundations. Asset prices stopped being about the mean-variance whims of a hypothetical investor and started being about the fundamental economics of scarcity. That is a much deeper place to stand.
• It created the stochastic discount factor framework. One equation, price equals expected payoff times the discount factor, now covers essentially every asset class. The CAPM, the APT, and option pricing are all special cases of it, differing only in what you assume about the discount factor.
• It explained why the equity premium exists at all, rather than just measuring it. Stocks pay badly in recessions, so they must be cheap. That is a reason, not a correlation.
• It set up the great puzzles. By making the theory precise, Lucas made it falsifiable, and within a decade Mehra and Prescott used exactly this machinery to show that the equity premium in the data is far larger than the model can generate. That failure has been enormously productive.

The honest limitations

• The economy has one tree and identical people. There is no heterogeneity, no unemployment, no market incompleteness, no borrowing constraints, nobody who is poor. Real-world risk-sharing failures are exactly the sorts of things that might explain asset prices, and they are assumed away.
• It fails quantitatively. The consumption-based model, taken to data, predicts an equity premium a fraction of the size we actually observe unless you assume implausibly extreme risk aversion. This is the equity premium puzzle, and it is the direct descendant of this paper.
• Aggregate consumption is smooth, and stock prices are not. Measured consumption barely wobbles from year to year while equity markets swing violently. Getting one from the other requires a leap the data does not support.
• Consumption data is bad. It is measured with lag, revised, aggregated, and partly imputed. Testing a theory whose central variable is poorly observed is a chronic difficulty.

The one-line takeaway

Lucas showed that an asset's price depends not on how risky it is in isolation but on whether it pays you in the states of the world where you are already suffering, which is why stocks, which crash in recessions, must be sold cheap enough to earn a premium.