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Robert Hall showed in 1978 that the permanent income hypothesis with rational expectations implies that consumption changes should be unpredictable. With quadratic utility and an interest rate equal to the rate of time preference, the consumption Euler equation becomes E_t[C_{t+1}] = C_t, so consumption follows a martingale, loosely called a random walk. Consumption moves only when news arrives that changes expected lifetime resources. The prediction can be tested by checking whether information known in advance, such as lagged income, helps predict consumption growth.
Derive the result in three lines from the Euler equation with quadratic utility before discussing evidence. Then simulate a consumer who receives an anticipated raise and one who receives a surprise raise, and plot both consumption paths. The point that anticipated income changes should not move consumption when they arrive becomes concrete, and it sets up the excess-sensitivity tests.
You know the permanent income hypothesis: consumption depends on expected lifetime resources, not on current income. Hall asked what this implies about how consumption changes over time. His answer, published in 1978, is that the changes should be unpredictable.
Start with the consumption Euler equation for a consumer who can borrow and save at real rate r and discounts the future at rate ρ: u′(C_t) = [(1 + r)/(1 + ρ)] E_t[u′(C_{t+1})]. Assume r = ρ, so the bracket equals one. Assume quadratic utility, u(C) = C − (a/2)C², so marginal utility is u′(C) = 1 − aC, a linear function. Substituting gives 1 − aC_t = 1 − aE_t[C_{t+1}], which simplifies to E_t[C_{t+1}] = C_t. Today's consumption is the best forecast of tomorrow's. Write C_{t+1} = C_t + e_{t+1}, where e_{t+1} is a forecast error with mean zero given anything known at time t. In probability language, consumption is a martingale; economists often say random walk.
The intuition is the same as in efficient-markets theory. If the consumer could predict that consumption would rise next year, that would mean she expects more resources than her current plan uses, and she would raise consumption now. Optimal planning removes every predictable change. What remains is the response to news. A surprise promotion moves consumption at once. A raise announced two years ahead moves consumption on the day of the announcement, and not when the higher paychecks start.
This gives a clean test. Regress consumption growth on variables dated t or earlier. Under the hypothesis, none of them should matter. Hall found that lagged income added little predictive power, which supported the theory, though lagged stock prices had some. Later work found more trouble. Marjorie Flavin in 1981 reported excess sensitivity, meaning consumption responds to predictable income changes. John Campbell and Gregory Mankiw in 1989 estimated that roughly half of income may go to households who consume their current income rather than smoothing.
Two departures explain much of this evidence. Some households face borrowing limits and cannot raise consumption ahead of an expected raise. And when marginal utility is convex rather than linear as under quadratic utility, uncertainty itself changes saving, which the topic of precautionary saving takes up. Hall's result remains the benchmark these models are measured against.