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Unformatted text preview: Problem Set 1 - Econ 202A (Second Half) Prof. David Romer GSI: Victoria Vanasco Exercise 1. Consumption under Uncertainty The basic model of consumption under uncertainty (with quadratic utility, and uncertainty only about labor income) predicts that: A. The change in income will not be predictable on the basis of past changes in consumption. B. The change in consumption will not be predictable on the basis of past changes in income. C. The change in consumption will not be correlated with the current change in income. D. (A) and (B). E. (A) and (C). F. (B) and (C) Answer. Note that option (A) is false. Our basic model of consumption under uncertainty predicts that changes in con- sumption occur when news about changes in income are received. In this sense, changes in consumption due to an expected future increase in income would help us predict the change in income. In other words, if we see consump- tion jumping, with no change in current income, we should expect an increase in future income. Thus, changes in income can be predicted on the bases of past changes in consumption. This rules out answers (A), (D), and (E). Option (B) is true. When β (1 + r ) = 1 , our basic model of consumption with quadratic utility tells us that consumption follows a random walk, which means that the expected change in consumption is zero. Therefore, any changes in consumption should be unpredictable, and thus we should not be able to predict them on the basis of past changes in income. If β (1 + r ) negationslash = 1 , we might be able to predict that consumption is growing or decreasing, but note that this expected changes in consumption are unrelated to changes in income and have more to do with the relative prices of consumption today vs. tomorrow. Option (C) is tricky. On the one hand, if the changes in current income were unpredictable, we should observe a positive correlation between the changes in consumption today and the change in current income. On the other hand, if the changes in current income were predictable, consumption already adjusted when news about this change arrived, and thus should not react to the actual change in current income; in this case, the correlation should be zero. Since both cases are possible, I would say we should observe some correlation between changes in consumption and current income, and that this correlation re ects the unanticipated shocks to income. So I would say (C) is false. Finally, (B) is the correct answer. Exercise 2. Consumption with state-contingent good Consumer maximizes the expected present value utility: E bracketleftBigg ∞ summationdisplay t =0 parenleftbigg 1 1 + δ parenrightbigg t U ( c st ) bracketrightBigg ,U ′ > and U ′′ < 1 Consumer can purchase state-contingent goods, where the price of one consumption good at time t state s is given by p st . Let S be the space of all possible states. Given that the probability of state s at time t is given by π st , and assuming agents' income is also contingent on the state, we can write the agent's problem as:...
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