Created 8/24/1998
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(Sections W 10-12 Evans 6, W 12-2 Evans 2)

Problem Set #2: Consumption

(a) Time-Averaging. Suppose that consumption follows a true random walk: C(t) = C(t-1) + e(t), where e(t) is white noise. However, the data do not report values of C(t): instead, the data report non-overlapping two-period averages of consumption only: that is, the data consist of observations of:

[C(t)+C(t+1)]/2, [C(t+2)+C(t+3)]/2, [C(t+4)+C(t+5)]/2, [C(t+6)+C(t+7)]/2,

and so on. Then:

(i) Find an expression for the change in measured consumption from one two-period interval to the next. Write this change in terms of the shocks, the e's.

(ii) Is the change in measured consumption from one two-period interval to the next uncorrelated with the previous change in measured consumption over two-period intervals? In light of this, does time-averaged measured consumption follow a random walk?

(iii) Suppose that measured consumption is not the average of consumption in a two-period interval, but is instead consumption in the second period of each two-period interval only: that is, one observes:

C(t+1), C(t+3), C(t+5), C(t+7), and so forth

In this case, is measured consumption a random walk?

(b) Saving and Taxation. Consider a two period model in which a representative consumer earns fixed and exogenous amounts Y(1) and Y(2) in each period. The consumer can save some of her period 1 income, invest it to yield a net return r, and thus (in the absence of taxes) consume (1+r) in period 2 for every 1 unit of income saved in period 1. Suppose, however, that the government levies a tax at rate t on all interest income: the government's revenue is thus equal to zero in period 1 and to tr(Y(1) - C(1)) in period 2.

Now suppose that the government eliminates the taxation of interest income, and instead imposes lump-sum taxes of amounts T(1) and T(2) on the representative consumer in th two periods. Assume that the interest rate r is also exogenous.

(i) Write down the representative consumer's budget constraint in the presence of proportional taxes on interest income.

(ii) Write down the representive consumer's budget constraint in the absence of interest taxes but in the presence of lump-sum taxes.

(iii) What condition must the new taxes satisfy if the change in tax regime from interest income taxes to lump sum taxes is to leave the present value of government revenues unchanged?

(iv) If the new lump-sum taxes satisfy the condition of part (c)--that the tax-law change be revenue-netural--is the optimal consumption bundle that the representative consumer chose in the presence of interest income taxes affordable under lump sum taxes? If it is affordable, is it just affordable or is there room to spare?

(v) If the new lump-sum taxes satisfy the condition of part (c), is first-period consumption higher, lower, or the same as under interest income taxes?

(c) Mankiw's Durable Goods Model. Suppose that the instantaneous utility function of the representative consumer is quadratic, and suppose that the interest and discount rate are both zero. However, the flow of consumption services to the consumer C(t) is not given by purchases of consumption goods in period t, but instead by:

C(t) = (1-b)C(t-1) + E(t)

Where E(t) is the amount of consumption goods purchased in period t, and b (for "breakage," b between zero and one) is the depreciation rate on the stock of consumer goods. Consumer goods are durable: they yield a consumption service flow to the consumer of 1 unit in the period when purchased, (1-b) units in the subsequent period, and so forth.

Consider small deviations from a path of consumption service flows over time, C(t), C(t+1), C (t+2), and so forth.

(i) Consider a marginal reduction dE(t) in consumer-good purchases in period t. Find values of dE(t+1) and dE(t+2) such that the combined changes in E(t), E(t+1), and E(t+2) leave the present value of purchases of consumption goods unchanged, and leave the flow of consumption services in period t+2 (C(t+2)) and in all subsequent periods unchanged from the original path as well.

(ii) What is the effect of the deviations in consumption-good purchases dE(t), dE(t+1), and dE(t+2) from the initial path on C(t), C(t+1), C(t+2)? What is the effect on the consumer's expected utility?

(iii) What condition must C(t), Et[C(t+1)], and Et[C(t+2)] satisfy in order for the marginal changes in spending patterns to have no first-order effect on expected utility?

(iv) Does the flow of consumption services to the consumer follow a random walk?

(v) Does the level of consumption-goods purchases by the consumer follow a random walk?

(d) Precautionary Saving. Suppose that the utility function of the representative consumer exhibits constant relative risk aversion, and suppose that the conditional distribution of C(t+1) given information available at time t is log-normal: that is, that the log of consumption in period t+1 has a normal distribution. Show that the relationship between consumption this period, C(t), and the distribution of C(t+1) depends on the risk-free interest rate, the subjective discount rate, and the conditional variance of period t+1's consumption.

Explain why the variance of next period's consumption matters for this period's consumption.

Professor of Economics J. Bradford DeLong, 601 Evans
University of California at Berkeley; Berkeley, CA 94720-3880
(510) 643-4027 phone (510) 642-6615 fax

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