Created 8/24/1998
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Problem Set #3: Investment: DRAFT

(a) Capital Levy. Consider the q-theory model of investment with adjustment costs. Suppose it becomes known at some moment t1 that at a point in the future t2 there will be a one-time capital levy: capital holders will be taxed a fraction f of the value of their capital holdings.

Assume that the economy is initially in long-run equilibrium with respect to its capital stock.

(i) What happens to the value of q and to the rate of investment at the time the news is learned?

(ii) How do K, I, and q behave between the time of the news and the time of the capital levy?

(iii) How do K, I, and q behave after the time of the capital levy?

(iv) When does the level of q jump discontinuously? When the news is announced? When the capital levy is imposed? Both? Neither?

(b) Investment and Real Wages. Consider a firm with a constant-returns-to-scale Cobb-Douglas production technology in labor and capital, with capital share a, with a constant level of total factor productivity A, and with a constant period-to-period depreciaton rate d. Suppose that the firm has quadratic costs of adjustment of its capital stock: an investment that increases the capital stock by I requires the purchase of I(1 + bI) capital goods.

Suppose that the price of new capital goods in terms of overall output is fixed at one, that the firm operates in competitive product and labor markets, and thus takes the real wage as given. The owners of the firm are risk-neutral and have a subjective discount rate r.

(i) Suppose that the firm can choose its level of employment freely in each period. Solve for the profit in period t as a function of the capital stock K(t) and the real wage w(t).

(ii) Determine optimal investment as a function of current and expected future wage levels.

(iii) Suppose that the wage follows a two state Markov process. The wage can take on one of two values, wL or wH, with transition probabilities given by:

P(w(t+1) = wL | w(t) = wL) = p

P(w(t+1) = wH | w(t) = wH) = q

Derive the process followed by optimal investment. Characterize the effects of changes in p and q on investment. Explain.

(c) Lucas Tree Asset Pricing. Suppose that the only assets in the economy are infinitely-lived "trees" that yield fruit: the representative consumer owns one representative "tree". Output is equal to the fruit of the trees, which is exogenously-determined and is unstorable from period to period. Thus C(t) = Y(t), where Y(t) is the exogenously-determined fruit that drops from the representative tree in period t.

Note that asset prices must always be such in equilibrium that the representative consumer wants to hold onto her tree, and neither accumulate nor decumulate ownership of trees.

Let P(t) be the price of a tree in period t (the ex-dividend price: after the fruit has dropped and while the fruit is being eaten).

Finally, the representative consumer's instantaneous utility function is equal to log(C(t)), and the representative consumer has a constant rate of subjective time preference.

(i) Suppose that the representative consumer reduces her consumption in period t by an infinitesimal amount, uses the resulting savings to increase her ownership of trees, and then sells those extra trees in period t+1 (and boosts period t+1 consumption by the amount of the sale). Find the condition that C(t) and expectations of Y(t+1), C(t+1), and P(t+1) must satisfy in order for this deviation from the initial path of consumption to have no first-order effect on expected utility.

(ii) Solve this condition for P(t) as a function of Y(t) (or C(t)) and expectations of Y(t+1), P(t+1), and C(t+1)

(iii) Iterate your answer to part (ii) forward in time to solve for P(t). Are there any free parameters? If so, howe would you interpret them?

(iv) What effect does an increase in expectations of future dividends have on the current price of a tree in this model? Can you explain why?

(d) The q-Theory Model. Consider the q-theory model of investment. Use the phase diagram to qualitatively discuss the effects of each of the following shocks on the behavior of K, I, and q over time. Assume that--before the shock--K and q are at their long-run equilibrium values:

(i) Suppose that the government suddenly imposes a tax that taxes income from capital at a rate t.

(ii) Suppose that a war destroys half the capital stock.

(iii) Suppose that the government suddenly enacts an investment tax credit: the government pays firms an amount g for each unit of capital they acquire (and an amount -g for each unit of disinvestment, each unit by which I is less than zero).

(iv) Suppose that the government announces at t1 that at t2 it will enact the investment tax credit of (iii).

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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