and = 1. With this production function, since individuals supply 1 unit of labor when they are young, an individual born m f obtains A umts of the good. And each unit saved yields 1 + r = 1 + dF(K,AL)lf)K - = 1 + -1= units of second-period consumption.

Finally, assume that m the mitial penod, period 0, in addition to the N() young individuals each endowed with A units of the good, there are [1/(1 + m)]No individuals who are alive only in period 0. Each of these "old" individuals is endowed with some amount Z of the good; their utility is simply their consumption m the initial period, C20.

(a) Describe the decentralized equilibrium of this economy. (Hint: given the overlappmg-generations structure, will the members of any generation engage in transactions with members of another generation?)

(b) Consider paths where the fraction of agents endowments that is stored, ft, is constant over time. What is total consumption (that is, consumption of all the young plus consumption of all the old) per person on such a path as a function of f ? If x < 1 + n, what value of f satisfying 0 < f < 1 maximizes consumption per person? Is the decentralized equilibrium Pareto-efftcient m this case If not, how can a social planner raise welfare?

2.18. Stationary monetary equilibria in the Samuelson overlapping-generations model. (Again this follows Samuelson, 1958.) Consider the setup described m Problem 2.17. Assume that x < 1 + n. Suppose that the old individuals in period 0, in addition to being endowed with Z units of the good, are each endowed with M units of a storable, divisible commodity, which we will call money. Money is not a source of utility.

(a) Consider an individual born at t. Suppose the price of the good in units of money is Pf m f and Pt+i in t -1-1. Thus the individual can sell umts of endowment for P, units of money and then use that money to buy I Pt+i units of the next generations endowment the following period. What is the individuals behavior as a function of P,/Pt+i

ib) Show that there is an equilibrium with Pt+i = Pt/(1 + n) for all f > 0 and no storage, and thus that the presence of "money" allows the economy to reach the golden-rule level of storage.

(c) Show that there are also equilibria with Pt+i = Pt/x for all f > 0.

id) Finally, explain why P, = for all f (that is, money is worthless) is also an equilibrium. Explain why this is the only equilibrium if the economy ends at some date, as in Problem 2.19(b), below. (Hint: reason backward from the last period.)

2.19. The source of dynamic inefficiency. There are two ways in which the Diamond and Samuelson models differ from textbook models. First, markets are mcomplete: because individuals cannot trade with individuals who have not been born, some possible transactions are ruled out. Second, because time goes on forever, there is an infinite number of agents. This problem asks you to investigate which of these is the source of the possibility of dynamic inefficiency. For simplicity. It focuses on the Samuelson overlapping-generations model (see the previous two problems), again with log utility and no discounting. To simplify further, it assumes n = 0 and 0 < x < 1. The basic issues, however, are general.

(a) Incomplete markets. Suppose we eliminate incomplete markets from the model by allowing all agents to trade in a competitive market "before" the beginning of time. That is, a Walrasian auctioneer calls out prices Qo, Qi, Q2, for the good at each date. Individuals can then make sales and purchases at these prices given their endowments and their ability to store. The budget constraint of an individual born at f is thus QtQt + Qt+iCzti = QAA-St)+Qt+ixSt, where St (whichmust satisfy 0 < St < A) is the amount the individual stores.

(i) Suppose the auctioneer announces Qt+i = Qt/x for all > 0. Show that in this case individuals are indifferent concerning how much to store, that there is a set of storage decisions such that markets clear at every date, and that this equilibrium is the same as the equilibrium described in part (a) of Problem 2.17.

Ui) Suppose the auctioneer announces prices that fail to satisfy Qf+i = Qt / X at some date. Show that at the first date that does not satisfy this condition the market for the good carmot clear, and thus that the proposed price path carmot be an equilibrium.

ib) Infinite duration. Suppose that the economy ends at some date T. That is, suppose the individuals born at T live only one period (and hence seek to maximize Cit), and that thereafter no individuals are born. Show that the decentralized equilibrium is Pareto-efficient.

(c) In light of these answers, is it incomplete markets or infinite duration that is the source of dynamic inefficiency?

2.20. Explosive paths in the Samuelson overlapping-generations model (See Black, 1974; Brock, 1975; and Calvo, 1978a.) Consider the setup described in Problem 2.18. Assume that x is zero, and assume that utility is constant-relative-risk-aversion with e < 1 rather than logarithmic. Finally, assume for simplicity that n = 0.

(a) What is the behavior of an individual born at as a function of Pt/Pt+i7 Show that the amount of his or her endowment that the individual sells for money is an increasing function of Ft/Pt+i and approaches zero as this ratio approaches zero.

(b) Suppose Pq/Pi < 1. How much of the good are the individuals born in period 0 planning to buy in period 1 from the individuals born then? What must Pi / 2 be for the individuals born in period 1 to want to supply this amount?

(c) Iterating this reasoning forward, what is the qualitative behavior of Pt/Pt+i over time? Does this represent an equilibrium path for the economy?

(d) Can there be an equilibrium path with Po/Pi > 1?

Chapter 3

BEYOND THE SOLOW MODEL: NEW GROWTH THEORY

The models we have seen so far do not provide satisfying answers to our central questions about economic growth. The models principal result is a negative one: if capitals earnings reflect its contribution to output and if its share in total income is modest, then capital accumulation carmot account for a large part of either long-run growth or cross-country income differences. And the only determinant of income in the models other than capital is a mystery variable, the "effectiveness of labor" (A), whose exact meaning IS not specified and whose behavior is taken as exogenous.

This chapter therefore investigates the fundamental questions of growth theory more deeply. It considers two broad views. The first view is that the driving force of growth is the accumulation of knowledge. This view agrees ulth the Solow model and the models of Chapter 2 that capital accumulation is not central to growth. But it differs from these models in explicitly interpreting the effectiveness of labor as knowledge and in formally modeling Its evolution over time. This view is the subject of Part A of the chapter. We win analyze the dynamics of the economy when knowledge accumulation is modeled exphcitly and consider various views concerning how knowledge is produced and what determines the allocation of resources to knowledge production.

The second view is that, contrary to the Solow model and the models of Chapter 2, capital is central to growth. Specifically, we will consider models that take a broader view of capital than we have considered so far-most importantly, extending it to include human capital. These models imply that physical capitals income share may not be a good guide to the overall importance of capital. We will see that, as a result, it is possible for capital accumulation alone to have large effects on real incomes. These models are the subject of Part of the chapter.