R.E. Lucas, Jr, On the mechanics of economi where the discount rate p and the coefficient of (relative)risk aversion o are both positive Production per capita of the one good is divided into consumption c(t)and capital accumulation. If we let K(r) denote the total stock of capital, and K () its rate of change, then total output is N((c(1)+k(). [Here k(t)is net investment and total output N(()c()+K(r)is identified with net national product. Production is assumed to depend on the levels of capital and labor inputs and on the level A(t)of the technology, according to N()c(n)+k(1)=A(t)k(t)3N(1}- where 0<B<l and where the exogenously given rate of technical change, A/A,isμ>0 The resource allocation problem faced by this simple economy is simply to choose a time path c(r)for per-capita consumption. Given a path c(t)and an initial capital stock K(O), the technology(2)then implies a time path K(r)for capital. The paths A(n )and N(t) are given exogenously. One way to think about this allocation problem is to think of choosing c(t)at each date, given the values of K(1), A(() and N(t) that have been attained by that date Evidently, it will not be optimal to choose c(t)to maximize current-period utility, N(o(/(1-o)lc(r)-1]-, for the choice that achieves this is to set net investment K(t) equal to zero(or, if feasible, negative): One needs to set some value or price on increments to capital. A central construct in the stud of optimal allocations, allocations that maximize utility(1)subject to the technology (2), is the current-value hamiltonian h defined by N H(K AcDI-olc-0-1]+0[AK] which is just the sum of current-period utility and [from(2)] the rate of increase of capital, the latter valued at the price(0). An optimal allocation must maximize the expression H at each date t, provided the price a(t)is correctly chosen The first-order condition for maximizing H with respect to c is 日, which is to say that goods must be so allocated at each date as to be equally valuable, on the margin, used either as consumption or as investment. It is 7The inverse o-i of the coefficient of risk aversion is sometimes called the intertemporal elasticity of substitution. Since all the models considered in this paper are deterministic, this latter terminology may be more suitable
known that the price 8(()must satisfy 6(1)=p6(t)-xH(K(1),6(2),c(),) p-BA()N()2k(1)21]( at each date t if the solution c(r) to(3)is to yield an optimal path(c(t)ao (3 )is used to express c(t)as a function A((), and this fur is substituted in place of c(t)in(2)and (4), these two equations are a pair of first-order differential eq K(r)and its 'pricea(t). Solving th system, there will be a one-parameter family of paths(K(o), e(t), satisfying the given initial condition on K(O). The unique member of this family that satisfies the transversality condition ime-°(t)K(r)=0 is the optimal path. I am hoping that this application of Pontryagin's Maxi mum Principle, essentially taken from David Cass(1961), is familiar to most of you. I will be applying these same ideas repeatedly in what follows For this particular model, with convex preferences and technology and with no external effects of any kind, it is also known and not at all surprising that the optimal program characterized by(2),(3),(4)and(5)is also the unique competitive equilibrium program, provided either that all trading is consum- mated in advance, Arrow-Debreu style, or(and this is the interpretation I favor)that consumers and firms have rational expectations about future prices n this deterministic context, rational expectations just means perfect fore- sight. For my purposes, it is this equilibrium interpretation that is most interesting: I intend to use the model as a positive theory of U.S. economic growth In order to do this, we will need to work out the predictions of the model in more detail, which involves solving the differential equation system so we can see what the equilibrium time paths look like and compare them to observa tions like Denison,s. Rather than carry this analysis through to completion, I will work out the properties of a particular solution to the system and then just indicate briefly how the rest of the answer can be found in Cass's paper Let us construct from(2), (3)and(4)the systems balanced growth path: the particular solution(K(1), 8(r),c(r)such that the rates of growth of each of these variables is constant. (I have never been sure exactly what it is that is balancedalong such a path, but we need a term for solutions with this constant growth rate property and this is as good as any. Let k denote the rate of growth of per-capita consumption, c(r)/c(o), on a balanced growth
R.E. Lucas, Jr, On the mechanics of economic development path. Then from(3), we have 8()/0(t)=-oK. Then from(4), we must have BA(t)N(t)bK()=p+aκ (6) That is, along the balanced path, the marginal product of capital must equal the constant value p+oK. With this Cobb-Douglas technology, the marginal product of capital is proportional to the average product, so that dividing(2) through by K(r)and applying(6)we obtain K()K(o) A(t)K()2-N() B B By definition of a balanced path, K(()/K(r)is constant so(7) implies that N(c(t)/K(r)is constant or, differentiating, that K(1 N(1) c(r) K(t)N(t)c(t)=k+λ (8) Thus per-capita consumption and per-capita capital grow at the common rate K. To solve for this common rate differentiate either(6)or(7)to obtain Then(7)may be solved to obtain the constant, balanced consumption-ci ratio N(rc(o)/K(ror, which is equivalent and slightly easier to interpret constant, balanced net savings rate s defined by B(K+入) (10) N(t)c(t)+R(t)p+oκ Hence along a balanced path, the rate of growth of per-capita magnitudes is simply proportional to the given rate of technical change, l, where the constant of proportionality is the inverse of labor's share, 1-B. The rate of time preference p and the degree of risk aversion o have no bearing on this long-run growth rate. Low time preference p and low risk aversion o induce a high savings rate s, and high savings is, in turn, associated with relatively high output levels on a balanced path. a thrifty society will, in the long run, b wealthier than an impatient one, but it will not grow faster In order that the balanced path characterized by(9)and(10) satisfy the transversality condition(5), it is necessary that p+ox >x+X[From(10),one sees that this is the same as requiring the savings rate to be less than capital's
R.E. Lucas, Jr, On the mechanics of economic share. Under this condition, an economy that begins on the balanced path will find it optimal to stay there. What of economies that begin of the balanced path-surely the normal case? Cass showed- and this is exactly hy the balanced path is interesting to us-that for any initial capital K(O)>0, the optimal capital-consumption path(K(t), c(r) will converge to the balanced path asymptotically. That is, the balanced path will be a good approximation to any actual path 'mostof the time Now given the taste and technology parameters(p, o, x, B and u)(9)and (10) can be solved for the asymptotic growth rate x of capital, consumption and real output, and the savings rate s that they imply. Moreover, it would be straightforward to calculate numerically the approach to the balanced path from any initial capital level K(O). This is the exercise that an idealized planner would go through Our interest in the model is positive, not normative, so we want to go in the opposite direction and try to infer the underlying preferences and technology from what we can observe. I will outline this, taking the balanced path as the model's prediction for the behavior of the U.s. economy during the entire (1909-57)period covered by Denisons study. From this point of view, Denisons estimates provide a value of 0.013 for A, and two values, 0.029 and 0.024 for K +A, depending on whether we use output or capital growth rates (which the model predicts to be equal). In the tradition of statistical inference, let us average to get x+x=0.027. The theory predicts that 1-B should equal labor's share in national income, about 0.75 in the U.S., averaging over the entire 1909-57 period. The savings rate (net investment over NNP)is fairly constant at 0. 10. Then(9)implies an estimate of 0. 0105 for p. Eq.(10) implies that the preference parameters p and o satisfy p+(0.014)o=0.0675 The parameters p and o are not separately identified along a smooth consumption path, so this is as far as we can go with the sample averages I have provided These are the parameter values that give the theoretical model its best fit to the U. S. data. How good a fit is it? Either output growth is underpredicted or capital growth overpredicted, as remarked earlier(and in the theory of growth a half a percentage point is a large discrepancy ). There are interesting secular changes in manhours per household that the model assumes away, and labor's share is secularly rising(in all growing economies), not constant as assumed There is, in short, much room for improvement, even in accounting for the secular changes the model was designed to fit, and indeed, a fuller review of with the parameter values described in this paragraph, the half-life of the approximate linear system associated with this model is about eleven years
R.E. Lucas, Jr, On the mechanics of economic development the literature would reveal interesting progress on these and many other fronts.A model as explicit as this one, by the very nakedness of its simplify ing assumptions, invites criticism and suggests refinements to itself. This is exactly why we prefer explicitness, or why I think we ought to Even granted its limitations, the simple neoclassical model has made basic contributions to our thinking about economic growth. Qualitatively, it empha sizes a distinction between'growth effects'-changes in parameters that alter growth rates along balanced paths- 'level effects'-changes that raise or lower balanced growth paths without affecting their slope - that is fundamen- tal in thinking about policy changes. Solow's 1956 conclusion that changes in savings rates are level effects(which transposes in the present context to the conclusion that changes in the discount rate, p, are level effects) was startling at the time, and remains widely and very unfortunately neglected today. The influential idea that changes in the tax structure that make savings more attractive can have large, sustained effects on an economys growth rate sounds so reasonable and it may even be true, but it is a clear implication of the theory we have that it is not Even sophisticated discussions of economic growth can often be confusing as to what are thought to be level effects and what growth effects. Thus Krueger(1983)and Harberger(1984), in their recent, very useful surveys of the growth experiences of poor countries, both identify inefficient barriers to trade as a limitation on growth, and their removal as a key explanation of several rapid growth episodes. The facts Krueger and Harberger summarize are not in dispute, but under the neoclassical model just reviewed one would not expect the removal of inefficient trade barriers to induce sustained increases in growth rates. Removal of trade barriers is, on this theory, a level effect, analogous to the one-time shifting upward in production possibilities, and not a growth effect. Of course, level effects can be drawn out through time through adjustment costs of various kinds, but not so as to produce increases in growth rates that are both large and sustained. Thus the removal of an inefficiency that reduced output by five percent(an enormous effect) spread out over ten years in simply a one- half of one percent annual growth rate stimulus Inefficiencies are important and their removal certainly desirable, but the familiar ones are level effects, not growth effects.( This is exactly why it is not paradoxical that centrally planned economies, with allocative inefficiencies of legendary proportions, grow about as fast as market economies. ) The empirical connections between trade policies and economic growth that 9In particular, there is much evidence that capital stock growth, as measured by Denison understates true capital growth due to the failure to correct price deflators for quality improve ments. See, for example, Griliches and Jorgenson( 1967)or Gordon (1971). These errors may well account for all of the 0.005 discrepancy noted in the text(or more varable, and which has the potential (at least) to account for long- run changes in manhours y is Boxall(1986)develops a modification of the Solow-Cass model in which labor suppl