denotes the first derivative of f{-), and f"{) the second derivative.

lAL is the amount of capital per unit of effective labor, and F(K, AL)IAL is Y/AL, output per unit of effective labor. Define = K/AL, = Y/AL, and f(k) = F{k. 1). Then we can rewrite (1.3) as

= ). (1.4)

That is, we can write output per unit of effective labor as a function of capital per unit of effective labor.

To see the intuition behind (1.4), think of dividing the economy into AL small economies, each with 1 unit of effective labor and /AL units of capital. Since the production function has constant returns, each of these small economies produces 1 /AL as much as is produced in the large, undivided economy. Thus the amount of output per unit of effective labor depends only on the quantity of capital per unit of effective labor, and not on the overall size of the economy. This is what is expressed mathematically in equation (1.4). If we wish to find the total amount of output, as opposed to the amount per unit of effective labor, we can multiply by the quantity of effective labor: = ALf(k).

The intensive-form production function, f(k), is assumed to satisfy f(0) = 0,f{k) > 0,f"{k) < 0 It is straightforward to show that f{k) is the marginal product of capital: since FiK,AL) = ALf(K/AL), dF(K/AL)/dK = ALf{K/AL)(l/AL) = f{k). Thus these assumptions imply that the marginal product of capital is positive, but that it declines as capital (per unit of effective labor) rises. In addition, [(•) is assumed to satisfy the Inada conditions (Inada, 1964): limfc-o f(k) = oo,limfcoo fik) = 0. These conditions (which are stronger than is needed for the models central results) state that the marginal product of capital is very large when the capital stock is sufficiently small and that it becomes very small as the capital stock becomes large; their role is to ensure that the path of the economy does not diverge. A production function satisfying /"(•) > 0, f"{-) < 0, and the Inada conditions is shown in Figure 1.1.

A specific example of a production function is the Cobb-Douglas:

F(K,AL) = "(AL) 0 < a < 1. (1.5)

This production function is easy to use, and it appears to be a good first approximation to actual production functions. As a result, it is very useful.

It is easy to check that the Cobb-Douglas function has constant returns. Multiplying both inputs by gives us

F(cK, cAL) = (cKricAL)-"

= cc-K"(AL)- (1-6)

= cF{K,AL).

figure 1.1 An example of a production function

To find the intensive form of the production function, divide both inputs by this yields

(1.7)

Equation (1.7) implies f(k) = ak"". It is straightforward to check that this expression is positive, that it approaches inhnity as approaches zero, and that it approaches zero as approaches infinity. Finally, f"{k) = -(1 ~ a)ak"~, which is negative.

Note that with Cobb-Douglas production, labor-augmenting, capital-augmenting, and Hicks-neutral technological progress (see n. 2) are all essentially the same. For example, to rewrite (1.5) so that technological progress is Hicks-neutral, simply define A = A-"; then Y = "!!").

1.2 Assumptions 11

The Evolution of the Inputs into Production

The remaining assumptions of the model concem how the stocks of labor, knowledge, and capital change over time. The model is set in continuous time; that is, the variables of the model are defined at every point in time. The initial levels of capital, labor, and knowledge are taken as given. Labor and knowledge grow at constant rates:

Ut) = nUt), (1.8)

A(t) = gA(t), (1.9)

where n and g are exogenous parameters and where a dot over a variable denotes a derivative with respect to time (that is, X{t) is shorthand for dX(t)/dt). Equations (1.8) and (1.9) imply that I and A grow exponentially. That is, if 1(0) and A(0) denote their values at time 0, (1.8) and (1.9) imply 1(f) = 1(0) ", ( ) = A(0)eSK

Output is divided between consumption and investment. The fraction of output devoted to investment, s, is exogenous and constant. One unit of output devoted to investment yields one unit of new capital. In addition, existing capital depreciates at rate S. Thus:

kit) = sY(t) - SK(t). (1.10)

\lthough no restrictions are placed on n, g, and 6 individually, their sum !s assumed to be positive. This completes the description of the model.

Since this is the first model (of many!) we will encounter, a general comment about modeling is called for. The Solow model is grossly simplified a host of ways. To give just a few examples, there is only a single good; government is absent; fluctuations in employment are ignored; production IS described by an aggregate production function with just three inputs; and the rates of saving, depreciation, population growth, and technological progress are constant. It is natural to think of these features of the model as defects: the model omits many obvious features of the world, and surely some of those features are important to growth. But the purpose of a model is not to be reaUstic. After all, we already possess a model that IS completely realistic-the world itself. The problem with that "model" is that it is too complicated to understand. A models purpose is to provide

The alternative is discrete time, where the variables are defined only at specific dates usually t = 0,1,2,...). The choice between continuous and discrete time is usually based on convenience. For example, the Solow model has essentially the same implications in discrete as in continuous time, but is easier to analyze in continuous time.

To verify this, note that L(t) = L(0)e" implies that L(f) - l(0)e"n = nUX) and that the initial value of I is l(0)e°, or 1(0) (and similarly for A).