The exact value of y, constrains the total amount that can be consumed; that is, at every period ? it must be the case that (18.1) The amount of the resource not extracted, therefore, is y, - (1 + ca.). This is the investrow that can generate future growth; call it x, From the preceding equation it follows that x, 2 0. Investment produces next period's stock Jie, through a production function, In Chapter 7 we examined the case of an exhaustible resource (with no growth possibility); that is, we assumed J,| = x7. By way of contrast, let us now consider a renewable resource. that is, a resource for which Ju > x, (at least for some investment levels), In fact, in order to do some actual computations, we will specify particular forms for the utility and production functions. Suppose that player /'s utility from consuming amount c is given by log c. utility increases with the amount consumed, although the utility increase is smaller, the larger is the base from which consumption is further increased. The utility function is pictured in Figure 18,I. We are also going to assume that investment r, results in a period / + 1 stock Jaj of size ya = 10 x /x. Again higher investments produce higher stocks, although additional investment becomes less and less productive, as the base investment grows larger. The production function is pictured in Figure 18.2." Note that if investment x, is equal to 0 in any period, then so is the stock y, in the next period. This fact suggests a natural horizon for the game-it continues as long as there is a positive resource level and hence can potentially go on forever