18.2 THE COMMONS PROBLEM: A MODEL The game environment at period / is the size of the resource stock in that period, y; y, 20. The resource can be accessed by any player, and let us continue to assume that there are two players. Denote player i's consumption or extraction of the resource in period by c. Again, it will be natural to only consider c0. Consumption gives player i a payoff or utility. log c FIGURE 18.1 The exact value of y, constrains the total amount that can be consumed; that is, at every period / it must be the case that Ct + Cat Syr (18.1) The amount of the resource not extracted, therefore, is y,- (c1, c2). This is the investment that can generate future growth; call it x,. From the preceding equation it follows that x, 20. Investment produces next period's stock y+1 through a production function. In Chapter 7 we examined the case of an exhaustible resource (with no growth possibility); that is, we assumed y+1 = x. By way of contrast, let us now consider a renewable resource, that is, a resource for which y+1>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 i'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 18.1. We are also going to assume that investment x, results in a period + 1 stock y+1 of size y+1=10xxt. 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 18.2.3 Note that if investment x, is equal to 0 in any period, then so is the stock y+1 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. 18.5 If the current size of the resource is 60, how much investment would there need to be to regenerate this stock level? What if the current resource stock is 90? 18.6 From the picture of the production function (Figure 18.2), and your answer to exercise 18.5, what can you conclude about sustainability of higher and higher stocks? Explain your answer. 18.7 The marginal productivity of investment is the additional stock that would result from a small increase in investment (for example, a unit increase). Graph the marginal productivity function 5/x, where x is the investment level. Can you explain its shape? 18.8 Explain the answers that you got in exercise 18.6 in light of what you concluded in exercise 18.7. Is it correct to say, "It becomes increasingly difficult to sustain larger resource levels