Plays a dual role: as consumer and as producer.
He chooses between two goods: leisure and coconuts as a customer. If he sits on the beach watching the ocean, he is consuming leisure. If he spends his time gathering coconuts, he has less time for leisure but gets to eat the coconuts. We can depict Robinson production opportunities and preferences over the two goods.
At this point, the slope of the indifference curve must equal the slope ofthe production function by the standard argument: if they crossed, therewould be some other feasible point that was preferred.
The utility maximizing choice for Robinson must be the point at which the highest indiﬀerence curve just touches the production function. Why? At any point inside the production function, Robinson could choose a diﬀerent point that involved less labor and/or more coconuts. Given that he prefers to be on higher indiﬀerence curves, he should choose the highest one that is possible. This means an indiﬀerence curve that just touches the production function. Assuming the production function and the indiﬀerence curves are both diﬀer¬entiable at this point, we can conclude that at the optimum choices for labor and coconuts, the marginal product of labor equals the marginal rate of substitution between leisure and coconuts. This makes sense. The marginal product of labor is the extra amount of coconuts Robinson would get from giving up one unit of leisure. The MRS is the marginal utility gets from coconuts per unit of marginal utility from leisure. So, imagine that initially the MRS waw greater than the MP . Then think about what happens if Robinson spends one less hour gathering coconuts. His consumption of coconuts goes down by the MP of labor. His utility from leisure goes up by the marginal utility of leisure. His utility from coconuts goes down by the marginal utility of coconuts divided by the marginal utility of leisure. Under the assumption that the MRS initially exceeds the MP , this causes a net increase in utility, so the initial choice of labor and coconuts could not have been optimal. II.Crusoe Inc.
So far we have looked at Robinson problem in a way that takes into account both his producer role and his consumer role. Now let’s think about what happens if Robinson decides to alternate between his two roles. One day he behaves as a producer, while the next day he behaves as a consumer. Imagine that Robinson sets up a labor market and a coconut market. Robinson also creates a ﬁrm, which he owns. The ﬁrms uses labor to gather coconuts, which it sells in the coconut market. The ﬁrm will consider the prices for labor and coconuts and then decide how much labor to hire and how many coconuts to produce. The ﬁrm’s decisions will be determined by proﬁt maximization. Robinson, as a worker, earns wages from the ﬁrm. Robinson, as the owner of the ﬁrm, gets proﬁts. Robinson, as a consumer, decides how much of the ﬁrm’s output to purobinsonhase. To keep track of these transactions, Robinson invents a currency, called dollars. Assume that we set the price of coconuts at $1 a piece; that is, we make coconuts the numeraire. This means we only need to determine the wage rate. We will think about this from the perspective of the ﬁrm (Crusoe, Inc.), and then from the perspective the consumer (Robinson). Speciﬁcally, we want to derive the equilibria in the markets for labor and coconuts.
Each evening, Crusoe, Inc. decides how much labor it wants to hire thenext day, and how many coconuts it wants to produce. Given a price ofcoconuts of 1 and a wage rate of labor of w, we can solve the firm’sprofit maximizationproblem in Figure 32.2. We first consider all combinationsof coconuts and labor that yield a constant level of profits, π. This meansthat
π = C − wL.
Solving for C, we have
C = π + wL.
Just as in Chapter 19, this formula describes the isoprofit lines—all combinations of labor and coconuts...