The Robinson Crusoe economy

In this lecture we consider a peculiar economy formed by just one person who acts both as a producer and, therefore, like a firm and as a consumer. This particular kind of economic models are named Robinson Crusoe Economies in honor of the famous character created by Daniel Defoe. Robinson can either gather coconuts or rest. If this is the case, because labor is undesirable (or at least we all prefer to rest rather than gathering coconuts) Robinson’s indifference curves will be upward sloping and not downward sloping as we are used to (we have dedicated an entire lecture on labor-leisure choices). We can also represent how much coconuts Robinson can gather using labor thanks to the production function (if you do not know what a production function is you may find it useful to go through our lectures about production). So which will be the best choice for Robison?

WHAT HAPPENS WHEN ROBINSON BEHAVES AT THE SAME TIME AS A CONSUMER AND A PRODUCER

As we studied in our lecture on consumer choice (lecture that you can find here) Robinson will choose the point on his budget set that helps him reach the highest indifference curve. This point is the point where the two curves are tangent to each other so mathematically where the marginal product of labor (the slope of the production function) is equal to the marginal rate of substitution between labor and coconuts (the slope of the indifference curves) .

WHAT HAPPENS WHEN ROBINSON BEHAVES FIRST AS A PRODUCER AND THEN AS A CONSUMER

Suppose that Robinson decides to create a market for coconuts and one for labor. The price for coconuts is equal to 1 making coconuts the numeraire. A numeraire is a good which is price is equal to 1. This means that the only price we have to care about is the price of labor. As a producer Robinson will act as a normal profit maximizing firm (we have a lecture dedicated to the behaviour of profit maximizing firms). The firm Robinson represents will have to pay the worker for his labor services with a wage ω. But Robinson is not only the only worker of the firm but he is also its owner and, because of that, he will collect some of the firm’s profits.

THE POINT OF VIEW OF THE FIRM

If we denote with C the number of coconuts gathered and with L the amount of labor employed we can determine the firm’s profits in the usual way: total revenue minus total costs.

π = C – wL

we can then rewrite the equation to isolate C

C = π + wL

This gives us the equations for the isoprofit lines for our firm. As a profit maximizing firm, Robinson will choose to employ the quantity of labor that makes his production set tangent to the highest possible isoprofit line. This condition is satisfied when the marginal product of labor MPL (the slope of the production function) is exactly equal to w (the slope of the isoprofit line).

THE POINT OF VIEW OF THE CONSUMER

Because Robinson is the owner of the firm we will receive some of the firm’s profits that we are going to indicate with π*. Robinson can choose to consume his share of its profits to gather coconuts and doing no labor or he can decide to work and gather other coconuts all his consumption choices can be represented with a budget line. Because of that, the point (0, π*) represents the vertical intercept of Robinson’s budget line. The line has a slope of w, the wage he receives by working that he can spend acquiring coconuts (remember that the price of coconuts is equal to 1 therefore $20 in wage means that Robinson can buy exactly 20 coconuts). As a consumer, Robinson’s objective is to reach the highest indifference curve given his budget constraint. This task is accomplished where the budget line and his indifference curve are tangent. Mathematically this condition is satisfied in the point where his marginal rate of substitution (the slope of the indifference curve) is exactly equal to w (the slope of the budget line).

WHAT HAPPENS WHEN YOU PUT THE CONSUMER AND THE PRODUCER TOGHETER

When you put the two perspectives together you can see that we reach the same solution we reached in the beginning when we considered Robinson as a producer and consumer at the same time meaning that given a set of prices, it is indifferent which time frame you consider because the competitive equilibrium is always the same. This is because, from the perspective of the firm, the production function must be tangent to the isoprofit line while, from the perspective of the consumer, the indifference curve must b e tangent to the budget line. Because the budget line and the isoprofit line are the same line and, because both the indifference curve and the production function must be tangent to it, the indifference curve and the production function must be tangent to each other.