This Demonstration depicts the Robinson Crusoe production decision with two states of nature and one consumption good, with and without trade in Arrow–Debreu securities.

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First, suppose that Robinson Crusoe is stranded on an island where he can only consume what he produces. There are two possible states of nature and there is a single consumption good. State 1 eventuates with probability and state 2 eventuates with probability .

Robinson derives utility from his consumption plan in each state. Robinson's only source of consumption at date is what he can produce by then. He must choose his production plan at date . For example, what mix of crops will he plant? We denote Robinson's production possibility set by .

Robinson is alone on the island, so he has no opportunity to trade assets and thereby alter his state-contingent consumption. In each state he can consume only what he has produced. Provided he believes that “more is better”, Robinson will consume all that he produces. That is, consumption in state will be and consumption in state will be . His utility will therefore equal if he chooses the production plan at date .

Therefore Robinson's constrained optimization problem is

.

The optimal production plan occurs where the indifference curve is tangent to the boundary of the production set. Because Robinson does not have any opportunity to trade assets, his consumption plan coincides with the production plan. This situation is represented in the Demonstration as the "without trade" graph.

Now consider a situation where Robinson can trade Arrow–Debreu securities with Friday, who is stranded on a nearby island. We denote the date price of an Arrow–Debreu security that pays one unit of the consumption good in state by . Robinson continues to maximize , but the ability to trade Arrow–Debreu securities means that consumption no longer has to equal production in every state. He can achieve his consumption plan by acquiring a portfolio of Arrow–Debreu securities (denoted ) that satisfies the condition . That is, his consumption in state comes partly from his Arrow–Debreu securities and partly from his production.

The can be positive or negative, so that he can consume more than he produces in some states and less than he produces in other states. This trading between states creates a budget set of possible consumption plans that lets Robinson produce on his production possibility frontier but consume off it as depicted in the "with trade" graph.

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## Details

is Robinson's utility function, where is the perceived probability of state , is consumption in state , and is Robinson's level of Arrow–Pratt relative risk aversion. Robinson maximizes this function subject to the constraint that the production plan is feasible, .

If the production function is (this function dictates the feasible production possibility set), and as stated previously, (with no trade), then Robinson maximizes the function with respect to to find the optimal state consumption and production plan . Given this, . In equilibrium, the marginal rate of substitution (MRS) is equal to the marginal rate of transformation, which is equal to the slope of the indifference curve at the equilibrium point.

When trade is introduced, the maximization methodology is consistent with that of the no-trade case. However, now Robinson chooses and to maximize , subject to the constraints that production is feasible, , and consumption is affordable, . This results in a boundary of possible consumption plans (the budget line) that is a straight line running through the production plan with a slope (the relative state price).

References

[1] H. Varian, *Intermediate Microeconomics: A Modern Approach*, 8th ed., New York: W. W. Norton, 2009.

[2] R. Starr,* General Equilibrium Theory: An Introduction*, 2nd ed., New York: Cambridge University Press, 2011.