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There is a one-person production economy called the Robinson Crusoe model. Robinson Crusoe is a profit-maximising producer, who produces a consumption good from labour inputs. He is also a utility-maximising consumer who owns the profit-maximising firm and provides his labour to the firm. He has a decreasing returns to scale production function ??x=L0.5??. Crusoe gains utility over consumption of and leisure through the Cobb-Douglas utility function ??u(x;R)=αln(x)+(1-α)ln(R)??. He is endowed with one unit of time, which can either be spent on labour ??L?? or leisure ??R??.
a. Solve for Robinson Crusoe's equilibrium wage, profit, output, and utility.
b. Another person arrives: Man Friday. He is endowed with a single unit of labour and has the same utility function as Crusoe. He goes to work for Crusoe, but owns none of the firm. Solve for Crusoe's equilibrium wage, profit, output, and utility.
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