Question

Robinson Crusoeis the sole inhabitant of an uncharted island. in this paradise, Robinson's only scarce commodities are food and leisure. the only scarce resource is his labor. Use the following notation: X=quantity of food P = price per unit of food Y= quantity of leisure hours W= wage rate (per hour) L=24-Y = labor hours Pi=profit as a consumer, Robinson has a utility function U(X,Y) expressing his preferences between food and leisure. as a firm Robinson can transform his labor hours into food according to the production function X=f(L) a. list the two markets in this economy and indicate which decision-making unit(i.e. consumer or firm) is the source of each demand and supply function in these markets. Assume both markets act as if they were perfectly competitive. b. Show how the demand functions for X and Y are derived. [Note that budget constraint must reflect the fact that he receives income as both wages and profits and spends it all on food] What are the important assumptions you must make this analysis valid. Form the demand for Y derive the supply of L.

Answer 1

a).

Consider the given problem here the utility function of the consumer is given by, “U(X, Y)”, “X=Food” and “Y=leisure”, where consumer want to maximize their utility given the budget line. So, here the budget line is given by.

=> P*X = W*(24-Y)+P, where “W=wage rate” and “P=profit”.

=> P*X + W*Y = W*24 + A. So, the maximization problem is given by.

=> Maximize “U(X, Y)” subject to “X + W*Y = W*24 + A”. Now, the producer want to maximize profit where profit function is given by.

=> A = P*X – W*L= P*f(L) – W*L, => A = P*f(L) – W*L. So, here there are two market “output market” and “labor market”.

b).

Now, in the utility maximization problem the choice variables are “X” and “Y” and “W”, “P” and “A” are parameters. So, if we maximize the utility function subject to the budget constraint, we will get the optimum value of “X” and “Y”. So, here the demands for “X” and “Y” are given by.

=> “Xd = X(W, P, A)” and “Yd=Y(W, P, A)”. Now, the labor supply is given by “L=24-Y”, => Ls(W, P, A)=24-Yd().

Now, form the profit maximization problem we will get the “labor demand” and “output supply function. So, the labor demand and output supply functions are given by, “Ld=L(W, P)” and “Xs=X(W, P)”. Now, by using “output demand” and “output supply” we will get the equilibrium “P” and “X”. Similarly, by “labor demand” and “labor supply” we get the equilibrium “W” and “L”.