Business Application: Valuing Land in Equilibrium: Suppose we consider a Robinson Crusoe economy with one worker who has preferences over leisure and consumption and one firm that uses a constant returns to scale production process using inputs land and labor.
A: Suppose that the worker owns the fixed supply of land that is available for production. Throughout the problem, normalize the price of output to 1.
(a) Explain why we can normalize one of the three prices in this economy (where the other two prices are the wage w and the land rental rate r ).
(b) Assuming the land can fetch a positive rent per unit, how much of it will the worker rent to the firm in equilibrium (given his tastes are only over leisure and consumption)?
(c) Given your answer to (b), explain how we can think of the production frontier for the firm as simply a single-input production process that uses labor to produce output?
(d)What returns to scale does this single input production process have? Draw the production frontier in a graph with labor on the horizontal and output on the vertical axis.
(e) What do the worker€™s indifference curves in this graph look like? Illustrate the worker€™s optimal bundle if he took the production frontier as his constraint.
(f) Illustrate the budget for the worker and the iso profit for the firm that lead both worker and firm to choose the bundle you identified in (e) as their optimum. What is the slope of this budget/is opfrofit? Does the budget/iso pfrofit have a positive vertical intercept?
(g) In the text, we interpreted this intercept as profit which the worker gets as part of his income because he owns the firm. Here, however, he owns the land which the firm uses. Can you re-interpret this positive intercept in the context of this model (keeping in mind that the true underlying production frontier for the firm has constant returns to scale)? If land had been normalized to 1 unit, where would you find the land rental rate r in your graph?
B: Suppose that the worker€™s tastes can be represented by the utility function u(x, (1ˆ’„“)) = xα (1ˆ’„“) (1ˆ’α) (where x is consumption, „“ is labor, and where the leisure endowment is normalized to 1.) Suppose further that the firm€™s production function is f (y,„“) = y0.5„“0.5 where y represents the number of acres of land rented by the firm and „“ represents the labor hours hired.
(a) Normalize the price of output to be equal to 1 for the remainder of the problem and let land rent and the wage be equal to r and w. Write down the firm€™s profit maximization problem, taking into account that the firm has to hire both land and labor.

(b) Take the first order conditions of the firm€™s profit maximization problem. The worker gets no consumption value from his land€”and therefore will rent his whole unit of land to the firm. Thus, you can replace land in your first order conditions with 1. Then solve each first order condition for „“ and from this derive the relationship between w and r.
(c) The worker earns income from his labor and from renting his land to the firm. Express the worker€™s budget constraint in terms of w and then solve for his labor supply function in terms of w.
(d) Derive the equilibrium wage in your economy by setting labor supply equal to labor demand (which you implicitly derived in (b) from one of your first order conditions).
(e) What€™s the equilibrium rent of land?
(f ) Now suppose we reformulate the problem slightly: Suppose the firm€™s production function is f (y,„“) = y(1ˆ’β)„“β (where y is land and „“ is labor) and the worker€™s tastes can be represented by the utility function u(x, (Lˆ’„“))= xα(Lˆ’„“)(1ˆ’α) ,where L is the worker€™s leisure endowment. Compare this to the way we formulated the Robinson Crusoe economy in the text. If land area is in fixed supply at 1 unit, what parameter in our formulation in the text must be set to 1 in order for our problem to be identical to the one in the text.
(g) True or False: By turning land into a fixed input, we have turned the constant returns to scale production process into one of decreasing returns to scale.

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