Julia lives on an island. Her preferences are over food (c) and leisure (l)such that: U(c, l) = ln(c) +ln(l). She is endowed with 1 unit of time that she can dividebetween leisure and labor (L = 1 - l). Suppose that Julia can produce food according to the following production function F(L) = 2L^(1/2).

a) Write down Julia's maximization problem. How much food and leisure does sheconsume?

b) Now suppose that a food production company is created on the island. Juliais still the only source of labor on the island, she is still the only consumer, and theproduction function stays the same. The firm offers her a wage w for each unit of laborL that she provides. Assume that Julia and the firm are price-takers. This means thatJulia and the firm take wage (w) as given. The price of food can be normalized to 1, sothe wage w is the only price. We also assume that the profits of the firm are fully taxedand redistributed to Julia as unearned income (y). Importantly, Julia takes unearnedincome y as given; she does not take into account the effect of her labor on the firm'sprofit.

(i) Write down Julia's utility maximization problem, and derive the labor supply asa function of the wage (w) and unearned income (y). Note that her budget constraintis now: c = w × L + y.

(ii) Write down the firm's profit maximization problem, and derive the labor demand as a function of the wage, and the profits as a function of the wage.

(iii) Find the competitive equilibrium (markets must clear). What is the equilibriumwage? What are the equilibrium values of labor, leisure, profits, and consumption?(iv) Does the condition that the marginal rate of substitution equals the price ratioholds in this economy?