# Budget constraint

1. James has a utility function of the following: U(L, X) = L + X, where L is leisure and X is consumption. If he works, he receives real wage w. Outside of the labor market, he has non-labor market income V. And his endowment of time, T, is normalized to 1. And the price of goods, p, is also normalized to 1. (a) Please write down James’ budget constraint. (b) Consider the case where w = 0.8, and V = 50. Please draw the budget constraint with consumption on the y axis and hours spent in leisure on the x axis, named it BC0, and show the interceptions on both axises and the slope of the budget constraint. (c) Next, show the slope of the indierence curve. (d) Solve for the optimal hours of working for James, and the optimal consumption. (e) Show the solution on the graph, and denote it as e0. And note the indierence curve with optimal solution as IC0. (f) Now, the wage increase from 0.8 to 2. Draw the new budget constraint. Name it BC1, and show the interceptions on both axises, and show the slope. (g) Solve for the new optimal hours of working for James, and the optimal consumption. (h) Denote the new solution on the graph and name it e1. And note the new indierence curve with optimal solution as IC1. (i) Do you think, in this case, which eect dominates? Income eect, substitution eect or none of them? Why? (j) In the graph, show total eect, income eect, and substitution eect. (Isolate income and substitution eects as in the slides.) Hint: To nd the income eect, you would want to nd the ”income adjustment” such that you reach the same utility level as the new one from the old utility level. en remember that total eect is composed of income and substitution eect. 2. Jack has a utility function of the following: U(L, X) = L 0.9X0.1 , where L is leisure and X is consumption. If he works, he receives real wage w. Outside of the labor market, he has non-labor market income V. And his endowment of time, T, is normalized to 1. And the price of goods, p, is also normalized to 1. (a) Please write down Jack’s budget constraint. And draw the budget constrain on the graph and denote it with interactions, and slope. (b) Consider the case where w = 0.8, and V = 0. (c) Solve for the optimal hours of working for Jack, and the optimal consumption. (d) Do you think, in this case, which eect dominates? Income eect, substitution eect or none of them? Why? (e) Now, Jack divorces Kate. As a result, Jack needs to spend 0.2 unit of time taking care of their kid, Arron. (at is, Jack now has only T minus 0.2 hours for him to decide freely on working or doing leisure.) Please write down and draw the new budget constraint. (f) What is the new optimal hours of working and the new optimal consumption for Jack? (g) Compare the answer in (c) and (f). Explain how the divorce aects Jack’s optimal working hours. 3. Hurley owns a restaurant. To operate the store, he needs to hire labor (L) and capital (K). e short run production function he is facing is the following: Q(L, K) = L 0.5K¯ 0.5 , where capital is non-adjustable. Moreover, a xed cost, F, is imposed on Hurley if the restaurant is open. e product’s price is p, assumed to be 1. e price of labor and capital are w, and r. We also assume that the market of the restaurant and the labor and capital market is perfectly competitive. Hurley is price taker. (a) Please write down Hurley’s prot. (b) Please solve for Hurley’s labor demand. (Express labor demand as a function of wage.) (c) Show that the labor demand is downward slopping. (d) Now, because of COVID-19, delivery service is rising. To adapt to the change. Hurley decides to add delivery service to his restaurant. And in order to do so, Hurley needs to hire another type of labor that only does delivery. erefore, he is facing a new short run production function Q(LS, LD, K) = L 1 4 SL 1 4 DK¯ 1 2 , where Ls is the labor that used in the store, and LD only does delivery. And K is still non-adjustable. Assuming Hurley cannot switch workers between two dierent types. Wage for two types of labor is ws, wD respectively. And there is also a xed cost F. Please write down the new prot function for Hurley. 2 (e) Please derive the rst order condition for labor used at the store, LS. (f) Please derive the rst order condition for labor used for delivery, LD. (g) Can you tell if Hurley would like to hire more LS, or more LD? If not, why?