This is a labor economics question regarding the utility function.
3. Dwight is an employee in Dunder Mifflin. Beside working as a salesman, he enjoys spend- ing time in his beets farm as leisure. He has a utility function of the following: (70%) U(L, X) = min {5L, 3X}, where L is the time spent in beets farm and X is his consump- tion. If he works, he receives a real wage w. His endowment of time is T. And the price of goods, p, is also normalized to 1.(a) Consider the case where T = 100, w = 0.5, please write down and draw the budget constraint, denote it with BC, with L on the r-axis and X on the y-axis, and show the slope and intersections. (b) Please solve for Dwight's optimal labor supply and optimal consumption. (c) Draw the solution {L*, X*} in the graph and denote it as co, and an indifference curve ICo that provides the optimal utility level. (d) Now, Dwight realizes that he is eligible for a social benefit program that provides monetary benefit payment to beet farmers of $20 per period. Please write down the new budget constraint. (e) Please draw the new budget constraint, and denote it as BC1 (f) Please solve for the new optimal labor supply. (g) Draw the new solution { LX, X}} in the graph and denote it as en, and a new indiffer- ence curve IC, that provides the new optimal utility level. (h) Please compare co and e1. Explain why there is a difference or no difference. (i) Please decompose the change into total effect, substitution effect and income effect on the graph. Which effect dominates? (j) What if alternatively, instead of giving subsidy of $20, the government subsidizes by increasing (10%) of the wage that beets farmers earns in the labor market. (That is, whatever Dwight earns, he receive 10% increase in wage. ) What is the new optimal labor supply and comsumption? And please denotes the new solution as ez. (k) Please decompose the change into total effect, substitution effect and income effect on the graph. Which effect dominates? (Compare co and ez.) (1) Consider a new case relative to the original problem (part(d)). Now, instead of receiv- ing $20 anyways, the $20 social benefit can only be obtain if Dwight does not work at all. If he decide to work any hour greater than zero, he can only obtain $5 benefit. Do you think Dwight would choose not to work at all and receive $20 or work and receive $5? Why?(m) What if instead of a utility function U = min {5L, 3X }, Dwight has a utility function of U = 5L + 3X. Do you think he will make a different decision than problem (1)? Why? (i.e. Given the new utility function. Would he choose to receive $20 without working or $5 working?) (n) What is the optimal working hour and consumption given the new utility function and the optimal choice of policy from part (m)
