Hello. Kindly solve clearly illustrating each step

1. (15 points) Marginal rate of substitution and utility maximization For each of the following utility functions, do the following: 1. Find the marginal rate of substitution. 2. Then find the value for MRS at the bundle X = 1, Y = 2. Describe in one sentence the interpretation of this value for MRS. 3. Find the bundle that maximizes utility subject to the budget constraint if the price of good X is $2, the price of good Y is $2, and income is $20. (a) U = X1/2y1/4 (b) U = In(X) +2Y (c) U = X +4Y1. Peach-Lemon problem Consider the market of used cars. Sellers have two types of cars: peaches (cars in good condition) and lemons (cars in bad condition). If a potential buyer is able to identify quality of a car, his is willing to pay $10000 for a peach and $5000 for a lemon. On the other hand, sellers is willing to sell a peach at $7500 and a lemon at $4000. The game unfolds as below: first, sellers choose to sell or not, and at which price to sell, After that, potential buyers choose to buy or not. Find equilibrium decisions in the following situations: a. There are four times as many peaches as lemons (q=4/5); (15 pts) b. There are four times as many lemons as peaches (q-1/5). (15 pts)1) (4 pt) Consider a firm that produces good r from labor and is the only demander of labor. This firm is a price taker on output market and a price maker on labor market. The output price is 2 per unit. The production function of the firm is given by (L) - 50L - 1.5L', The labour supply function is L,(w) = 0.5w. a) Find equilibrium wage rate and employment. Illustrate graphically. b) Analyze the effect of imposition of a minimum wage rate at w = 28. Illustrate by graph. c) Repeat part (b) for the minimum wage rate set at w) = 46. d) "When firms have some monopsony power, a minimum wage slightly above the free market equilibrium is good for jobs - it offsets the distortion caused by the market power of employers - but a minimum wage substantially above the free market equilibrium is bad for jobs." - BVFD, p.239. Find all possible levels of minimum wage that are bad for jobs in considered economy.4 One-Period Economy Consider a one-period economy where we have a representative consumer, a representative firm, and a government. The representative consumer derives utility from a consumption good (de- noted by C) and leisure (denoted by (), according to the separable utility function U(C, () = u(C) + v(!) where both u(C) and u(() are increasing functions of their respective arguments. As usual, the consumer's endowment consists of h hours, which he can devote either to work or leisure. If he decides to work, he supplies labor to the representative firm and earns a wage rate to per hour worked. It also enjoys a dividend income x and has to pay a lump-sum tax to the government of size T. The representative firm produces the consumption good according to a production function given by =F(K, N) = =KN- where z stands for total factor productivity, K' denotes capital, /V denotes labor input, and a is a positive real number. Finally, the government collects T in taxes and then uses these resources to finance government expenditures G. a. Define a CE for this economy. b. Let a = 1/2, = = 1, A = 1025 and h = 24 assume that consumer's preferences are given by U(C.D) = CM where vil) = 0 ( consumer does not value leisure). Suppose that the size of government expen- ditures is G - 40. Find the competitive equilibrium