Question

1. Assume that you have exactly 100 hours of labor to allocate between producing goods X and Y. Your output of X and Y depends solely on the hours of labor you spend so the production functions, qi=fLifor i=X and Y $B1n8ngDGypRzcpwAAAABJRU5ErkJggg==$ , are:

X=LX.5 and Y=LY.5

1. If you can sell your output of X and Y at the fixed prices P_X = 10 and

P_Y = 5, how much of goods X and Y would you produce to maximize your income? (Hint: to do this you need to find the marginal products of labor for X and Y. Since this involves a bit of calculus, I provide these for you. The MP_L for X is MPLX=.5LX-.5 and MP_L for Y is MPLY=.5LY-.5. You will use these to show your profit maximizing level of output of X and Y by equalizing the value of the marginal products, i.e. P_X*MPL_X = P_Y*MPL_Y. The value of the marginal product is called the Marginal Revenue Product (MRP). This can be used to solve for L_X. Next, combine this value with your constraint of 100 hours of labor to be allocated between X and Y, i.e. L_X + L_Y = 100. Solve for L_X and L_Y, and X and Y. Note that your numbers for L_X and L_Y are integers, but X and Y are not. You can check to see if you have the right numbers for X and Y by the income you might receive from selling them this is: 10X + 5Y ≈ 111.50.

2. Now assume further that you have the following utility function:

      U=10X.5Y.5 $Bo1dHZN4FQoAAAAASUVORK5CYII=$   

If you can trade a bundle of goods X and Y that you produce in the market at fixed prices P_X = 10 and P_Y = 5, what bundle would you consume to maximize your utility? Now your problem is to consume quantities of X and Y that maximize your utility subject to your income of $111.50. To do this, you will need marginal utilities for X and Y. They are as follows, MUX=5X-.5Y.5 $eb3u2gAAAABJRU5ErkJggg==$ and MUY=5X.5Y-.5 $jNMPbX9M2l8EYcMAAAAASUVORK5CYII=$ . Utility maximization is achieved by equating the marginal rate of substitution (MRS) with the relative prices, i.e. MSR_YX = MU_X/MU_Y = P_X/P_Y = 10/5 = 2. Now substitute into the budget constraint of 10X + 5Y = 111.80. Solve for X and Y. Again these are not integers.

Are you a net demander or a net supplier of the two goods? Draw a Production Possibilities frontier diagram to depict what is happening.

Answer 1

First, equate the MRPs. (10)(0.52)(5)(0.5) PrMPLY PMPL 2 1 L 932125 L 4L The constraint is that the total labor allocation be 100. Lr L100 5Ly 100 4L L 100 Ly 20 Lx 4L 4(20) = 80 X Production of X and Y: X L(80) Y=L(20) =8.94 =4.47 Now to the second part. Write the budget constraint. 10X 5Y 10(8.94)+5 (4.47) 10X5Y 111.80 Write the condition of utility maximization: 5X уos MUx MUr 10 10 5 5 Y Y 2X X Substitute this in the budget constraint. 10X5(2X 111.80 20X 111.80 X 5.59 Y 2(5.59) 11.18 Since the consumption (5.59) of X is less than the production (8.94) of X, the individual is a net supplier of X. Since the consumption (11.18) of Y is greater than the production (4.47) of Y, the individual is a net demander of Y