Question 1- Part 1: There are 100 people who comprise a market for good x. They

Question

Question 1-

Part 1: There are 100 people who comprise a market for good x. They have utility functions given by U = (xy)0.5 and incomes to spend on the two goods equal to 60. Assuming the price of y stays constant at 10 and the supply for the market is Qs = 50 + 10P, show the equilibrium price and quantity in this market for x is 15 and 200, respectively.

Part 2: Ben’s Bonanza is a firm in this competitive market for x. The firm’s production function is x = K1/4L1/2, where K = number of units of capital employed and L = number of units of labor employed. The price of both inputs is 1. In the short run, assuming the firm has only 100 units of capital with which to work, show that the firm earns in the short run profits of approximately 462.

Explanation / Answer

At equilibrium,

            MRS    = PX/PY

            y/x       = PX/10

            10y      = xPX                                       …… (1)

The budget constraint is

            xPX      +          yPY      =          60

            xPX      +          10y      =          60

Substitute (1) in budget constraint.

            10y      +          10y      =          60

            y          =          3

Substitute 3 for y in (1).

            10(3)    =          xPX

                x          =          30/PX                           ….. (2)

Equation (2) gives individual demand. Since there are 100 consumers, the market demand is

            xm        =          100x

            xm        =          100(30/PX)

            xm        =          3000/PX                       …… (3)

At equilibrium market demand equals market supply. Therefore,

            3000/PX           =          50 + 10PX

Solving the above expression, we get

            PX        =          15

Substitute 15 for PX in eq. (3).

xm        =          3000/15
xm        =          200

(b)

Substitute 100 for K in the given production function to get the short run production function. The short run production function is

            x          =          1001/4L1/2                                 …… (4)

From this we have,

            L1/2      =          x/1001/4

Square both sides.

            L          =          x2/10                                        …… (5)

The long run total cost is

            LRTC = wL + rk

Since w = 1 and r = 1,

            LRTC = L + K

Since K = 100 in short run, the short run production function is

            SRTC = L +100

Substitute eq.5 in the above expression.

            SRTC = x2/10 +100                                       …… (6)

Differentiate the above expression with respect to x to get the short-run marginal cost.

            SRMC = x/5

Find the output supplied by the firm as follows:

            SRMC             =          PX

            x/5                   =          15
            x                      =          75

Calculate the profit of the firm.

            Profit = TR – SRTC

                        = xPX   – (x2/10 +100)

                        = (75)(15)        – (752/10 +100)

                        = 1125 – 562.5 – 100
                        = 462.5

Profit is approximately 462.

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