You are the manager of a firm that sells a "commodity" in a market that resembles perfect competition, and your cost function is C(Q) = 2Q + 3Q2. Unfortunately, due to production lags, you must make your output decision prior to knowing for certain the price that will prevail in the market. You believe that there is a 70 percent chance the market price will be $200 and a 30 percent chance it will be $600. a. Calculate the expected market price. $ b. What ouptut should you produce in order to maximize expected profits? units c. What are your expected profits? $

Explanation:

a)Their is a 70 percent chance the market price will be $200 and 30 percent chance it will be $600. Thus , expected market price is calculated as follows

Expected market price = 0.7 * 200+0.3*600

                                     = 320

b) The expected profit maximixing output is set were margina cost is equal to marginal revenue .The margina revenue is calculated

           Expected market price = 2 + 6Q

                                             320 = 2 + 6Q

                                              6Q = 318

                                              6Q = 53 unit

53unit is the expected profit maximixing output

c)The expected profit is calculated below

profit = Total revenue – Total expense

        = 14,405

a. $320

b. 53 units

c. $8,427

Explanation:

a. Calculate the expected market price.

The expected market price is obtained by adding the multiplications of each of the two market prices and their respective probability as follows:

Expected market price = (0.7 × $200) + (0.3 × $600) = $320

Therefore, the expected market price is $320.

b. What output should you produce in order to maximize expected profits? units

Note that the correctly stated cost function is C(Q) = 2Q + 3Q^2.

Also, in a perfect competitive market, profit is maximized when price (P) is equal to marginal cost (MC) ( i.e. when MC = P)

Differentiating the cost function with respect to Q to obtain the MC as follows::

dCCQ)/dQ = MC = 2 + 6Q (1)

Since, in a perfect competitive market, profit is maximize MC = P, and we know that expected market price (P) is $320 in a above, we therefore equate equation (1) to 320 and solve for Q as follows:

2 + 6Q = 320

6Q = 320 - 2

6Q = 318

Q = 318 ÷ 6

Q = 53 units

Therefore, output that will maximize expected profits is 53 units.

c. What are your expected profits? $

Expected total revenue = Expected market price × Expected units

Expected total revenue = 320 × 53 = $16,960

To obtain expected total cost, substitute 53 units for Q in the cost function as follows:

Expected total cost = C(53) = 2(53) + 3(53^2) =  106 + 8,427 = $8,533

Expected profit = Expected total revenue - Expected total cost

Expected profit = $16,960 - $8,533 = $8,427.

Therefore, expected profits is  expected profits is $8,427.