A company operates in a perfectly competitive market and produces two ranges of

Question

A company operates in a perfectly competitive market and produces two ranges of ice cream (denoted as 0 for organic and B for basic). The company faces the inverse demand functions for its two ranges and total cost (TC) function as follows: Find the total profit function for the firm. Find the profit maximising levels of output for each type of ice cream. Find the level of prices corresponding to the profit maximising level of output for each type of ice cream. Calculate the own price elasticity for each type of ice cream at the profit maximising level of output.

Explanation / Answer

NOTE : My junior from college helped me in solving this, although his problem statement was related to Sony Companies. I although replaced the relevant terms with QB,PO,QO as & when necessary, if there are few errors, please do check them at the earliest & change them to the ones above(i.e. in the question).

A. TR = POQO + P2QB

TC = 10 +8(QO+QB)

profit = TR - TC = POQO +P2QB - (10 + 8(QO+QB)) = (60-2QO)QO + (40-QB)QB - 10 - 8(QO+QB)

= 60QO - 2(QO)^2 + 40QB - (QB)^2 - 10 - 8QO - 8QB

= 52QO - 2(QO)^2 + 32QB - (QB)^2 - 10

B. profit maximising P for retail outlets @ d(profit)/dQO = 0; 52 - 4QO =0, 52 = 4QO, QO=13;

PO = 60 - 2QO; PO = 60 - 2(13) = 60 - 26 = 34

profit maxmising Q for superior @ d(profit)/dQB = 0; 32 - 2QB = 0, 32 = 2QB, QB = 16

p2 = 40 - 16 = 24

C. MR = dTR/dQ; dTR/dQO = PO = 34; dTR/dQB = p2 = 24

D. you can use the original formula for this, but I am going to use the profit function I derived in A:

profit = 52QO - 2(QO)^2 + 32QB - (QB)^2 - 10

= 52*13 - 2*(13)^2 + 32*16 - 16^2 - 10

= 676 - 338 + 512 - 256 -10

= 1188 - 604 = 584

E. TR = PQO + PQB = P(QO+QB)

TC = 10+8(QO+QB)

profit = TR - TC = P(QO + QB) - 10 - 8(QO + QB) = (P-8)(QO+QB) -10

profit max retail = d(profit)/d(QO) = 0; P-8 = 0; P =8

profit max superior = d(profit)/d(QB) = 0; P = 8 (confirmed)

using the demand functions above for P? and P2...

PO = 60 - 2QO = 8; 52 = 2QO; QO = 16

P2 = 40 - QB = 8; 32 = QB

However, as you know, if P = 8, (P-8)(QO+QB) = 0, so profit = -10

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