# Math Help - Microeconomics Price Optimization Problem

1. ## Microeconomics Price Optimization Problem

I have an econ exam tomorrow and there are a few questions that I have difficulty solving. They all relate to finding the optimum quantity or price that a company should produce at.

The correct answers are in brackets after the question , but I am more interested in explaining how the answer was obtained.

Glyde Air Fresheners is the dominant rm in the solid room aromatizer industry, which has a total market demand given by Q = 80 - 2P. Glyde has competition from a fringe of four small rms that produce where their individual marginal costs equal the market price. The fringe rms each have total costs given by TCi = 10Qi +2Q2 . If Glyde's total costs are given by TC.G = 100 + 6QG, what price should Glyde establish for air fresheners? -

If a representative rm with long-run total cost given by TC = 50 + 2q + 2q2 operates in a competitive industry where the market demand is given by QD = 1410 - 40P, the long-run equilibrium output of the industry will be:

(The correct answer is 530 units)

If labor and capital produce output according to Q = 6KL^(1/2), labor costs $12, capital costs$240, and output sells for \$2, the optimal level of L is:

(The correct answer is L= 400)

I know that I should try to equate MR = MC (or mrp = me) but it does not always work (the answer I get is different from the correct answer)

Any help is greatly appreciated

2. Glyde Air Fresheners is the dominant rm in the solid room aromatizer industry, which has a total market demand given by Q = 80 - 2P. Glyde has competition from a fringe of four small rms that produce where their individual marginal costs equal the market price. The fringe rms each have total costs given by TCi = 10Qi +2Q2 . If Glyde's total costs are given by TC.G = 100 + 6QG, what price should Glyde establish for air fresheners? -
Step 1:Start by finding the amount produced by each small firm, if the market price is P.
You are told that they produce at Marginal Cost = Price
MC = 10 + 4Q
P = 10 + 4Q
Q=(P-10)/4

There are 4 identical small firms, so the total they produce is
Q=4*(P-10)/4
Q=P-10

Step 2 Find the demand curve faced by the large firm
The demand curve faced by the large firm is the totla market demand, less the amount produced by the other firms:
Qd = 80 - 2P - (P-10)
Qd = 90 - 3P
P = (90-Qd)/3

Step 3 Find the Total revenue function of the large firm.
The demand curve is the average revenue. Muliply by Q to get the Total Revenue.

P = (90-Q)/3
PQ = TR = Q(90-Q)/3
TR = 90Q - Q^2

Step 4
Now you have the total revenue and total cost functions for the large firm. Find the price by equating MC = MR

3. oops there is a shocking algebra error in the last step of line 3.

should say:
PQ = TR = Q(90-Q)/3
TR = 30Q - (Q^2)/3