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Answer to Question #51164 in Microeconomics for Paul Muchira

Question #51164
1.Given the following monotonically transformed utility function faced by the consumer
U(X1X2) = X_1^0.5 X_2^0.5
The price of good X1 is P1 and the price of good X2 is P2. Derive the optimal demand (Marshallian demand) function for X1 and for X2.

2.Under a perfect competition the price as sh. 6 per unit has been determined. An individual firm has a total cost function given by C=10+15Q - 5Q^2+Q^3/3. Find:
Revenue function
The quantity produced at which profit will be maximum profit
Maximum profit
Expert's answer
Question 1
(a) U(X1X2) = X1^0.5 X2^0.5
The price of good X1 is P1 and the price of good X2 is P2. 
Optimal demand (Marshallian demand) function for X1 and for X2 will be:
X = (0.5I/P1, 0.5I/P2)
Question 2
P = 6 per unit
C=10+15Q - 5Q^2+Q^3/3.
i) Revenue function is:
TR = P*Q = 6Q
ii) The quantity produced at which profit will be maximum profit is in the point, where marginal revenue equals marginal cost: MR = MC
MR = TR' = 6
MC = C' = 15 - 10Q + Q^2
15 - 10Q + Q^2 = 6
Q^2 - 10Q + 9 = 0
Q1 = 9 units, Q2 = 1 unit (may not be profit maximizing).
iii) Maximum profit is:
TP1 = TR - TC = 6*1 - (10+15-5+1/3) = -$14.33
TP2 = TR - TC = 6*9 - (10 + 15*9 - 5*81 + 729/3) = 54 - 17 = $37

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