Question

10. Given the demand function of good 1: Q1=200-5P1+(P2)^2

Q1: the demand for good 1

P1: the price for good 1

P2: the price of good 2

Answer the following questions

What is the inverse demand function of good 1?

Derive the expression of total revenue in terms of quantity of good 1(Q1) and price of good 2(P2).

If total cost function C(Q1) = 30+(Q1)^2 , derive the profit function in terms of quantity of good 1(Q1) and price of good 2(P2).

If P2=2, what’s the optimal quantity to produce for good 1? What’s the corresponding price to charge?

If P1=30 and P2=2, what’s the own price elasticity of good 1? Write out the formula, calculate and interpret.  

Are good 1 and good 2 complements or substitutes? Why?

Answer 1

10. Given the demand function of good 1: Q1=200-5P1+(P2)^2

The inverse demand function of good 1 is given by

5P1 = 200 - Q1 + P2^2

P1 = 40 - 0.2Q1 + 0.2P2^2

This is the inverse demand function.

Total revenue in terms of quantity of good 1(Q1) and price of good 2(P2) is given by

TR = P1Q1

= (40 - 0.2Q1 + 0.2P2^2)Q1

= 40Q1 - 0.2Q1^2 + 0.2Q1P2^2

If total cost function C(Q1) = 30+(Q1)^2 , find profit function as the difference between revenue and cost

profit = 40Q1 - 0.2Q1^2 + 0.2Q1P2^2 - 30 - Q1^2

= 40Q1 - 1.2Q1^2 + 0.2Q1P2^2 - 30

If P2=2, profit function is 40Q1 - 1.2Q1^2 + 0.8Q1 - 30 and so maximizing the profit will result in

40 - 2.4Q1 + 0.8 = 0

Q1 = 17 and price P1 = 40 - 0.2*17 + 0.8 = 37.4

If P1=30 and P2=2, the own price elasticity of good 1 is given by ed = price coefficient x price / quantity

Q1=200-5*30+(2)^2 = 54 units

elasticity = -5*30/54 = -2.778. The demand is highly elastic at this price.

Good 1 and good 2 are substitutes because the price coefficient (of P2) has a postive sign showing a rise in price of P2 will increase quantity demanded of Q1.