A firm\'s demand function for product X has the following question: Qx = 1420 -2

Question

A firm's demand function for product X has the following question:
Qx = 1420 -20Px- 10Py + 0.02M + 0.04A
where Qx is the quantity purchased, Px is the price of X, Py is the price charged for a related good, M is per capita income, and A is the dollar spent on advertising.

Suppose the firm spends $1200 per week on advertising, that Py=$40 and that income is $8,000 per capita.

a.) Write the equation of the demand curve for product X.
b.) Briefly explain how product X is related to product Y. (Is Y a substitute or a complement, and how can you tell?)
c.) Given the stated values of the other independanct variables, calculate the point price elasticity of demand for X at Px = $50.
d.) Given the stated values of M, A and Py at what price and quantity demanded will total revenue maximized? What will the maximum revenue be?

Explanation / Answer

A)
Qx = 1420 -20Px- 10Py + 0.02M + 0.04A
Qx = 1420 -20Px - 10*40 + 0.02*8000 + 0.04*1200
Qx = 1420 + - 10*40 + 0.02*8000 + 0.04*1200 - 20Px
Qx = 1228 - 20Px

B) Here, we see that in increase in Py decreases Qx. This implies that x and y are compliments. When Py increases, consumers purchase less of y and less of x.

C)
Qx = 1228 - 20Px
Qx = 1228 - 20*50
Qx = 228
Qx = 1228 - 20Px
dQx/dPx = -20
E = dQx/dPx * Px/Qx
E = -20 * 50/228
E = -4.38596

D)
R = Px*Qx
Qx = 1228 - 20Px
R = Px*(1228 - 20Px)
Take the derivative and set equal to zero to maximize.
MR = 1228 - 40Px = 0
Px* = 1228/40
Px* = 30.7
Qx = 1228 - 20Px
Qx = 1228 - 20*30.7
Qx* = 614
R* = Px*Qx
R* = 30.7*614
R* = 18849.8

Price is 30.7
Quantity is 614

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