We know that:
M = 20 (in thousands)
Py = 2
Px = 14
Q = 14000
Q = 20000 - 500*Px + 25*M + 250*Py
From Using Calculus To Calculate Cross-Price Elasticity of Demand we see that:
We can calculate any elasticity by the formula:
Elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)
In the case of cross-price elasticity of demand, we are interested in the elasticity of quantity demand with respect to the other firm's price P'. Thus we can use the following equation:
Cross-price elasticity of demand = (dQ / dPy)*(Py/Q)
In order to use this equation, we must have quantity alone on the left-hand side, and the right-hand side be some function of the other firms price. That is the case in our demand equation of Q = 20000 - 500*Px + 25*M + 250*Py.
Thus we differentiate with respect to P' and get:
dQ/dPy = 250
So we substitute dQ/dPy = 250 and Q = 20000 - 500*Px + 25*M + 250*Py into our cross-price elasticity of demand equation:
Cross-price elasticity of demand = (dQ / dPy)*(Py/Q)
Cross-price elasticity of demand = (250*Py)/(20000 - 500*Px + 25*M + 250*Py)
We're interested in finding what the cross-price elasticity of demand is at M = 20, Py = 2, Px = 14, so we substitute these into our cross-price elasticity of demand equation:
Cross-price elasticity of demand = (250*Py)/(20000 - 500*Px + 25*M + 250*Py)
Cross-price elasticity of demand = (250*2)/(14000)
Cross-price elasticity of demand = 500/14000
Cross-price elasticity of demand = 0.0357
Thus our cross-price elasticity of demand is 0.0357. Since it is greater than 0, we say that goods are substitutes (if it were negative, then the goods would be complements). The number indicates that when the price of margarine goes up 1%, the demand for butter goes up around 0.0357%.
We'll answer question (b) on the next page.
