Question

A. Based on previous studies, you believe the linear demand function for your good is:

                                    Q_X^d = 20,000 -10P_X + 7P_Y + 0.5M + 250A_X

                  where P_Xis the price of X

                                    P_Y is the price of a related good Y

                                    M is the income of the buyers in the market

                                    and A_X is advertising for X.

                  The good currently sells for $25, the related good sells for $40, the company is spending $50 on advertising, and average consumer income is $25,000. The marketing manager wants to know the own price elasticity and income elasticity for this good. Compute them.

B. The marketing manager also wants to know how much sales will increase if she increases the advertising budget by 10%. Compute this from the information given.

Answer 1

(A)

Demand function is as follows -

QXd = 20000 - 10PX + 7PY + 0.5M + 250AX

QXd = 20000 - (10*25) + (7*40) + (0.5*25000) + (250*50)

QXd = 20000 - 250 + 280 + 12500 + 12500

QXd = 45,030

Calculate ∆QXd/∆PX -

∆QXd/∆PX = d(QXd)/dPX = d(20000 - 10PX + 7PY + 0.5M + 250AX)/dPX = -10

Calculate own price elasticity of demand -

e_p = (∆QXd/∆PX) * (PX/QXd)

e_p = -10 * (25/45030) = -0.005

The own price elasticity of demand is -0.005.

Calculate ∆QXd/∆M -

∆QXd/∆M = d(QXd)/dM = d(20000 - 10PX + 7PY + 0.5M + 250AX)/dM = 0.5

Calculate income elasticity of demand -

e_I = (∆QXd/∆M) * (M/QXd)

e_I = 0.5 * (25000/45030) = 0.277

The income elasticity of demand is 0.277.

(B)

Calculate ∆QXd/∆AX -

∆QXd/∆AX = d(QXd)/dAX = d(20000 - 10PX + 7PY + 0.5M + 250AX)/dAX = 250

Calculate advertising elasticity of demand -

e_A = (∆QXd/∆AX) * (AX/QXd)

e_A = 250 * (250/45030) = 1.388

The advertising elasticity of demand is 1.388.

Calculate percentage change in sales =

Percentage change in sales = Advertising elasticity of demand * percentage increase in advertising budget

Percentage change in sales = 1.388 * 10 = 13.88

The sales will increase by 13.88 percent.