Question

Suppose the market demand for Cashews is as follows: Qc = 500 + (.01)I – 100Pc - 25Pb Qc = Quantity of Cashews demanded I = Income Pc = Price of Cashews Pb = Price of Beer Find the own-price, cross-price, and income-elasticities of cashews as a function of prices, income and quantity. (3 pts) Assume that Income = 50,000 Pc = $6, and Pb = $10 Solve for the elasticities above by plugging these values in. In each case, note whether elastic, inelastic, or unit elastic. Are cashews a normal or inferior good? Are beer and cashews complements, substitutes, or unrelated goods?

Answer 1

Q_c = 500 + 0.01 I - 100 P_c - 25 P_b

Own-price elasticity of Cashews = (Change in Q_c / Change in P_c ) (P_c /Q_c )

Now, to calculate( change in Q_c/ Change in P_c) , take the first partial derivative of demand function with respect to P_c, we get:
Change in Q_c / Change in P_c = -100

Own price elasticity of cashews = -100 (P_c / Q_c)

Cross price elasticity of cashews = (change in Q_c/ Change in P_b) (P_b / Q_c)

To calculate (change in Q_c / Change in P_b ) , take the first partial derivative of Q_c with respect to P_b,we get:

Change in Q_c / Change in P_b = -25

Cross price elasticity = -25 (P_b / Q_c)

Income elasticity of cashews = (change in Q_c / Change in I ) ( I / Q_c )

To calculate (change in I / change in Q_c ) , take the first partial derivative of Q_c with respect to I, we get:

Change in Q_c / change in I = 0.01

Income elasticity = 0.01 (I / Q_c )

Assume I = 50,000 , P_c = $ 6 and P_b = $10.

Firstly , put these values in Q_c , we get

Q_c = 500 + 0.01 (50000) - 100(6) -25(10)

Q_c = 500 + 500 -600 -250 = 150

Now, put these values in elasticities we get ,

Own price elasticity = -100 (P_c /Q_c)

= -100 (6/150 )

= -4

Because own price elasticity is less than 1 . It means that demand is inelastic.

Cross price elasticity = -25 (P_b / Q_c)

= -25 (10/150 )

= -1.66 .

Cross price elasticity is less than zero. Therefore, it implies that beer and cashews are complements.

Income elasticity = 0.01 (I /Q_c)

= 0.01 (50000/ 150)

= 3.33

If income elastcity is greater than 1 . Then, it represents normal goods. It means cashews are a normal good.