Question

The demand function for your brand X shirts is estimated as

Q^d_x   =   1,000 - 5 P_x - 10 P_y + 9 P_Z+ 0 .001 I

The price of X is $ 10, Y is $ 4, Z is $ 10 and incomes are $ 20,000.

                  

1) What are your sales?                                                                                                               ______________

2) What is the elasticity of demand for X?                                                                               _______________                                                                 

3) What is the cross elasticity between X and Y?                                                                       ______________

                                     

4) Are X and Y substitutes or complements?                                                                            ______________

5) What is the income elasticity for X?                                                                                      ______________

6)   If incomes increased by $10,000, what would be the increase in sales of X?                      ______________

                                               

Answer 1

The demand function for your brand X shirts is estimated as

Q^d_x   =   1,000 - 5 P_x - 10 P_y + 9 P_Z+ 0 .001 I

The price of X is $ 10, Y is $ 4, Z is $ 10 and incomes are $ 20,000

[1]

Sales value = [price *quantity ] of x

quantity of x = Q^d_x   =   1,000 - 5 [10] - 10 [4] + 9 [10]+ 0 .001[20000]

Qx= 1000-50-40+90+20 = 1020

so, sales value = 1020*$10 = $10200

[2]

price elasticity = dQx/dPx * Px/Qx

dQx/dPx = derivative of Qx with respect to Px = -5

price elasticity = -5* [10/1020] = -5/102 = -0.049

[3]

cross price elasticity = dQx/dPy * Py/Qx

Qx= 1020

Py= $4

dQx/dPy = derivative of Qx with respect to Py = -10

so, cross price elasticity = -10 * [4/1020] = -4/102 = -0.039

[4]

Thus , the cross price elasticity is negative, therefore , it can be concluded that the good X and good Y are complementary goods. They are not substitutes.

[5]

Income elasticity of X: [dQx /d income ]*[income /Qx]

dQx/dM= derivative of Qx with respect to income = 0.001

Income elasticity of x= 0.001 [20000/ 1020]

=20/1020 = 0.0196

As income elasticity lies between 0 to 1 , it is normal goods.

[6] If income is increased by $10000

Thn new income is $30000

so, Qx = 1,000 - 5 P_x - 10 P_y + 9 P_Z+ 0 .001 I

Qx = 1000 -5[10] -10[4] +9[10] +0.001[30000]

Qx= 1000-50-40+90+30 = 1030

Therefore, new sales value = Px*Qx= 10*1030 = $10300

Therefore sales value increased by = $10300 -$10200 = $100