Question

Suppose the consumer's utility function is given by U(x1, x2) = xq x , where a=6.5, and b=4.0. Suppose further, the price of good 1 is 2.4, price of good 2 is 1.1 and income is 87 Finally suppose the price of good 1 has decreased to 0.4 What is the substitution effect for good 1 Round your answer to two decimal places Your Answer:

Answer 1

U = x1^6.5x2⁴

Utility is maximized when MU1/MU2 = p1/p2

MU1 = $\dpi{80} \partial$ U/$\dpi{80} \partial$x1 = 6.5 x (x1^5.5x2⁴)

MU2 = $\dpi{80} \partial$ U/$\dpi{80} \partial$x2 = 4 x (x1^6.5x2³)

MU1/MU2 = [6.5 x (x1^5.5x2⁴)] / [4 x (x1^6.5x2³)] = (6.5/4) x (x2/x1) = (13/8) x (x2/x1)

With initial prices,

(13/8) x (x2/x1) = 2.4 / 1.1

x2/x1 = 1.34

x2 = 1.34x1

Substituting in initial budget line,

87 = 2.4x1 + 1.1x2

87 = 2.4x1 + 1.1(1.34x1)

87 = 2.4x1 + 1.474x1

87 = 3.874x1

x1 = 22.46

x2 = 1.34 x 22.46 = 30.09

U = x1^6.5x2⁴ = (22.46)^6.5(30.09)⁴

After price change,

(13/8) x (x2/x1) = 0.4 / 1.1

x2/x1 = 0.22

x2 = 0.22x1

Substituting in new budget line,

87 = 0.4x1 + 1.1x2

87 = 0.4x1 + 1.1(0.22x1)

87 = 0.4x1 + 0.242x1

87 = 0.642x1

x1 = 135.51

x2 = 0.22 x 135.51 = 29.59

To find substitution effect (SE), we keep U unchanged and substitute x2 = 0.22x1 in utility function:

x1^6.5(0.22x1)⁴ = (22.46)^6.5(30.09)⁴

x1^6.5.x1⁴ (0.22)⁴ = (22.46)^6.5(30.09)⁴

x1^10.5 = (22.46)^6.5(30.09/0.22)⁴

Taking (1/10.5)-th root on each side,

x1 = [(22.46)^6.5(30.09/0.22)⁴ ]^(1/10.5)

x1 = 44.70

Substitution effect = Decomposition bundle value for x1 - Original value for x1 = 44.70 - 22.46 = 22.24