Question

Assume that a consumer’s preferences are given by u(x1, x2) = 10x₁^1/2 *  x₂^1/2

Currently, m = 200 and p₁ = 10 and p₂ = 20. Suppose now that p₁ increases to p'₁ = 20. What is the total effect of this price change in the optimal consumption of the two goods for the consumer, and what are the substitution and income effects?

Step 1:Solve the consumer’s problem given her preferences (described by u) and under the assumptions that m = 200, p₁ = 10, p₂ = 20.

Please help me solve this. The answer is (10, 5) but I am not getting that.

Answer 1

u = 10x1^1/2x2^1/2

Utility is maximized when MU1/MU2 = p1/p2

MU1 = $\dpi{80} \partial$ u/$\dpi{80} \partial$x1 = 10 x (1/2) x (x2/x1)^1/2

MU2 = $\dpi{80} \partial$ u/$\dpi{80} \partial$x2 = 10 x (1/2) x (x1/x2)^1/2

MU1/MU2 = x2/x1 = p1/p2

Initially,

x2/x1 = 10/20 = 1/2

x1 = 2x2

Substituting in budget line,

200 = 10x1 + 20x2

20 = x1 + 2x2

20 = x1 + x1 = 2x1

x1 = 10

x2 = x1/2 = 10/2 = 5

u = 10 x (10)^1/2(5)^1/2 = 10 x (50)^1/2 = 10 x 7.07 = 70.7

After price change,

x2/x1 = 20/20 = 1

x1 = x2

Substituting in new budget line,

200 = 20x1 + 20x2

10 = x1 + 2x2

10 = x1 + x1 = 2x1

x1 = 5

x2 = 5

For good x1, Total effect (TE) = new value of x1 - original value of x1 = 5 - 10 = - 5

For good x2, Total effect (TE) = new value of x2 - original value of x2 = 5 - 5 = 0

To find substitution effect (SE), we keep u = 70.7 and substitute x1 = x2 in utility function:

10x1^1/2x1^1/2 = 70.7

10x1 = 70.7

x1 = 7.07

x2 = 7.07

For good x1, SE = decomposition bundle value of x1 - original value of x1 = 7.07 - 10 = - 2.93

For good x2, SE = decomposition bundle value of x2 - original value of x2 = 7.07 - 5 = 2.07

For good 1, Income effect (IE) = TE for x1 - SE for x1 = - 5 + 2.93 = 2.07

For good 2, Income effect (IE) = TE for x2 - SE for x2 = 0 - 7.07 = - 7.07