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.

The answer is (10, 5).

Step 2:

Step 2a: Find the amount of income m' that you will use in step 2b.

Step 2b: Solve the consumer’s problem given her preferences (described by u) and under the assumptions that m' = 300 (as we found in 2a) , p₁ = 20, p₂ = 20 (new prices).

The answer is = (7.5, 7.5).

Step 3: Solve the consumer’s problem given her preferences (described by u) and under the assumptions that m' = 200 (original income), p₁ = 20, p₂ = 20 (new prices).

The answer is = (5, 5).

Please help me solve this.

Answer 1

Munz ! * u(x,,H2) = 10x, 42 x 42 budget line 200 = 1000,+ 2002 at optimal point slope of u = stopte slope of budget line il MRS = x on MUNI MUXL - 10x/221,-1/2 x 42 10x / x2-1/2 x 1/2 + 12 = 2 2 2 = 242. put nizan, in the budget line we will get, 200 = 10x212 + 20 X2 On 4002= 200 2 12 = 200 = 5 a ni=245=10 so ni=10 and n 25 when p, increases to 20 then budget line becomes. 200 = 200, + 2002 10= nituz at optimal per point, MRS = 1 1 1 P I an 2 = % :. 2 = 21 i nitni =10 40

this n2zn, halds an = 10 . Ni 5 So, tabal effect - n"-a = 5-10 = -5 Now amount of n needed to buy old bung 10 at neue price is 20x10+ 20x 5 = 200+100 a 300 ml and at neue m'=300 , 20, 22 So 30o =200, +200, On 22 0 or non, = 300 and a2 = n = 7.5 so substitution effect = n'a = 7.5-10 2-2.5 • Income effect = 0,"_n,' = 5-7.5 = -2.5 ni = 7.5