Question

Slutsky Equation:

Exercise 12:

Assume preferences can be represented by the following utility function: u(x1, x2) = x1 x22

a. Are preferences monotonic? Justify.

b. Set up the consumer’s utility maximization problem for prices p1, p2 and income m (the general case)

c. Solve the problem. You will obtain demand functions x∗1 (p1 , p2 , m) and x∗2 (p1, p2, m) in terms of the parameters (p1, p2, m) . Obtain price elasticity of demand for good one. Obtain income elasticity of demand for good 2.

d. Assume that, originally, the consumer faces:
prices p1 =2, p2 =5 and income m=30(A+1).

where A is 4. Now assume the price of good 1 increases to p,1 = 3. Obtain the income and substitution effects for good 1 with Slutsky compensation (that is, compensating the individual so that it can still buy the old bundle at the new prices).

Need help with D.

Answer 1

The utility function is U (x1,x2) = x1*x2

d). Given P1 = 2 and P2 = 5

New P1n= 3.

And income(M) = 30 (A+1) where A=4

So, M = 30 (4+1) = 30*5 = 150.

Our budget line :- 2x1 + 5x2 = 150

And B.L for P1n:- 3x1 + 5x2 = 150

First we have to find out optimal bundle.

Max U (x1,x2) = x1*x2

S.t 2x1 + 3x2 = 150

For utility maximization Mux1/P1 = Mux2/P2

MUx =x2 and MUy = x1 And P1 =2 and P2 =5

So, x2/2= x1/5 ; 5x2 = 2x1

Putting it in budget line:- 5x2 + 5x2  = 150

So, x2= 150/10 =15 and x1 =5*15/2 =75/2 = 37.5

(x1*,x2*) = (37.5,15)

Now new price of x1 is 3

Budget line for this prices:- 3x1 + 5x2 = 150

With U(x1,x2) = x1x2

Now with the same method as above we will get optimal bundle =(25,15)

In this image orange colour shows Utility function

And blue :- 2x1 + 5x2 = 150

And red:- 3x1 + 5x2 = 150.

Now we have to get the income so that we can get the old bundle with prices P1n = 3 and P2 = 5

So firstly U = x1x2 = 37.5*15 (the old bundle)

Min(x1,x2) 3x1 + 5x2

S.t.. U (x1,x2) = x1*x2>= 37.5*15

So for utility maximization MUx1/P1=MUx2=P2

x2/3 =x1/5; 5x1 = 3x1.

Putting it in Utility function,

X1*3*x1/5 = 37.5*15

x1^2 = 37.5*15*5/3 So, x1=75/6^0.5

x2 = 45/6^0.5

This is the bundle which will give the same amount of satisfaction as old bundle with new prices.

And the purple curve gives old bundle with new prices.

The original choice means old bundle(x10'x20) =(37.5,15)

The bundle with change in price(x1c,x2c) = (25,13

The final choice(x1f,x2f) =(75/6^0.5,45/6^0.5)

Now Substitution effect = x1c - x10 = 25 - 37.5

And income effect = final choice - original choice = x1f - x10 = (75/6^0.5) - 37.5.