Question

An agent chooses between two goods, x and y, with prices px and py, respectively. She has an income I and her preferences are represented by the utility function U (x, y) = lnx + y.

A. Suppose that I = 100, px = 2 and py=1. How much of good x and y will the agent choose?

B. If the price of good x rises to px=4, with income and the price of good y remaining the same, what quantities does he/she buy and what is his/her resulting utility? Illustrate graphically

C. Find the income and substitution effect for good x due to this price change. Illustrate graphically.

D. Find the compensating variation for the price change and illustrate graphically.

E. What is the equivalent variation of the price increase?

Answer 1

A) Consider U (x,y) = lnx + y

MRS = y / x

The Income given in the question is as follows: I = 100, px = 2, py = 1. Her Income constraint is

I = p_xx + p_yy

100 = x + y

Step 1: Set MRS equal to price ratio

MRS = p_{x /} p_y

y/x ₌   1 / 1

y = x

this relationship must hold at the utility maximizing point.

Step 2: Substitute step 1 into income constraint

Since y = x, the income constraint becomes

100 = x + y

= x + x

= 2x

Solving for x yields x = 100 / 2 = 50

Therefore, y = 50 and u = (50)(50) = 2500

Change the price of x

Now suppose the price of x falls to 0.5 or 1/2, Re-do steps 1 and 2,

MRS = P_x / P_y

y / x = 0.5 / 1 = 1 / 2

y = 1 / 2x

Substitute this new relationship into the budget constraint

100 = x + y

100 = x + 1 / 2x

100 = 1.5x

x = 100 / 1.5 = 66.7

y = 33.3

So, the agent will choose 66.7 of Good x and 33.3 of Good y.

B) In case, if the price of good x rises to px = 4, here is how it goes:

Solving for x yields x = 100 / 4 = 25

Therefore, y = 50 and u = (25)(50) = 1250.

C) 1) The substitution effect involves the substitution of good x₁ and good x₂ or vice-versa due to a change in relative prices of two goods.

2) The income effect results from an increase or decrease in the consumer's real income or purchasing power as a result of the price change.

3) The sum of these two effects is called the price effect.

D) Compensating variation is the adjustment in income that returns the consumer to the original utility after and economic change has occured.

E) Equivalent variation is the adjustment that changes the consumer's utility equal to the level that would occur if the event had happened.