Question

Ahn’s utility function for goods X (pizzas) and Y (cola) is represented as
U(X, Y) = 2ln(X)+ln(Y).
The prices of X and Y are $1 and $1, respectively. Ahn’s income is $12.
1) Calculate Ahn’s optimal consumption bundle (X*, Y*).
(X*, Y*)= .
2) Suppose there is an increase in the price of X. Illustrate the net effect, income effect, and substitution effect on Ahn’s optimal consumption choice.

Answer 1

ax ay Ann's Hility function for goods X (pizzas) and (cola ) is represubed as U(X,Y) = 2 en (X) + lncy) - Now, marginal utility of x, mux = JU(XY) = 2 and Maginal utility of y, muy = a0(x,y) = m Now price of x Px = $1 and pull of y Py = 42. and income M= $12 Hice budget constant je мг х Рxt у Ру -) 12 = x+y condition will be? Now atlity Manimising 1 + + 2 3 4 5 2Y = 1 => Xay in the budget constraint we Now substituting x=2y get? ) 12- Xty 2y+ y = 12 3y=12 Y = 4 2) AS X = 2y = 2*4 = 8 optinel Consumption bundle will be Hnce A ha's cxoys = (84) Scanned with CamScanner

. M qy q B & F substitution effect 4-testethit To fal ebbet q, q. substitution effect (SE) q 2 en come effect (IE) Total et bedt (TE) - 9593 =/SE+IE) Scanned with CamScanner
The following diagram describes the effect of price increase of good X. The initial budget line is AC and the initial indifference curve is I1 which is tangent to the budget line AC at point M and the initial consumption of good X is q1. Now due to a price increase of good X, the budget line rotates inward from AC to AB the new budget line becomes tangential to a new indifference curve I2 at point N.  Where the consumption of good X is q3. Now the total effect of the movement from M to Ncan be decomposed into two parts- a substitution effect and an income effect.

To understand the substitution effect, we will hold the real Income unchanged. Hence consumer will stay on the same indifference curve I1 but as the price ratio has changed hence the consumer will stay on the budget line with the same slope as the new budget line AB. Hence we draw a hypothetical budget line DF which is tangent to the initial indifference curve curve I1 at point O and is parallel to the new budget line AB. Hence the movement from M to O which leads to a decline in the quantity demanded of good X from q1 to q2 is the substitution effect.

Now to understand the income effect we consider real income as variable. Now as price of good X increased, consumers will feel poorer and there will be a parallel shift of budget line from DF to AB which becomes tangent to the new indifference curve I2 at point N. Hence the movement from point O to point N which leads to a decline in the quantity demanded of good X from q2 to q3 is the income effect. And the sum total of substitution effect and income effect is the total effect.