Question

14. Suppose Jack has an income of $12 to buy two goods: sandwiches and sodas. The price of a bottle of soda is $1, and the price of a sandwich is $2. Draw Jack’s budget line (BL1) given his income is $12. (Measure sodas on the X-axis and sandwiches on the Y-axis.) Assume Jack’s utility function is U(x,y)=xy (x is the consumption amount of sodas and y is the consumption amount of sandwiches). Jack’s marginal utility of consuming sodas and sandwiches at consumption bundle (x, y) are denoted by MUx(x, y) and MUy(x, y) respectively. Jack’s preferences are depicted by typical ICs (the left graph). The consumption bundle (x, y) which maximizes Jack’s utility satisfies: MUx(x, y)/MUy(x, y)=y/x.

(A) Please find the numerical values of x and y of the utility maximization point (x, y). Draw a typical indifference curve (IC1) through this utility maximization point.

(B) Suppose the price of a bottle of soda increases from $1 to $4, draw Jack’s new budget lines (BL2) and find his new utility maximization consumption bundles.

(C) Draw an imaginary budget line (BL3) parallel to the new budget line (BL2) and make it tangent to the initial indifference curve (IC1). Show the income and substitution effect of the decrease in the consumption of soda as the price of soda increases. At the new price level, at least how much income should Jack get to achieve the original utility level? (Hint: find the tangent point of BL3 and initial indifference curve (IC1)) (4) Determine the demand curve for Good X.

Answer 1

Utility is maximized when MUx/MUy = Px/Py

MUy/MUy = y/x = Px/2

x.Px = 2y

(A)

When Px = 1,

x.1 = 2y

x = 2y

Substituting in budget line,

12 = x + 2y

12 = x + x = 2x

x = 6

y = x/2 = 6/2 = 3

Initial budget line: 12 = x + 2y

When x = 0, y = 12/2 = 6 (vertical intercept) and when y = 0, x = 12 (horizontal intercept).

In following graph, BL1 is above budget line and IC1 is initial indifference curve tangent to BC1 at point E with optimal bundle being (x1, y1) = (6, 3).

3 BLI BVS T3 12 3 LL

(B)

MUx/MUy = y/x = 4/2 = 2

y = 2x

Substituting in new budget line,

12 = 4x + 2y

12 = 4x + 2.(2x)

12 = 4x + 4x = 8x

x = 1.5

y = 2 x 1.5 = 3

New budget line: 12 = 4x + 2y

When x = 0, y = 12/2 = 6 (vertical intercept) and when y = 0, x = 12/4 = 3 (horizontal intercept).

In above graph, BL2 is above budget line and IC2 is new indifference curve tangent to BC2 at point F with optimal bundle being (x2, y1) = (1.5, 3).

(C)

In case (A), U = 6 x 3 = 18

For good x,

Total effect (TE) of price change = 1.5 - 6 = - 4.5

To find substitution effect (SE), we keep U = 18 and substitute y = 2x in utility function:

x.2x = 18

2x² = 18

x² = 9

x = 3

y = 2 x 3 = 6

SE = 3 - 6 = - 3

Income effect (IE) = TE - SE = - 4.5 + 3 = - 1.5

Cost of old bundle at new prices = 4 x 6 + 2 x 3 = 24 + 6 = 30 (required income to achieve same utility)

In graph, BL3 is parallel to BL2 and tangent to IC1 at point G with decomposition bundle being (x3, y3) = (3, 6). Total effect (TE) is movement from E to F, SE is movement from E to G and IE is movement from G to F.