U(x,y)=3x+y, where x and y represent the quantities of two goods, X and Y. The o

Question

U(x,y)=3x+y, where x and y represent the quantities of two goods, X and Y. The ocnsumer has I=60 , to spend on the two goods, and good Y costs Py=2 per unit. The price of good x, Px is also exogenous.

1. with good x on the horizontal axis and good Y on the vertical axis, draw indifference curve for utlity leve U=6. Show the exact coordinates of at least two points on the curve. On the same diagram similarly draw the indifference curves for the utlity levels U=18 and U=30.

2 Drw consumer's budget line and calculate the first order condition for this problem. On your budget line, show optimal point. Justify your answer.

3. If optimal point = 10 dollar, calculate the consumer's demand function which shows Qd as a function of its price.

4. Graph demand function that you found in part 3, showing the specific coordinates of the points with Px=2, 4, 6, 8, 10.

5. Calculate the consumer's elasticity of demand for X, at the point Px=4 and Px=8.

Explanation / Answer

U = 3x + y

(1)

An additive utility function signifies that goods x & y are perfect substitutes and the indifference curve (IC) is a straight line cutting both axes.

(i) U = 6

6 = 3x + y

When y = 0, x = 2 (horizontal intercept of IC)

When x = 0, y = 6 (vertical intercept of IC)

(ii) U = 18

18 = 3x + y

When y = 0, x = 6 (horizontal intercept)

When x = 0, y = 18 (vertical intercept)

(iii) U = 30

30 = 3x + y

When y = 0, x = 10 (horizontal intercept)

When x = 0, y = 30 (vertical intercept)

The IC are drawn as follows.

(2)

Budget line: I = x. px + y. py

60 = 4x + 2y [Note: Price of X is given with question (5) as 4]

Or, 30 = 2x + y [Dividing by 2]

When y = 0, x = 15 (horizontal intercept of budget line)

When x = 0, y = 30 (Vertical intercept of budget line)

When consumption bundle is (0, 30), U = 3x + y = 30

When consumption bundle is (15, 0), U = 3x + y = 45

Since 45 > 30, utility is higher with bundle (15, 0) which is the optimal point.

The first order condition for general utility maximization problems is: (MUx / MUy) = (Px / Py)

But when the goods x & y are perfect substitutes, this condition may or may not hold, therefore is not applicable.

To check:

U = 3x + y

MUx = dU / dx = 3

MUy = dU / dy = 1

MUx / MUy = 3

But Px / Py = 4 / 2 = 2

So the 1st order condition doesn't hold.

(3)

Question not clear. Optimal point = $10 means your total income is $10, or your (x, y) bundle is (10, 0) or (0, 10)?

Without this clarification, (3) & (4) cannot be answered.

(5)

When Px = 4, x = 15

When px = 8, Budget line is: 60 = 8x + 2y

30 = 4x + y [Dividing by 2]

When x = 0, y = 30 & U = 3 x 0 + 30 = 30

When y = 0, x = 7.5 & U = 3 x 7.5 + 0 = 22.5

So, optimal bundle is (0, 30).

So, when Px = 8, x = 0

Change in price = $(8 - 4) / $4 = 1

Change in demand = (0 - 15) / 15 = -1

Elasticity of demand = Change in demand / Change in price = -1 / 1 = - 1 [Unitary elastic demand]

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