2. Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and
ID: 1190243 • Letter: 2
Question
2. Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y represent the quantities of two goods, X and Y. For parts (a)-(h), assume that good X costs pX=$3 per unit and good Y costs pY=$4 per unit. “Wealth” and “income” have the same meaning in this problem.
(a) With good X on the horizontal axis and good Y on the vertical axis, draw the consumer’s budget line if she has W=$31 to spend, Show the exact coordinates of the intercepts.
(b) On your budget line, show the optimal point (which you calculated in Homework #6). How much utility does that point produce? Draw the indifference curve that passes through that optimal point, showing the exact coordinates of at least three points on the curve.
(c) On the same diagram, draw three new budget lines, representing the consumer’s budget constraint for other possible values of her wealth: W=$7, W=$55, and W=$79. For each budget line, show the exact coordinates of its intercepts.
(d) If the consumer’s wealth is W=$7, then calculate the optimal point and show it on the appropriate budget line. Do the same for W=$55 and W=$79. Your diagram should now show four possible budget lines and four optimal points.
(e) Calculate the marginal rate of substitution of X for Y, at each of the four optimal points. Explain the pattern that emerges.
(f) Connect the four optimal points to show the income expansion path. Are X and Y normal or inferior goods?
Explanation / Answer
(a) The maximum quantity that can be purchased of good X is 10.33 (=31/3) units. The maximum quantity that can be purchased of good Y is 7.75 (31/4) units.
(b) MRSXY= MUX/MUY
= (3Y)/(3X+1)
The budget constraint is
PXX + PYY = W
3X + 4Y = W
At the optimal bundle,
MRS = PX/PY
(3Y)/(3X+1) = 3/4
This implies Y = (3X+1)/4. Remember this result.
The budget line becomes 3X + 4[(3X+1)/4] = W
This implies X = (W – 1)/6. Remember this result.
When W = 31, X = (31 – 1)/6 = 5, and Y = (3(5)+1)/4 = 4.
Therefore, the optimal bundle is (5, 4).
The utility at this bundle is U(5, 4)=[3(5)+1]4 = 64.
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