Question

1. A consumer has the utility function U = min(2X, 5Y ). The budget constraint isPXX+PYY =I.

(a) Given the consumer’s utility function, how does the consumer view these two goods? In other words, are they perfect substitutes, perfect complements, or are somewhat substitutable? (2 points)

(b) Solve for the consumer’s demand functions, X∗ and Y ∗. (5 points)

(c) Assume PX = 3, PY = 2, and I = 200. What is the consumer’s optimal bundle?

(2 points)

2. Assume a consumer has a “standard” utility function that we have worked with in class. Draw a graph that shows:

(a) The consumer’s optimal consumption bundle at X = 10 and Y = 4. This should include the indifference curve and budget constraint (1 point)

(b) Now assume the price of Y decreases so that the new consumption bundle isX = 5 and Y = 12. Draw the new budget constraint, indifference curve, and optimal bundle on the same graph. (1 point)

(c) On the same graph, show/label the total effect, income effect, and substitution effect. You should use different colored pens/pencils to make the graph easier to read. (4 points).

3. Suppose there are 3 buyers in the market. Buyer “A” has the demand functionQA = 10 − 2P, “B” has the demand function QB = 6 − 0.5P, and “C” has the demand functionQC =12−3P.

(a) Derive the market demand. (3 points)

(b) Graph the demand curves for each consumer and the market demand when P is 1, 2, 4, 6, 8, and 10. (2 points)

Answer #1

1.a) Given the Consumer's utility function, U = min (2X, 5Y), we can say that consumer view these two goods x, Y as perfect complements. This type of utility can be represented as 'L' shaped Indifference curve where the optimal bundle lies at the kink which touches the budget line.

b) At point of maximum utility,

2X = 5Y

or, X = 5Y/2... equation (1)

putting this in the budget constraint, P_{X}X +
P_{Y}Y = I, we get,

P_{X}. (5Y/2) + P_{Y}Y = I

or, Y[(5P_{X}/2)+P_{Y}] = I

or, Y[5P_{X}+ 2P_{Y}] = 2I

or, Y = 2I / [5P_{X}+ 2P_{Y}]

Again putting this in equation (1), we get,

X = 5Y/2 = (5/2).(2I / [5P_{X}+ 2P_{Y}] )

or, X = 5I / [5P_{X}+ 2P_{Y}]

So the demand function for X,

X* = 5I / [5P_{X}+ 2P_{Y}]

The demand function for Y,

Y* = 2I / [5P_{X}+ 2P_{Y}]

c) P_{X} = 3, P_{Y} = 2, and I = 200

then optimal bundle, (X*, Y*) = (5I / [5P_{X}+
2P_{Y}] , 2I / [5P_{X}+ 2P_{Y}] )

or, (X*, Y*) = (5x200 / [5x3 + 2x2] , 2x200 / [5x3 + 2x2] )

or, (X*, Y*) = (1000/19, 400/19)

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