Problem Set 4 Solutions PDF

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ECON 3100 LMU Economics Problem Set 4 Solutions 1. The Calculus of Demand (Cobb-Douglas Utility Function) Suzie purchases two goods, food and clothing. She has the utility function U(X, Y) = 2XY , where X denotes the amount of food consumed and Y the amount of clothing. Hint: You may find it easier to begin by solving a Lagrangian for generic I, PX, and PY.

ECON 3100 LMU Economics a. Use a Lagrangian to find an expression for her demand curve for clothing when I = 200 and PX = 2. Does this satisfy the law of demand?

ECON 3100 LMU Economics b. Use a Lagrangian to find an expression for her Engel Curve for clothing when PX = 2, PY = 2. Is clothing normal or inferior?

ECON 3100 LMU Economics 2. The Calculus of Demand (Special Utility Functions) Refer to Q3 of Problem set 3. a. Josiah Bartlett collects 2 goods, antique maps (M) and antique books (B). His utility function is U(M, B) = 2M + B (they are perfect substitutes). Find Josiah’s demand curve for maps if he has $1500 to spend and the price of antique books is $100. Answer: Since the goods are perfect substitutes, we are looking for a corner solution. Therefore we compare the utility Josiah receives if he spends all of his money on maps with the utility he receives from spending all of his money on books. If he spends all of his money on maps, he buys 𝑀 =

utility of 𝑈(𝑀, 𝐵) = 2 (

𝑀 = 0 maps and 𝐵 = 2(0) + 15 = 15.

1500

𝑃𝑀 1500 100

)+0

maps and 𝐵 = 0 books. This provides him with a

𝑃𝑀 3000 = 𝑃 . 𝑀

If he spends all of his money on books, he buys

= 15 books. This provides him with a utility of 𝑈(𝑀, 𝐵) =

Therefore he chooses maps if curve will be

1500

3000 𝑃𝑀

> 15 or 𝑃𝑀 < 200. This means that the demand

1500 𝑖𝑓 𝑃𝑀 < 200 𝑃𝑀 𝑀 = 0 𝑖𝑓 𝑃𝑀 > 200 {∈ [0,7.5] 𝑖𝑓 𝑃𝑀 = 200 =

b. Josiah decides to focus on collecting books and improve the way he displays the books. For each book, he needs 2 bookends (E), so his utility function is U(E, B) = min{E,2B} (they are perfect complements). Find his demand curve for bookends if he has $1500 to spend and the price of antique books is $100.

Answer: We know that he buys 2 bookends for each book, so 𝐸 = 2𝐵 or 𝐵 = 2 𝐸. The equation 1

for his budget line is 𝑃𝐸 𝐸 + 100𝐵 = 1500. Substituting 𝐵 = 𝐸 into this gives 1 𝑃𝐸 𝐸 + 100 𝐸 = 1500 2 𝑃𝐸 𝐸 + 50𝐸 = 1500 𝐸(𝑃𝐸 + 50) = 1500 1500 𝐸= 𝑃𝐸 + 50

1

2

ECON 3100 LMU Economics

3. The Calculus of Income and Substitution Effects Consider the following three graphs, which illustrate the preferences of three consumers (Bob, Carol, and Ted) regarding two goods, apples and peaches. Each consumer has an income of $30, and each consumer pays $2 for apples and $3 for peaches. (There are some extra graphs at the end of the problem set if you need.)

a. Suppose that the price of peaches falls to $2. Draw a new budget line for each consumer and find the new optimal bundle of apples and peaches each would buy. How does the new quantity of peaches compare to the original quantity? Indicate the change in the first column of the table below (an increase of 1 unit might be denoted as a +1). Answer: Bob’s peach consumption increases by 4 units, Carol’s peach consumption increases by 1 unit and Ted’s peach consumption increases by 1 unit due to this price change. See

ECON 3100 LMU Economics the graphs below (the lines that passes through point B indicate the new budget line). Also see the table below.

b. For each consumer, determine the substitution effect of the price change. Hint: Draw a hypothetical budget line with the same slope as your new budget line, but just tangent to the consumer’s original indifference curve. Indicate that change in the second column of the table below. Answer: See the graphs above (the lines that passes through point A’ indicate the hypothetical budget line). Bob’s peach consumption increases by 1 unit, Carol’s peach consumption increases by 2 units and Ted’s peach consumption increases by 1 unit due to the substitution effect of price change. See the table below. c. Now add the income effect. Compare each consumer’s peach consumption in (b) to his or her final peach consumption in (a). Indicate the difference in column 3 of the table below. Answer: Bob’s peach consumption increases by 3 units, Carol’s peach consumption decreases by 1 unit and Ted’s peach consumption does not change due to the income effect of price change. See the table below.

ECON 3100 LMU Economics

d. Do Bob, Carol, and Ted consider peaches normal or inferior? Answer: 





Bob considers peaches normal. As the price of peaches fall, he feels relatively richer. From the table, we can see that he buys 3 more peaches due to income effect of price change. Carol considers peaches inferior. As the price of peaches fall, she feels relatively richer. From the table, we can see that she buys 1 less peach due to income effect of price change. Ted considers peaches neither inferior nor normal since his peach consumption does not change due to the income effect of price change.

4. Deriving Market Demand Three students have different demands for doughnuts. André’s demand is given by Q = 5 – P; Carlene’s demand is given by Q = 6 – 2P; Cooper’s demand is given by Q = 4 – 0.5P. a. Derive the market demand curve for doughnuts algebraically. Answer: Adding all 3 individual demand curves together, we get Q = (5 – P) + (6 – 2P) + (4 – 0.5P) = 15 – 3.5P However, André enters the market at prices below 5; Carlene at prices below 3; and Cooper at prices below 8.

ECON 3100 LMU Economics Hence for prices between $5 and $8, the market demand curve is equal to Cooper’s demand curve (Q=5-P). For prices between $3 and $5, the market demand curve is equal to Cooper’s and André’s demand curves combined (Q = (5 – P) + (4 – 0.5P) = 9 – 1.5P) and for prices below $3, it is equal to the demand curve we calculated above (Q=15 – 3.5P). Hence the market demand curve is 15 − 3.5𝑃 𝑖𝑓 𝑃≤3 9 − 1.5𝑃 𝑖𝑓 3 < 𝑃 ≤ 5 𝑄 = {4 − 0.5𝑃 𝑖𝑓 5 < 𝑃 ≤ 8 0 𝑖𝑓 𝑃>8

b. Graph the market demand curve for doughnuts. Pay special attention to any kinks in the market demand! Answer:...