Department of Economics Fall 2019 Problem Set 2 - Solutions

EC 203 - INTERMEDIATE MICROECONOMICS Bo˘gazi¸ciUniversity - Department of Economics Fall 2019 Problem Set 2 - Solutions

1. Consider a consumer who wants to consume only two commodities and has an income of $100. Assume the price of good 1 is $10 per unit and the price of good 2 is $20 per unit. Now, inflation causes the price of good 1 to increase to $20 per unit, while the price of good 2 increases to $25 per unit. On the other hand, the consumer also gets a raise of $100 (so her new income is $200). Is she better off or worse off? Solution: Originally, the consumer’s budget constraint is

10x1 + 20x2 ≤ 100.

The budget line has horizontal intercept m = 100 = 10 and vertical intercept m = 100 = 5. After the p1 10 p2 20 change, her budget constraint becomes

20x1 + 25x2 ≤ 200,

and so the new budget line has horizontal intercept m = 200 = 10 (the same as before) and vertical p1 20 intercept m = 200 = 8 (higher than before), and so the consumer’s budget set has grown: she can p2 25 now afford bundles she previously could not, and she can still afford all bundles she previously could. Therefore she is better off.

2. Suppose prices are Px = 2, Py = 1 and income is m = 100.

0 (a) Show the change in the budget set if price of x decreased and price of y increased such that Px = 1 0 and Py = 2. Solution: The new budget line is shown as the blue line in the ﬁgure belove. y

100

−2

−1/2

50 100

0 0 (b) Suppose that the government taxes (at the new set of prices Px = 1 and Py = 2) the consumer an extra 0.50 dollar for each unit of x he buys beyond 70 units. That is, no tax is collected for units x < 70, and 0.50 dollar tax for each x ≥ 70. Show the new budget set.

1 new Solution: Px = 1 for the ﬁrst 70 units, there is no tax. For each additional good x purchased after 70 units, consumers pay $0.50 per unit tax. There is no tax on the second good. This is shown in the following ﬁgure.

100

slope −1/2

slope −3/4 x

70 90

(c) Suppose that on top of part (b) government subsidizes the consumer with a 0.40 dollar for each unit of y he buys beyond 40 units. That is, there is no subsidy for y > 40, and consumer pays 0.40 dollar less for each y ≥ 40. Show the new budget set. new Solution: Py = 2 for the ﬁrst 40 units, there is no subsidy. For each additional good y purchased after 40 units, consumers pays $1.60. This is shown in the following ﬁgure.

100

52.5 slope −5/8

40 slope −1/2

slope −3/4 x

20 70 90

3. Consider a consumer who is choosing how many of two goods to buy: Footballs and cricket balls. The consumer has an income of $20, and the cost of a football is $4 and a cricket ball is $2.

(a) Write down the equation for the consumer’s budget constraint and graph it. Solution: Let F be the amount of footballs and C be the amount of cricket balls. The budget is 4F + 2C = 20, that is 2F + C = 10.

2 C

− pF = −2 pC

(b) The government decides that football is evil and needs to be taxed. They introduce a 50% tax on each football sold. Rewrite and re-graph the budget constraint. Solution: The new cost of buying a football is $6 after the tax. Here the budget line is 6F +2C = 20, or 3F + C = 10.

− pF = −3 pC

10/3

(c) A new government is elected that hates all sports. They now tax both footballs and cricket balls at 50%. What does the budget constraint look like now? Solution: Now the cost of buying a cricket ball is $3. The budget line now is 6F + 3C = 20.

3 C

20/3

− pF = −2 pC

10/3

(d) Due to a threat of revolt amongst sports fans, the government hands out a subsidy of $10 to the consumer. What does their new budget constraint look like? Solution: The budget now is 6F + 3C = 30.

(e) Revolution comes, and all taxes and subsidies are abolished. Even better, the consumer finds a new shop that offers bulk discounts. In this shop, footballs cost $4 each if you buy 3 or less. However, the cost of any additional football after 3 is $2. What does the budget set look like now? discount Solution: PF = $4 for the first 3 units. PF = $2 for any additional football after 3 units.

4 C

− pF = −2 pC

− pF = −1 pC F

3 7

4. William has $300 to spend on food and clothing. The price of food is $2 per unit and the price of clothing is $1 per unit.

(a) Write down the equation for William’s budget constraint and graph it. Solution: Let F denote the amount of food, and C denote the amount of closing. The budget line is 2F + C = 300.

300

− pF = −2 pC

150

(b) Suppose that the government gives William 100 free coupons as food stamps (assuming that William can get free unit of food for each coupon) and strictly enforce a rule which prohibits William from selling his food stamps to someone. Draw the new budget set. Solution: For the ﬁrst 100 units of Food, William can use coupons, and resale is prohibited.

5 C

300

− pF = −2 pC

100 250

(c) Suppose that the rule prohibiting the food stamps sale is diﬃcult to enforce and William can trade each food stamp in black market for $0.50. Draw his budget set. Solution: Now he has 100 coupons for food, but he can sell it in the black market at $0.5 per coupon.

− pF = −1/2 350 pC

300

− pF = −2 pC

100 250

(d) Suppose that in addition to 100 units of food stamps, government also gives 50 free coupons for clothing (each coupon can be used for a free unit of clothing). Assume that trading both coupons in black market can be prohibited. Draw the budget set. Solution: He has coupon for both C and F (trading in black market is prohibited).

6 C

350

− pF = −2 pC

50 F

100 250

√ 5. Mark has preferences for guns and banjos represented by the utility function u(xg, xb) = xg + xb.

(a) Write down the equation for an indiﬀerence curve, that is, for some utility levelu ¯, write down the number of guns as a function of the number of banjos. 1 1/2 Solution: u¯ = xg + (xb) 2 , that is xg =u ¯ − (xb) (b) Find the marginal rate of substitution of banjos for guns. Solution: We have MU = 1 x−1/2 and MU = 1, then we get MRS = MUb = 1 x−1/2. b 2 b g bg MUg 2 b (c) Determine whether these preferences are convex, that is, determine whether there is diminishing MRS property. Solution: Convexity of preferences is equivalent to the diminishing MRS property. You can check

whether MRS is decreasing, as xb increases and xg decreases: MRS is actually decreasing since 1 −1/2 2 xb is decreasing in xb. Alternatively, you can check the second order derivative of the indiﬀerence 2 dxg 1 −1/2 d xg 1 −3/2 curve you found: dx = − 2 xb , then 2 = 4 xb . Second order derivative is positive, so the b dxb indiﬀerence curve has a convex shape.

(d) Show that, for a budget constraint pbxb + pgxg = m, at any point of tangency between budget line pg 2 and indiﬀerence curves, it must be the case that xb = (1/4)( ) . pb pb 1 −1/2 pb 1/2 pg Solution: Tangency is equivalent to MRSbg = . Then x = , that is, x = , which pg 2 b pg b 2pb pg 2 implies, xb = (1/4)( ) . pb

(e) What is the optimal consumption bundle, if pg = $2 , pb = $1 and m = $8? Solution: Since this utility function is well behaved (it is monotone, represents convex preferences since MRS is diminishing, and continuous and diﬀerentiable) at an interior solution tangency should ∗ 2 hold. From the tangency condition above, plugging prices, we get xb = (1/4)(2/1) = (1/4)4 = 1. ∗ ∗ Then, from the budget equation, we get 1 · 1 + 2xg = 8, that is, xg = 7/2. Since both xb = 1 and ∗ ∗ ∗ xg = 7/2 are positive, we have an interior solution at (xb = 1, xg = 7/2).

(f) What is the optimal consumption bundle if pg = $5 , pb = $1 and m = $4?

7 Solution: From the tangency condition, we get xb = 25/4. However, this together with budget equation implies a negative amount of guns. Hence the best solution is to buy no guns and buy as much banjos as possible. Mark spends all income on banjos: x∗ = m = 4, and none spent on guns, b pb ∗ xg = 0. MUb MUg 1 −1/2 You can also see this through comparing and . We have MUb = x and MUg = 1, pb pg 2 b 1 x−1/2 that is, MUb = 2 b = 1/2 and MUg = 1/5. When spending all income on banjos, the most he pb 1 1/2 pg xb MUb 1/2 can buy is xb = 4. With this amount = = 1/4 which is larger than 1/5. Thus, he would pb 41/2 ∗ like to get more xb, but xb = 4 is the limit. Thus, this is the solution, xb = 4, together with no guns, ∗ xg = 0.

a 1−a 6. Suppose a consumer has a utility function u(x1, x2) = x1x2 , for goods x1 and x2.

(a) Find her demand for x1 and x2 in terms of her income, m, prices, p1, p2, and the positive constant 0 < a < 1. (Although we derived this in class, I included this problem here, since it may be useful for the rest of the problem set.) Solution: This is a Cobb-Douglas utility function, which is well-behaved (it is increasing in x and y, that is, it is monotone. It has DMRS, that is, it represents convex preferences. It is also continuous and diﬀerentiable. You should be able to check monotonicity and DMRS). And the budget line is

p1x1 + p2x2 = m. We can directly use the tangency condition: a−1 1−a MU1 ax1 x2 a x2 p1 MRS = = −a = . And the price ratio is simply . Thus, we get MU2 a 1−a x1 p2 (1−a)x1 x2 a x p 2 = 1 1 − a x1 p2 Then, we have x = x 1−a p1 . Now plug this into the budget line and solve for x . 2 1 a p2 1

1 − a p1 p1x1 + p2x1 = m a p2

1−a 1−a 1−a that is, p1x1 + x1 a p1 = m, which implies x1[p1 + a p1] = m, or x1p1[1 + a ] = m, that is, 1 x1p1[ a ] = m. Thus, we get ∗ m x1 = a p1 Plugging this into the equation x = x 1−a p1 , we get x = a m 1−a p1 , that is, 2 1 a p2 2 p1 a p2

∗ m x2 = (1 − a) p2

(b) What share of her budget does she spend on x1, and what share on x2? ∗ p1x1 Solution: The income share she is spending on good 1 is m = a, and the income share she is ∗ p2x2 spending on good 2 is m = (1 − a).

7. Suppose that the preferences of a typical household for quantities of electricity (E) and gasoline (G) are given by u(E,G) = aln(E)+(1−a)ln(G), where 0 < a < 1. Suppose the prices of gasoline and electricity in the units provided are both $1/unit and the household has an income of $100. Suppose in addition, the

8 government has chosen to ration electricity by allowing a maximum consumption of 50 units of electricity,

that is, Emax = 50.

(a) If a = 1/4, ﬁnd the optimal consumption bundle of gasoline and electricity. Solution: The utility function is a positive monotonic transformation of V (E,G) = EaG1−a. Thus the optimal choice bundle can be found through E∗ = a m and G∗ = (1−a) m , which imply E∗ = 25, pE pG G∗ = 75 (you should be able to solve this without applying these formulas). (b) If a = 3/4, ﬁnd the optimal consumption bundle of gasoline and electricity. 100 Solution: Calculation gives E = .75 1 = 75, but this is not allowed. Thus consumers will consume as much E as possible, that is the maximum possible: E∗ = 50, and the rest will be spent on gasoline G∗ = 50.

8. Suppose that a consumer consumes only food (f) and entertainment (e) where pf = $5, pe = $10 and m = $600. Suppose that the consumer has to have a minimum of 50 units of f, and a maximum of 20 units of e.

(a) Draw the budget set. Solution:

50 80 120 F

(b) Suppose the utility function is u(f, e) = f · e. Find the optimal consumption bundle. Solution: This is a monotonic transformation of u(f, e) = f 1/2 · e1/2. By Cobb-Douglas solution above, e = m 1 = 30 and f = m 1 = 60. But, e = 30 is not allowed. Thus, the consumer will pe 2 pf 2 consumer as much e as he can, the maximum allowed, that is, e∗ = 20. Then, from the budget equation we get f ∗ = (600 − (10 · 20))/5 = 80. You can see the optimal bundle in the graph below.

9 E

50 80 120 F

9. Suppose that a consumer only consumes good 1 and good 2 under the following prices and income:

p1 = $2, p2 = $1 and m = $100. Consumer’s preferences can be represented by the utility function,

u(x1, x2) = 3x1 + x2.

(a) Find the optimal bundle.

Solution: This is a perfect substitutes case. MU1 = 3 and P1 = 2, and MU2 = 1 and P2 = 1. Thus MRS = MU1 = 3 = 3 > P1 = 2 = 2, that is, MU1 > MU2 . Thus, this consumer will spend all 12 MU2 1 P2 1 p1 p2 ∗ ∗ income on good 1, which means x1 = 50, x2 = 0. (b) Now suppose that the consumer receives a coupon of $20 which can be spent only in good 2. Draw the new budget constraint and ﬁnd the new optimal consumption bundle. Solution: The budget set is drawn below. Since $20 coupon can only be spent on good 2, he will use the coupon to buy good 2, and use all income to buy good 1, as the prices did not change, that is MU1 > MU2 is still the case. Thus we get, x∗ = 50, x∗ = 20 as the optimal consumption bundle. p1 p2 1 2

120

50 x1

10. Pamir spends most of his time in Just Coffee shop. Pamir has $12 a week to spend on coffee and muffins.

Just Coffee sells muffins for $2 each and coffee for $1.2 per cup. Pamir consumes xc cups of coffee per k k week and xm muffins per week. His utility function for coffee and muffins is u(xc, xm) = xc xm, where k > 0.

10 (a) Find Pamir’s optimal consumption bundle. Does it depend on k? Solution: k k 1/2 1/2 A positive monotonic transformation of u(xc, xm) = xc xm gives u(xc, xm) = xc xm . Then, using the solution we found for the Cobb-Douglas utility function, we get

∗ 1 m 1 12 xc = = = 5 2 pc 2 1.2

∗ 1 m 1 12 xm = = = 3 2 pm 2 2 The solution does not depend on k! (b) Now Just Coffee has introduced a frequent-buyer card: For every five cups of coffee purchased at the regular price of $1.2 per cup, Pamir receives a free sixth cup. Draw Pamir’s new budget set. Solution:

xc 5 6 11 12

(c) With frequent-buyer card, find the new optimal consumption bundle. Solution: (5,3) is the optimal point without the frequent-buyer card. With the frequent-buyer card, the upper portion of the budget line (the portion above (5,3)) cannot be optimal because, all those bundles were affordable but Pamir chose (5,3), thus they all are on a lower indifference curve than (5,3) is on. Thus, Pamir has two possibilities for optimal choice: it might be at the kink (6,3), or it might be in the portion of the budget line to the right of the kink. Consider the modified m: Pamir now has m = 1.2 · 6 + 3 · 2 = 13.2. The Cobb-Douglas solution implies x = 1 m = 1 13.2 5.5, c 2 Pc 2 1.2 and x = 1 m = 1 13.2 = 3.3, however this is not feasible as this bundle lies above the flat portion m 2 Pm 2 2 ∗ ∗ of the budget set. Thus the optimal point should be at the kink (6, 3). Then xc = 6 and xm = 3.

11. A consumer can buy two goods: good 1 denoted by x1 and good 2 denoted by x2. Her utility function is given by u(x , x ) = x1x2 , and p and p are the prices of good 1 and good 2, respectively, and m is the 1 2 x1+x2 1 2 consumer’s income level.

(a) Is this utility function well-behaved? (Hint: It is continuous and diﬀerentiable. How about monotonicity and DMRS?)

11 2 2 x2 x1 Solution: Both marginal utilities are positive: MU1 = 2 , and MU2 = 2 . Thus, (x1+x2) (x1+x2) u(x1, x2) is increasing in both x1 and x2. It is monotone. For DMRS, ﬁnd MRS ﬁrst: using 2 2 2 x2 x1 x2 x2 2 MU1 = 2 and MU2 = 2 we get MRS12 = 2 = ( ) . This is decreasing as x1 (x1+x2) (x1+x2) x1 x1 increases and x2 decreases, implying DMRS. Thus this is well-behaved.

(b) Solve for her demand for x1 and x2 both as a function of p1, p2 and m, that is, x1(p1, p2, m) and

x2(p1, p2, m). Solution: Since u(·) is well-behaved, at an interior solution tangency should hold. Equating MRS q to price ratio gives: MRS = ( x2 )2 = p1 . Then x = x p1 . Plugging this into the budget line, 12 x1 p2 2 1 p2 p1x1 + p2x2 = m, and solving for x1 we get,

∗ m x1(p1, p2, m) = √ p1 + p1p2

Then solving for x2, we get ∗ m x2(p1, p2, m) = √ p2 + p1p2