Warwick Economics Summer School 2016 Problem set 1 Answers Question 1 The basic labour supply model assumes that wage income is untaxed. Suppose instead that a set of marginal tax rates is imposed such that at a wage of £10 per hour, the first 5 hours of labour is untaxed, the next 10 hours of labor are taxed at a 20% rate, and all labor thereafter is taxed at a 50% rate. Show on a graph how this would affect the budget line. How might this affect the number of hours worked? Question 2 Suppose we are interested in looking at Andrewβs consumption of wine. One way to do this is to assume Andrew buys wine, and a combination of all other goods: the composite good. 1. How would you adopt this interpretation in writing Andrewβs budget constraint? a. To adopt this interpretation it is useful to think about the composite good as being the amount of money (£) left for Andrew to spend on the combination of all the other goods. Therefore, the price of one unit of the composite good would be set to 1 (the price of one pound, is one pound) and the budget constraint would be such as ππ° π + π = π Where C represents the composite good (everything else Andrew can spend his money on that is not wine), W the quantity of wine Andrew purchase, and π·π the price of wine. At your high-school's fund-raising picnic, you pay for soft drinks with tickets purchased in advance -one ticket per bottle of soft drink. Tickets are available in sets of three types: ο· ο· ο· Small: £3 for 3 tickets Medium: £4 for 5 tickets Large: £5 for 8 tickets. The total amount you have to spend is £12. If fractional sets of tickets cannot be bought (and no resale of tickets is possible), graph your budget constraint for soft drinks and the composite good. Short way to answer is to assume \free-disposal" of drinks and money. This means that if any quantity (x*; y*) is available, then so is (xβ; yβ) where x*β€ xβ and y*β€yβ. Therefore, the budget line will include all these bundles: (0,12), (3,9), (5,8), (8,7), (11,4), (13,3), (16,2). And bundles that are in between. This yields a step-shaped budget line as shown in the figure below. Longer way to answer is to work out the budget for every quantity: (1) If no soft drinks are bought then there is £12 remaining to spend on the composite commodity. (2) One soft drink can be obtained at lowest cost by buying 3 tickets, 2 can obtained by buying 3 tickets, 3 can be obtained by buying 3 tickets. Thus, zero to 3 soft drinks cost £3 which leaves £9 to spend on the composite commodity. This is shown in the figure as the line segment (0,9) (3,9). (3) Four soft drinks can be obtained at lowest cost by buying 5 tickets at a cost of £4, five soft drinks cost the same. This is shown in the figure as the line segment (3,8)β(5,8). (4) And so on... Composite good (in money terms) 12 9 8 7 4 3 2 Soft drinks (tckts) 3 5 8 11 13 16 Question 3 What is a βbadβ good? Given that definition, draw an indifference curve for a consumer making a choice between Chocolate and Broccoli, where broccoli is assumed to be a βbadβ. a. A bad is a commodity that the consumer doesnβt like. There could still be a trade-off between a good and a bad, in which case, to consume more of the bad, the consumer would want to be given more of the βgoodβ Broccoli Chocolate Question 4 What is a neutral good? Given that definition, draw an indifference curve for a consumer making a choice between Chocolate and Broccoli, where broccoli is now assumed to be neutral b. Consumers are indifferent about neutral goods. They do not care about consuming them, neither do they care about not. In which case, the indifference curve will be vertical at the chosen amount of the other good, as increasing the consumption of the neutral good does not change the satisfaction of the consumer. Broccoli Chocolate Question 5 Suppose there are 3 different goods: cups of coffee (x1), ounces of milk (x2) and packets of sugar (x3). Assume that each of these goods costs 25p and you have an exogenous income of £15. (a) Suppose that the only way you get enjoyment from a cup of coffee is to have at least one ounce of milk and one packet of sugar in the coffee, the only way you get enjoyment from an ounce of milk is to have at least one cup of coffee and one packet of sugar, and the only way you get enjoyment from a packet of sugar is to have at least one cup of coffee and one ounce of milk. What is the optimal consumption bundle on your budget constraint? (b) How will your optimal consumption bundle change if the price of coffee rises to 50p? (c) Return to the original prices of 25p for all goods. Write down the budget constraint. (d) Write down a utility function that represents the tastes described above. (e) Suppose that instead your tastes are less extreme and can be represented by the utility function U (x1, x2, x3) = x1Ξ±x2Ξ²x3. Calculate optimal consumption of each good when your economic circumstances are described by the prices p1, p2 and p3 and income is given by I. At 25p each, 60 units of one good can be bought with £15, assuming nothing else is bought. The goods are perfect complements, implying with given prices and income x1 = x2 = x3 = 20 and L-shaped indifference curves. (f) Same quantities of each will still be bought. 0.5x1 + 0.25x2 + 0.25x3 = 15, but x1 = x2 = x3 at any optimum. (g) 0.25x1 + 0.25x2 + 0.25x3 = 15 (h) U(x1, x2, x3) = min{x1, x2, x3} (i) U (x1, x2, x3) = x1Ξ±x2Ξ²x3. Take logs and maximise utility function. Take FOCs: Ξ±/x1 = Ξ»p1 Ξ²/x2 = Ξ»p2 1/x3 = Ξ»p3 P1x1 + p2x2 + p3x3 = I Solve for x1 = Ξ±p3x3/p1 and x2 = Ξ²p3x3/p2 by solving equation 3 and substituting into equations 1 and 2. Substitute these into budget constraint and solve for x1 = Ξ±I/(Ξ±+Ξ²+1)p1 and x2 = Ξ²I/(Ξ±+Ξ²+1)p2 and x3 = I/(Ξ±+Ξ²+1)p3 Question 6 Suppose you solve a consumer's constrained 2-good optimisation problem for a given economic environment --- and your answer contains a negative consumption level of good 2. Which of the following is a valid conclusion on your part? A. The true optimum has the consumer consume none of good 1. B. The true optimum has the consumer consume none of good 2. C. There are multiple βtrueβ optimal consumption bundles. D. The consumer will sell good 2. E. None of the above. Answer is B. FOCs identify tangencies wherever they occur. If consumption is negative for one good, it means the true optimum has the consumer consume none of that good, with all income spent on the other good β corner solution. Question 7 Suppose Joe's utility for lobster (L) and squash (S) can be represented as U = L0.8S0.2. Joe walks into a restaurant with £90. Lobsters cost £24 each and glasses of squash cost £2 each. How much lobster and squash will Joe consume if he intends to spend all his money? (There are no taxes and no tips.) Maximizing Joe's utility subject to his budget constraint yields: U = L0.8S0.2 + Ξ»(90 - 24L - 2S) FOCS: 1) dU/dL = 0.8L-0.2S0.2 β Ξ»24 = 0 2) dU/dL = 0.2L0.8S-0.8 β Ξ»2 = 0 3) dU/dΞ» = 90 - 24L - 2S = 0 From 1) and 2), S/L = 3 or S = 3L. Substituting into 3) yields L = 3. Since S = 3L, S = 9. Thus, Joe will buy 3 lobsters and wash it all down with 14 glasses of squash. Question 8 Suppose the utility function for good x and y is given by π(π₯, π¦) = π₯π¦ + π¦. a. Calculate the uncompensated (Marshallian) demand functions for x and y, describe how the demand curves for x and y are shifted by changes in income M or the price of the other good. The Lagrange method yields π π·π = π + π π·π π·π π = π·π π + π·π By substituting into the budget constraint, we obtain π= π° β π·π ππ·π π= π° + π·π ππ·π Hence, changes in py do not affect x, but changes in px do affect y. b. Calculate the expenditure function for x and y. The indirect utility function is (π° + π·π )π π½= ππ·π π·π which yields an expenditure function of π¬ = βπ½πππ π·π β π·π c. Use the expenditure function calculated in (b) to compute the compensated demand functions for goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in income or by the changes in the price of the other good. Clearly the compensated demand function for x depends on py whereas the uncompensated function doesnβt. By Shepherdβs Lemma: ππ = ππ¬ = π½π.π πβπ.π ππ.π π π βπ ππ·π Question 9 A college student who loves chocolate has a budget of £10 a day, and out of that income she purchases chocolate x and a composite good y. The price of the composite good is £1. The studentβs preferences are represented by the quasi-linear utility function π(π₯, π¦) = 2βπ₯ + π¦. a. Find the marginal utilities π΄πΌπ = π βπ ; π΄πΌπ = π b. Suppose the price of chocolate is initially £0.50 per ounce. How many ounces of chocolate and how many units of the composite good are in the studentβs optimal consumption basket? π΄πΌπ π΄πΌπ π = β = π·π π·π π·π βπ The studentβs demand curve is therefore given by π = πβ . At a price of £0.50, the student (π·π )π buys πβ(π. π = π ounces of chocolate. From the budget constraint, we can obtain the amount of π) the composite good the student consumes: π·π π + π·π π = π° β π. π(π) + (π)π = ππ β π = π c. Suppose the price of chocolate drops to £0.20 per ounce. How many ounces of chocolate and how many units of the composite good are in the new optimal consumption basket? π΄πΌπ π΄πΌπ π = β = π·π π·π π·π βπ The studentβs demand curve is therefore given by π = πβ(π· )π . At a price of £0.20, the student π buys πβ(π. π = ππ ounces of chocolate. From the budget constraint, we can obtain the amount π) of the composite good the student consumes: π·π π + π·π π = π° β π. π(ππ) + (π)π = ππ β π = π d. What are the substitution and income effects that result from the decline in the price of chocolate? Illustrate these effects on a graph. Initially the consumer consumes π = π and π = π and obtains a utility level πΌπ = πβπ + π = ππ. At the compensated basket, it must therefore be true that πΌ(π, π) = πβπ + π = ππ. We also know that π΄πΌπ π·π = π΄πΌπ π·π must be such that π βπ = π. ππ. Solving for x and y from those two equations with two unknowns we get π = ππ and π = π. The substitution effect is therefore ππ β π = ππ ounces of chocolate and the income effect is the rest of the effect, that is, zero. Other goods A C π2 = 15 B C π΅πΏ2 π1 = 12 π΅πΏ1 Chocolate Question 10 Morgane purchases two goods, food and clothing. She has the utility function π(π, π) = π₯π¦, where denotes the amount of food consume and y the amount of clothing. Denote the price of π₯ as ππ₯ and the price of π¦ as ππ¦ . (a) Find the marginal utilities. π΄πΌπ = π ; π΄πΌπ = π (b) In terms of marginal utilities found above, write down the consumerβs optimum condition and rearrange to find an expression for x in terms of y. π΄πΌπ π΄πΌπ = π·π π·π π·π π π = β π = π( ) π·π π·π π·π (c) Write the budget constraint, substitute for x from the expression found in (b). π·π π + π·π π = π° π·π π ( π·π ) + π·π π = π° π·π ππ·π π = π° π= π° ππ·π (d) What can you say about the expression found above? It is the equation for Morganeβs demand curve. (e) Is clothing a normal good? Yes, it is a normal good. As her income increases, holding the price of y (clothing) constant, the quantity demanded by Morgane of clothing increases. Suppose now that Morgane has an income of £72 per week, that the price of clothing is ππ¦ = £1. Suppose also that the price of food is initially ππ₯1 = £9 per unit, and that the price falls to ππ₯1 = £4. a. Find the initial consumption basket when the price of food is £9, by using the two equilibrium conditions (budget constraint and tangency condition). π·π π + π·π π = π° β ππ + π = ππ π΄πΌπ π΄πΌπ = β π = ππ π·π π·π Solving for x and y we get: π = π and π = ππ b. Repeat this step to find the consumption basket when is price of food is £4 π·π π + π·π π = π° β ππ + π = ππ π΄πΌπ π΄πΌπ = β π = ππ π·π π·π Solving for x and y we get: π = π and π = ππ c. Find the utility level corresponding to the initial consumption basket. πΌ(π, π) = ππ = π(ππ) = πππ d. Find the compensated basket, and therefore the substitution effect. Remember that at that point, Morgane faces the new budget constraint, but is compensated to obtain the same level of utility as before the change in price. ππ = πππ π΄πΌπ π΄πΌπ = β π = ππ π·π π·π Solving for x and y we get: π = π and π = ππ The substitution effect is therefore π β π = π, or 2 units of food. e. Using your answer in b. and d., find the income effect. The income effect is the remaining change in quantity to obtain the new consumption basket, or π β π = π, that is 3 units of food. f. Graph your results. Clothing, y π2 = 324 36 24 π1 = 144 π΅πΏππππ π΅πΏ1 4 6 9 π΅πΏ2 Food, x