Intermediate Microeconomics Problem Set 1 to be handed in at the latest by 1PM on November 2nd 2021 Consumer Theory Exercise 1 Suppose the demand and supply for beer in Springfield is given by p = 120 − 0.7Qd p = 3 + 0.2Qs where the quantity is in million of liters and the price in cents per liter. Assume that Springfield does not import or export beer. a) Find the market equilibrium quantity Q∗ and the equilibrium price p∗ . b) Find the consumer and producer surplus at the market equilibrium. c) The Springfield breweries successfully lobby for a price floor of p̄ = 36 cents per liter. What will be the new quantity sold in the market, Q̄? d) Find the new consumer and producer surplus after the price floor. e) What is the deadweight loss from the price floor? f) Suppose the government doesn’t agree with a price floor, but proposes instead to give a subsidy of 3 cents to the breweries for each liter sold. Find the new equilibrium quantity Q∗∗ and the new equilibrium price p∗∗ . What is the total subsidy amount spent by the government? 1 Exercise 2 Barney spends all his monthly income of m = A C500 on two goods, beers (x) and burgers (y). Assume both goods are divisible. The price of a burger is px = A C8. Draw Barney’s budget constraint in the following three cases: a) One beer costs px = A C6 if he buys between 0 and 3 beers and px = A C5 per bottle if he buys more than 3 beers. b) One beer costs px = A C6 if he buys between 0 and 3 beers; if he buys more than 3 beers, each additional beer is half-price, i.e. px = A C3 c) One day, Barney sees an advertisement stating: ”Buy one beer, get one free! No limits per customer.” One beer costs px = A C6. Exercise 3 For each of the following utility functions describing Barney’s preferences for beers (x) and burgers (y) calculate the marginal rate of substitution for positive levels of both goods. a) u(x, y) = (x + 5y)2 b) u(x, y) = 5 ln x + 2 ln y √ c) u(x, y) = 12 x + 3 y d) Assume now that Barney’s preferences for beers and burgers can be described by the following utility function: u(x, y) = k(x − a)α (y − b)β , where k, a, b, α, β > 0. Show that the utility function u(x, y) = δ ln(x − a) + (1 − δ) ln(y − b) represents the same preferences let δ = α α+β . Exercise 4 • Barney invites his friends Homer and Moe for lunch. They can choose between burgers (x), beers (y) and ice cream (z). Prove the following results. a) If x y and y z, then x z. b) If x ∼ y and y ∼ z, then x ∼ z. 2 c) If x y and y ∼ z, then x z. d) If x ∼ y and y z, then x z In other words, if the ’parent’ binary relation % is transitive, what does it imply about its ’offspring’, and ∼? • Consider the above mentioned 3 alternatives x, y, z. How many comparisons have to be specified with rationality? (You can ignore comparisons with the alternative itself.) Consider now the general case with n alternatives. How many alternatives have to be specified without rationality and how many have to be specified with rationality. Exercise 5 For each of the following utility functions, calculate Barney’s demand for burgers and beers, hx (px , py , m) and hy (px , py , m), as functions of the prices and income. In addition to that, verify that the demand functions satisfy the budget exhaustion property and that they are homogeneous of degree zero in prices and income. a) u(x, y) = 3 ln x + 5 ln y b) u(x, y) = [min{x, 2y}]2 c) u(x, y) = 3x + 2 ln y √ √ d) u(x, y) = x + y Exercise 6 Find the demand functions for Barney if his preferences are given by the utility function u(x, y) = x− y1 and calculate the price elasticities εxx , εyy , εxy , εyx and income elasticities η1 and η2 as functions of px , py , and m. Exercise 7 We know that Barney loves eating burgers and drinking beers. This year, due to the mad cow disease, there was a sharp increase in the price of cattle meat - the main ingredient of burgers. This, in turn resulted in an increase in the price of burgers. Show how a change in the price of burgers effects Barney’s optimal consumption choice. 3 a) Illustrate the • the total effect, • the substitution effect, • and the income effect for an increase in the price of burgers (px ) graphically. b) Explain the logic behind the total effect, the substitution effect and the income effect. c) Suppose Barney’s preferences are given by the utility function u(x, y) = x + y. His monthly income is m = A C500, the price of a burger is A C8 and the price of a beer is A C5. Due to the mad cow disease, the price of a burger then increases, such that px = A C9. Decompose the price effects into the substitution and income effects using either the Hicks-Allen or Slutsky decompositions. Intertemporal choice Exercise 8 Barney lives in two periods (period 1 and period 2). Barney's utility is: U (c1 , c2 ) = u(c1 ) + βu(c2 ) where c1 and c2 are the consumptions in period 1 and period 2 respectively, β < 1 is the discount factor, and u(c) = ln(c) with ln the natural logarithm. Barney has incomes y1 in period 1 and y2 in period 2 with the possibility to borrow and lend at the market interest rate r. a) Write down the intertemporal budget constraint faced by Barney. b) Find out the Euler equation. c) Express the optimal saving as function of incomes and interest rate. 4