Microeconomics I
Mock Exam
1. Consider Charlie who only consumes chocolate (x1 ) and milk (x2 ). Charlie has a Cobb-Douglas utility function given by
The price of one bar of chocolate is $10 and the price of one carton of milk is $5. Suppose that Charlie has a monthly income of $600 to spend on chocolate and milk.
(a) [1 point] Write down Charlie’s budget set.
(b) [1 point] Write down Charlie’s utility maximization problem.
(c) [6 points] Find the optimal bundle of chocolate and milk using the La- grange method. Clearly indicate the irst order conditions of the problem.
(d) [4 points] Now suppose that Charlie wishes to achieve a utility of 50. Solve Charlie’s expenditure minimization problem and show that the optimal bun-dle is (approximately) the same as the bundle you’ve found in part (c). (e) [4 points] What is the elasticity of substitution for Charlie?
(f) [3 points] Suppose the price of chocolate drops to $7.5. Calculate a hypo- thetical bundle which would provide the same utility to Charlie as when the price is $10.
(g) [1 point] Again suppose the price of chocolate drops to $7.5. Calculate a hypothetical income which allows Charlie to consume his initial optimal bundle.
2. In each of the following parts, derive the optimal bundle for the given utility function, prices, and income.
(a) [7 points] Let u(x1 , x2 ) =px1 + 5 + x2 , p1 = p2 = 5€, and I = 200€ . What type of preferences does u represent?
(b) [7 points] Let u(x1 , x2 ) = 2x1 + 5x2 , p1 = 3€ , p2 = 5€, and I = 100€ . What type of goods are these?
(c) [6 points] A consumer only consumes kolbasz (x1 ) and potatoes (x2 ), and always eats 1 kolbasz with 3 potatoes. Let p1 = 1800 HUF, p2 = 200 HUF, and suppose the consumer has a daily income of I = 7200 HUF. What type of goods are these? Write the utility function and calculate the daily optimal choice of the consumer.
3. Suppose that preferences of a consumer over two bundles x, y E R2 are deined by
(a) [4 points] Determine whether the preference relation is complete.
(b) [4 points] Determine whether is transitive.
(c) [4 points] Does satisfy monotonicity (more is better)? Explain.
(d) [4 points] Suppose that the utility function is given byu(x1 , x2 ) = max{x1 , x2 }. Draw a couple of indiference curves for diferent levels of utility. Explain on the graph why does not satisfy convexity.
(e) [4 points] Show that 之 is not a continuous preference relation. Hint: fnd two sequences of bundles xn and yn such that xn yn for all n but y x as n 一 ∞.
4. Suppose Jules visits Big Kahuna Burger and wishes to purchase burgers (x1 ) and fries (x2 ) for everyone at a party. As this is his favorite burger joint, he knows that the price of a burger is $5 and 1 unit of fries is $2. Let Jules’ utility function be given by u(x1 ; x2 ) = x1 x2(2) and suppose he has an income of $210. Arriving at the restaurant, he immediately notices that the price of a burger has increased to $7.
(a) [7 points] Calculate the substitution and income efects on x1 using Hick- sian decomposition. Draw the graph and illustrate the efects on it. Note: you can use that 14001/3 = 11:18 when calculating the substitution efect.
(b) [7 points] Calculate the substitution and income efectson x1 using Slutsky decomposition. Explain the diference with Hicksian decomposition (you do not need to draw the graph).
(c) [3 points] Now suppose that burger is an inferior good. Show (without doing any calculation) on a graph where the new optimal bundle could be, how the income and substitution efects would look like, and explain the intuition.
(d) [3 points] Draw the own-price demand curve for burgers, using your graph from part (a). Indicate the initial optimal bundle, the inal optimal bundle, and the hypothetical optimal bundle on the graph.
5. Consider the Star Wars universe. Suppose that the demand and supply for blaster ri月es are given by
QD (P) = 720 - 20P; QS (P) = 60P:
Suppose that after the collapse of the Galactic Republic and the rise of the Galactic Empire, Emperor Palpatine wishes to create some revenue from tax and decides to impose a tax of 4 imperial credits (IC) on blaster riles: producers will pay the empire 4 IC for each rile sold.
(a) [5 points] Find the equilibrium price and quantity for riles before the tax. (b) [5 points] Find the equilibrium price and quantity after the taxis imposed.
(c) [5 points] How much is the empire’stax revenue? Why shouldn’t we calcu- late the revenue simply by multiplying the amount of tax with the quantity found in part (a)?
(d) [5 points] Draw a graph with pre-tax demand and supply curves, together with the post-tax supply curve. Indicate the area of dead-weight loss and calculate the amount.