Microeconomics I
Midterm Exam
1. Consider the utility function
which is similar to Cobb-Douglas utility but incorporates the income of the con- sumer, where γ > 0. me extra term γI implies that the consumer derives utility directly from income (on top of consuming x1 and x2) by a factor of γ. Let the prices of good 1 and good 2 be p1 and p2 , respectively.
(a) Write down the budget constraint and state the utility maximization problem of the consumer. Solve the maximization problem by the Lagrange method. What is the optimal bundle?
(b) Does the extra term have an erect on the optimal choice? Provide intuition why or why not.
(c) Derive the indirect utility function V(p1 ,p2 , I).
(d) Now suppose that γ can take negative values. For what values of γ would indi- rect utility decrease with income? Hint: check the derivative of the indirect utility function.
(e) Suppose the optimal utility of the consumer is Write down the expenditure minimization problem given that the consumer wants to achieve
(f) Now suppose that the utility is , which is similar to CES utility but incorporates the income of the consumer. For which values of α, β, and p does the solution coincide with what you have found in part (a)? Explain without calculations.
(g) How do we interpret the parameter p? What do conditions p > 0 and p < 0 imply under the CES utility function?
2. In each of the following parts, derive the optimal bundle for the given utility function, prices, and income.
(a) Let u(x1 , x2 ) =px1 +5 + x2 , p1 = p2 = 5€, and I = 200€. What type of preferences does u represent?
(b) Let u(x1 , x2 ) = 2x1 +5x2 , p1 = 3€, p2 = 5€, and I = 100€. What type of goods are these?
(c) A consumer only consumes kolb sz (x1) and potatoes (x2), and always eats 1 kolb sz 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 deaned by
(a) Determine whether the preference relation % is complete.
(b) Determine whether is transitive.
(c) Does satisfy monotonicity (more is better)?Explain.
(d) 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) Show that is not a continuous preference relation. Hint:，nd two sequences of bundles xn and yn such that xn yn for all n but y % xas n ∞.
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) Calculate the substitution and income erects on x1 using Hicksian decompo- sition. Draw the graph and illustrate the erects on it. Note: you can use that 14001/3 = 11.18 when calculating the substitution e，ect.
(b) Calculate the substitution and income erects on x1 using Slutsky decomposit- ion. Explain the direrence with Hicksian decomposition (you do not need to draw the graph).
(c) Now suppose that burger is an inferior good. Show (without doing any calcula- tion) on a graph where the new optimal bundle could be, how the income and substitution erects would look like, and explain the intuition.
(d) Draw the own-price demand curve for burgers, using your graph from part (a). Indicate the initial optimal bundle, the anal optimal bundle, and the hypotheti- cal optimal bundle on the graph.
5. Consider the Star Wars universe. Suppose that the demand and supply for blaster ri，es are given by
QD (P) = 720 − 20P, QS (P) = 60P.
Suppose that arer 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 rines: producers will pay the empire 4 IC for each rine sold.
(a) Find the equilibrium price and quantity for rines before the tax.
(b) Find the equilibrium price and quantity arer the tax is imposed.
(c) How much is the empire’stax revenue? Why shouldn’t we calculate the revenue simply by multiplying the amount of tax with the quantity found in part (a)?
(d) Draw a graph with pre-tax demand and supply curves, together with the post- tax supply curve. Indicate the area ofdead-weight loss and calculate the amount.
(e) You have calculated DWL using the usual demand curve instead of the marginal willingness to pay (MWTP) curve. What does this implicitly assume for the calculation to be correct?
(f) Suppose that blaster rine is a normal good. Draw the MWTP curve and state whether the usual demand curve overstates or understates the DWL relative to the MWTP curve. Note: do not derive the MWTP curve, only draw to indicate how it would be relative to the demand curve.