## Microeconomics

Duration: 1 hour 30 Minutes + 30 minutes upload time

No word count specified

Approved calculators are permitted

To achieve full points for a problem it is not sufficient to provide the correct final answer. I must be able to retrace every step that took you to your answer.

This is an Open Book exam

### 1. Problem – Bundles of goods (15 points) 代考Microeconomics

Let X = R 2 + be the set of all bundles of apples (good 1) and oranges (good 2). Alex has 1 apple and no oranges at home (bundle A = (1, 0)).

a) Marie is invited to Alex’ house. She brings a net of 5 oranges (bundle M). Calculate bundle A + M.

b) Ben arrives and brings 2 apples and 1 orange (bundle B). He eats 1/3 of bundle A + M + B. Calculate the remaining bundle λ(A + M + B) with λ = 2/3.

c) Taylor and Lily join the others. Taylor brings 4 apples and 2 oranges (bundle T). Lily brings 3 apples and 1 orange (bundle L). Since there is already a lot of fruit at Alex’ house now, Taylor and Lily decide to each take home half of their bundles and leave the other half at Alex’ house. Calculate bundle κT + (1 κ)L with κ = 0.5.

d) Draw a two-dimensional graph that contains all the friends’ bundles and the bundles you calculated, i.e. bundles A, M, B, T, L, A+M, λ(A + M + B), and κT + (1 κ)L. Make sure to include axis labels and grid information. Indicate the name of each bundle that you draw next to the bundle.

1. ### Problem – Common assumptions about consumer preferences (12 points)

Let X = R 2 + be the set of all bundles of two goods. Denote by x1 and x2 the amounts of good 1 and good 2 contained in bundle x X.

a) Show that the preference relation on X represented by the utility function u(x1, x2) = x1x2 satisfies strong monotonicity.

b) Show that the preference relation on X represented by the utility function u(x1, x2) = x1 + x2 satisfies convexity.

1. ### Problem – The consumer’s choice problem (20 points) 代考Microeconomics

There are two goods, bread and milk. Billie has wealth w = 15 and the prices of the two goods are pb = 3 for a loaf of bread and pm = 1.5 for a pint of milk.

a) State Billie’s budget constraint.

b) Sketch Billie’s budget line. Make sure to include axis labels and grid information indicating the location of vertical and horizontal intercept as well as the slope of the budget line.

c) What is Billie’s opportunity cost of consuming a pint of milk?

d) The government is contemplating whether to impose a value tax of 20% on both bread and milk. What would Billie’s new budget constraint be? Include a sketch of her budget line after the tax change into your graph from b). Make sure to include proper labeling such that I can distinguish the two lines in your picture.

e) What would Billie’s new budget constraint be if the government imposed a lump-sum tax of T on Billie’s wealth instead of the value tax ?

f) How high must T be for Billie to prefer the 20% value tax over the lump sum wealth tax?

### 4.Problem – Consumer demand (15 points)

a) For which values of α is good 2 an inferior good for Finneas? Explain your answer.

b) For which values of α is good 2 a regular good for Finneas? Explain your answer.

For the rest of this problem, assume that α = 1.

c) A consumer with Cobb-Douglas preferences always spends a fixed proportion of his wealth on each of the two goods irrespective of their prices. Show that Finneas’ demand functions fulfill this characteristic of Cobb-Douglas preferences.

d) Assume that p1 = 1, p2 = 1 and w = 90. Calculate Finneas’ demand for each of the two goods given these prices and wealth level.

1. ### Problem – Consumer demand (16 points) 代考Microeconomics

Susie has preferences on the set of alternatives consisting of all bundles of two goods X = R 2 +. Denote by x1 and x2 the amounts of good 1 and good 2 contained in bundle x X. Susie’s preferences are represented by the utility function

u(x1, x2) = 7 + min{3x1, 9x2}.

Let p1 and p2 denote the prices of good 1 and good 2 respectively. Susie has wealth w > 0.

a) To which general class of preferences do the ones of Susie belong?

b) Consider the alternative utility function ˜u(x1, x2) = min{x1, 3x2}. Does this utility function represent Susie’s preferences? Explain your answer.

c) Derive Susie’s demand function.

d) After a change in prices, Susie’s demand for good 2 decreases by three units. How does her demand for good 1 change?

1. ### Problem – WARP (12 points)

With wealth w and prices p1 and p2, Ruby’s optimal bundle is given by (x1, x2) = (5, 2). Suppose the price of good 1 decreases to p 1 < p1 and the price of good 2 increases to p 2 > p2 such that the bundle (y1, y2) = (4, 5) is just affordable, i.e. costs exactly w, both at the initial prices (p1 and p2) and the new prices (p 1 and p 2 ).

a) Provide a sketch of the two budget lines, i.e. for budget sets B(w, p1, p2) and B(w, p1 , p2 ). Make sure to properly label the budget lines and to indicate at which bundle they cross each other.

b) Assuming that Ruby’s demand function satisfies WARP, can you conclude from the given information that the bundle (z1, z2) = (3, 7) cannot be optimal for Ruby at the new prices, i.e. for budget set B(w, p1 , p2 )? Explain your answer. It might help to use your sketch from part a) for your explanation.

### 7.Problem – Monopoly (10 points) 代考Microeconomics

The housing market in a small village consists of 5 available apartments owned by a monopolist and 5 potential buyers. Assume that all apartments are the same and that the monopolist has to charge the same price for all of them. Out of the 5 buyers, 4 have a reservation price of 4, i.e. they are willing to pay up to 4 units of money for an apartment, and 1 has a reservation price of 3, i.e. is willing to pay up to 3 units of money for an apartment.

a) What price per apartment will the monopolist set in order to maximize her profit? How many apartments will be sold?

b) Compare the allocation in a) to the one under perfect competition. What would be the price per apartment and how many apartments would be sold? Is the monopoly solution efficient? Explain your answer. 