Download Differential Equations Final and more Exams Differential Equations in PDF only on Docsity!
Examination Cover Sheet
Princeton University Undergraduate Honor Committee
January 2017
Course Number: ECONOMICS 310 Course Name: Microeconomic Theory - A Mathematical Approach Instructor: Leonardo Pejsachowicz Date: January 19th, 2016 Time: 1:30 pm
- This examination is administered under the Princeton University Honor Code. Students should sit one seat apart from each other, if possible, and refrain from talking to other students during the exam. All suspected violations of the Honor Code must be reported to the Honor Committee Chair at [email protected].
- You may NOT use any of the following in this examination: course textbooks, course notes, other books/printed materials, calculators or phones. Please place items you will not need out of view in your bag or under your working space now. Students may not wear headphones during an examination.
- Students may only leave the examination room for a very brief period without the explicit permission of the instructor. The exam may not be taken outside of the examination room.
• This exam is a timed examination. You will have 3 hours only to complete this
exam. Students who do not finish the exam promptly when told to do so may be penalized. Follow the timing suggestions for the questions.
- Do not forget to write and sign the Honor Code pledge:
I pledge my honor that I have not violated the Honor Code during this exami- nation
Additional Instructions
- The exam has 7 pages (including the title page) and 6 questions. Make sure you have them all. All questions will receive equal credit.
- The exam has 3 sections, labeled A, B and C. Use a separate answer-book for each section.
- Print your name clearly on the front cover of the exam and of each answer-book. No (or unclear) information – no grade.
- Write clearly. Illegible answers will cost you points. Show the steps of your math. Use well-labeled diagrams and explain any symbols or notation you introduce.
- You will receive partial credit for answers that are not completely correct (but in the right direction) and questions that are not completely finished. Don’t get stuck!
- During the exam the Instructorr is reachable through the preceptors, that will be available outside of the examination room at set intervals.
- Good Luck!
u 1 (4) = u 1 (12)
we need
α128 + (1 − α)64 = α192 =⇒ α =
So {(^12 , 12 ), (^12 , 12 )} is a mixed Nash equilibrium of the game.
Question 2 (40 minutes)
Consider an economy that can be in in one of two possible states, a good state g with probability p, and a bad state b with probability (1 − p). The economy is populated by a large number of identical expected utility maximizers, with von Neumann-Morgenstern index u(.), that will have wealth wg in the good state and wb in the bad state. Individuals only care about their contingent levels of wealth in the two states. Let x be a project paying xg = 10 in the good state and xb = 2 in the bad state. Assume individuals can buy any share α in the project.
(a) Suppose u(.) is such that individuals are risk neutral. What is the value v of the project? (how much would a risk neutral individual be willing to pay for it?) (b) Go back to the case of a generic u(.). Let Q be the price of the project, so that buying α shares of the project costs αQ. For fixed Q, what is the value V (α) to any such individual of buying α shares of the project? Moreover, what equation must the equilibrium price Q∗^ of the project satisfy? Hint: Our approach in class was to solve for the equilibrium price by finding the maximum price Q∗^ at which an individual would be ”just willing” to buy an infinitesimal share. You can do the same here. (c) Suppose u(.) is now such that individuals are strictly risk averse. What is the equilibrium price Q 1 of this project if wg = wb? (d) Let wg = 100 and wb = 60, and let u be as in part c). Find the new equilibrium price Q 2. In particular, how does Q 2 compare to Q 1 and v? Provide some economic intuition for your answer.
Solution
(a) If individuals are risk neutral, then we can consider u to be linear, in particular, u (x) = x represents the preferences of any risk neutral consumer. In fact, this is the same as saying that the risk neutral individual cares about the expected value of the project. His expected utility of not buying the project is:
EUn (0) = p (wg) + (1 − p) (wb)
while his expected utility of buying it is:
EUn (1) = p (wg + xg − q) + (1 − p) (wb + xb − q)
For him to be willing to buy the project, it must be that: EUn (1) − EUn (0) ≥ 0 p (xg − q) + (1 − p) (xb − q) ≥ 0 pxg + (1 − p) xb ≥ q 10 p + 2 (1 − p) ≥ q
Therefore the risk neutral investor is willing to pay up to 10p + 2 (1 − p) for the project, that is, v = 10p + 2 (1 − p). Notice that this is exactly the expected (gross) return from the project.
(b) The utility of an investor who purchases a share α of the project is:
EUn (α) = p · u (wg + α (xg − Q)) + (1 − p) · u (wb + α (xb − Q))
The equilibrium price of the project would be such that individuals would be indifferent between buying an infinitesimal share or not. So we take the first order condition of the above utility, and find Q at α = 0. The FOC is: 0 = p · u′^ (wg + α (xg − Q)) (xg − Q) + (1 − p) · u′^ (wb + α (xb − Q)) (xb − Q)
Setting α = 0: 0 = p · u′^ (wg) (xg − Q∗) + (1 − p) · u′^ (wb) (xb − Q∗) p 1 − p
u′^ (wb) (xb − Q∗) u′^ (wg) (xg − Q∗)
(c) If wb = wg, then the equation above simplifies to: p 1 − p
(xb − Q 1 ) (xg − Q 1 ) 0 = p (xg − Q 1 ) + (1 − p) (xb − Q 1 ) Q 1 = pxq + (1 − p) xb which is exactly the same as we found in part (a), which is reasonable, since even risk averse investors are risk neutral at the margin of buying something (that’s uncorrelated to their other income/assets) or not.
(d) If wg = 100 and wb = 60, then equation (1) becomes:
p 1 − p
u′^ (60) (xb − Q 2 ) u′^ (100) (xg − Q 2 )
Q 2 =
pxgu′^ (100) + (1 − p) xbu′^ (60) u′^ (100) + u′^ (60)
Note that this is a weighted average of the returns (return on the good state weighted by u′^ (100) and on the bad state weighted by u′^ (60)). Since the investor is risk averse, u′′^ < 0, and u′^ (100) < u′^ (60), so Q 2 is putting more weight on the bad state. That means Q 2 < Q 1 = v. This makes economic sense, since the asset’s return is now possitively correlated to the investors income, so a risk averse consumer would be less willing to purchase it.
x
y
Marginal Revenue
Demand Function
x
y
Marginal Revenue
Demand Function
Max Revenue
(b) Since marginal revenue is always positive for the first demand function, there is no point at which revenue is maximized, as it grows indefinetely. For the second demand function, we can set marginal revenue equal to zero, and that happens at q = 2 (ignoring q > 4, as that has zero marginal revenue, but also zero revenue). That point is in green in the graph in part (a), (c) To calculate the elasticity of demand, we must get the demand functions, and apply the formula: =
∂q(p) ∂p
p q(p) So, for the first demand function,
d 1 (p) = p
− 3 2
d′ 1 (p) =
p−^
5 2
p−^
(^52) ·
p p
− 3 2
And for the second demand function:
d 2 (p) = 4 −
p 2
d′ 2 (p) = −
p 4 − p 2
p 8 − p
(d) Each monopolist solves: max p p(q)q − q
Which has first order condition:
p′(q)q + p(q) = 1
So, for market 1, we have: −
q
− 5 3 1 q^1 +^ q^
− 2 3 1 = 1
So that: q 1 = 3−^
3 (^2) and p 1 = 3 And for market 2: − 2 q 2 + (8 − 2 q 2 ) = 1 So that q 2 = 74 and p 2 = 94.
Question 4 (45 minutes)
Consider an exchange economy with two goods, x and y, and two individuals, Adam and Steve, with utility functions given by
UA(x, y) = min{ 4 x, 2 y} for Adam
and US (x, y) = x^2 y^2 for Steve
Adam is endowed with 20 units of x, while Steve has 30 units of y
(a) Draw the Edgeworth box for this economy, and identify the initial endowment on the graph (b) Solve for a competitive equilibrium of this economy (you can normalize py to 1). (c) Find the set of Pareto Optimal allocations. Describe the set both algebraically and on the Edgeworth box graph. (d) What does the first Welfare theorem say? Show that your answers from part b) and c) provide (partial) verification to the theorem. (e) Pick two Pareto optimal allocations such that: in the first one Adam is strictly better of tat at the competitive equilibrium allocation from part (b); in the second one Adam is strictly worse off than the competitive equilibrium allocation from part (b). Show how these two allocations can be supported as competitive equilibrium outcomes, by trans- ferring some of the endowment of Adam to Steve or the other way around. (this means you also need to find the competitive price vector for each of these two allocations.)
and consumptions (xA, yA) = 7. 5 , 15 (xS^ , yS^ ) = 12. 5 , 15
(c) The interior Pareto Optimal allocations lline in the line yA^ = 2 · xA^ as, outside that line, Adam can give up a bit of the good he has in excess without losing utility, while benefitting Steve. However, if Adam has an excess of x while Steve has no y, transferring x to Steve doesn’t increase either consumer’s utility, so that is also Pareto efficient: as long as Adam has yA^ = 30 and xA^ ≥ 15, then those extra units of x doesn’t affect either one of them. So we can express the set of PE allocations as:
P E = {(xA, yA, xS^ , yS^ )|yA^ = 2xA, xS^ = 20 − xA, yS^ = 30 − yA} ∪ {(xA, yA, xS^ , yS^ )|xA^ ≥ 15 , yA^ = 30, xS^ = 20 − xA, yS^ = 0} (2)
Which is represented in blue in the following figure:
Adam
Steve
(d) The First Welfare Theorem states that, under certain conditions, then every (interior) equilibrium allocation is Pareto efficient. It’s easy to check that the competitive equi- librium we found in part (b) is in the set of Pareto efficient points, as it is interior and satisfies yA^ = 2xA^ (it’s also market in green in the Edgeworth box in item (c)). (e) The two trivial points to choose are the ones where Adam has everything (in which case he is better off), and the one in which he has nothing (in which he is worse off). These two are Pareto efficient, as they are in the region found in (c) So now we have to find initial allocation and prices that make these points competitive equilibria. (i) (xA, yA) = (20, 30), (xS^ , yS^ ) = (0, 0). Let’s then make the initial allocation be this final allocation, and find prices that make either not want to trade: (xAe , yAe ) = (20, 30), (xSe , ySe ) = (0, 0). Steve will never want to trade anyway, as he is unable to sell any of the 2 goods (since his endowment is 0 for each). To make Adam unwilling to trade, it must be that px = 0: otherwise, he would be willing to sell a small amount of x to get some of y. Therefore prices of (px, py) = (0, 1) make this an equilibrium allocation.
(ii) (xA, yA) = (0, 0), (xS^ , yS^ ) = (20, 30). Again, let’s consider this the initial allocation as well: (xAe , yAe ) = (0, 0), (xSe , yeS ) = (20, 30). Adam now is unwilling to trade at any price, since his utility can only decrease from trade. For this allocation to be optimal for Steve, it must be that he is spending the same ammount on both goods (because of his Cobb Douglas utility function). Therefore, prices must be (px, py) = (^23 , 1) for this to be an equilibrium. In general, any point in the PE set found in part (c) can be chosen to set an equilibrium. If that point is in the yA^ = 2xA^ line, then prices must make Steve not wanting to trade (as Adam wouldn’t want for any price) so that it must hold that
pxxS^ = yS^ ⇒ px =
yS xS^
30 − yA 20 − xA
And if the chosen point is in the horizontal line where xA^ ≥ 15 and yA^ = 30, then px = 0 (note that in this case, px = 0 is also making Steve spend the same amount, 0 on each goods, so Steve’s optimality condition would be sufficient for most of that region
- except for the specific point (ii) we proposed)
SECTION C
Question 5 (35 minutes)
Let ω represent the quality of cars in a used car market. Assume quality is unobservable to buyers ( but known to sellers) and it is uniformly distributed on [0, 2]. Suppose used car sellers value their cars v(ω) where v(ω) = 34 ω. There are a large number of potential buyers who have value ω for a used car of expected quality ω. Assume that the market for used cars is competitive, so the marginal seller must be making zero profit.
(a) Fix a price p > 0. Which cars would be sold at this price? What would the expected quality of used cars offered at this price be? Is there a strictly positive price p > 0 at which the market is in equilibrium? Solve analytically and give a brief comment explaining in words what is happening in this market. (b) Now suppose that
v(ω) =
1 4 ω^ ω^ ∈^ [0,^ 1] 3 4 ω^ ω^ ∈^ [1,^ 2] As in part a) find for any price p > 0 the cars that would be sold at p and the ex- pected quality of used cars offered at p. Is there a p > 0 at which the market reaches equilibrium?
Solution
(a) For a given p > 0 only those producers who get non-negative profits will participate. That is, those with ω, such that
v (ω) 6 p.
Q(p)
S(p)
p
1 4
1 2
1 2
3 4
3 2
price 0
quality
For a price to constitute an equilibrium it must be that Q(p) > p, which is satisfied for p ∈ (0, 1 /2] (see figure). Note that for every p ∈ (0, 1 /2), the market excludes some sellers who would make positive profits under p = 1/2. Therefore, the only efficient equilibrium is the one with p = 1/2.
Question 6 (15 minutes)
Consider a monopolist facing two consumers, with utilities
u 1 (x, y) = 4v(x) + y u 2 (x, y) = 6v(x) + y
where v(x) = 1 −(1−x)
2
- Assume the monopolist, who has zero marginal cost of production cannot distinguish the two consumers.
(a) Find the optimal two part tariff (A, p). (you can assume the monopolist wants to sell to both consumers, and solve only for that case) (b) On a graph with total payments to the firm on the y axis and quantities bought on the x axis, graph the possible (quantity, payment) pairs induced by the two part tariff and indicate the points choosen by each consumer.
Solution
(a) We need to find an equilibrium where both types are served. The post-entry problem is
[x∗ 1 (p) + x∗ 2 (p)] p + 2A → max p,A subject to: ∀i ∈ { 1 , 2 } : ui(x∗ 1 (p), I − px∗ 1 (p) − A) > ui(0, I), (3)
(^1) For p > 1 .5 it becomes Q(p) = E[ω|v (ω) ≤ p] = E [ω|ω ∈ S(p)] = E [ω|ω ∈ [0, 2]] = E[ω] = 1 < p.
where (3) reflects the participation constraints. For consumer 1 we derive^2 the demand by equating MRS and the price ratio as follows (recall that px = p, py = 1)
p 1
4 v′(x) 1
= 4(1 − x) =⇒ x∗ 1 (p) = 1 −
p 4
Similarly, for consumer 2
x∗ 2 (p) = 1 −
p 6
For a given price p, the monopolist sets A to extract all the surplus from consumer 1 so that her participation constraint (3) binds as follows
0 = u 1 (x∗ 1 (p), I − px∗ 1 (p) − A(p)) − u 1 (0, I) = 4v(x∗ 1 (p)) + I − px∗ 1 (p) − A(p) − 4 v(0) − I,
=⇒ A(p) = 2
(p
4
− p
p 4
= 2 − p +
p^2 8 = (p − 4)^2 / 8.
Then the problem becomes
[x∗ 1 (p) + x∗ 2 (p)] p + 2A(p) → max p [ 1 −
p 4
p 6
]
p + 2 (p − 4)^2 / 8 → max p
It is a concave function, so FOC is necessary and sufficient [ 1 −
p 4
p 6
]
p −
p + (p − 4)/2 = 0
=⇒ p = 0
Thus, the optimal two part tariff is (2, 0), which can be interpreted as an all-you-can-eat buffet. (b) The total payments to the firm are given by
π(x) =
0 or 2, x = 0 2 , x > 0
Follow the figure below for an illustration
(^2) Formally, since the utility function is monotone only on [0, 1], the demands are x∗ 1 (p) = min { 1 − p 4 ,^1
} and x∗ 2 (p) = min
{ 1 − p 6 , 1
}
