Envelope Theorems
Let X Rn for some positive integer n, let T be an open real interval, and let f : X T R.
1
Constraint set depends on t
Let X = Rn , and let g : Rn Rm for some positive integer m. Consider the problem V (t) = max f (x, t) m
xR
(1)
subject
ECN 712 Fall 2009 Professor Schlee Problem Set X, Due 17 November
1. Price discrimination (of the third degree). A monopolist produces a single good under constant returns to scale. It sells the good in two markets, 1 and 2, and the demands are independen
ECN 712 Fall 2009 Problem Set XI, Due 24 November
1. MWG, Exercise 10.C.3. (The Herndahl index is often used to measure market concentration.) 2. Consider a Bertrand duopoly game with identical goods. Firm i s cost function given by ci (qi ) = kqi + F if
ECN 712 Fall 2009 Professor Schlee Problem Set XII, Due 1 December
1. Entry. Consider the following two-stage entry game with two rms. In the rst stage, the rms simultaneously decide whether or not to enter a market I for IN, O for OUT). If a rm chooses I
ECN 712 Fall 2009 Professor Schlee Problem Set XIIa, not to be turned in
1. Evaluate : In a nitely repeated prisoners dilemma, the strategy of defect in every period, no matter what the history is dominant for each player. 2. Prove that in a repeated pris
ECN 712: Microeconomic Analysis I
FALL 2009 E Schlee Oce: BAC 553; ph: 965-5745; email: [email protected] Hours: Mainly by appt; walk-in times are Mondays 3-350 and just after class. TA: Ran Shao, [email protected] Well begin a study of the core of microe
Practice Exam Two
Name
_
Chapter 6
1. The supply curves for the only two firms in a competitive industry are P = 2 Q1 and P = 2 + Q2,
where Q1 is the output of firm 1 and Q2 is the output of firm 2. Graph the supply curve for each
firm. What is the market
Practice Exam One
Name
_
Chapter 1
1. Residents of your city are charged a fixed weekly fee of $6 for garbage collection. They are
allowed to put out as many cans as they wish. The average household disposes of three cans of
garbage per week under this pl
Problem Set Nine Solutions
Chapter 13
1. Mountain Breeze supplies air filters to the retail market and hires workers to assemble the
components. An air filter sells for $26, and Mountain Breeze can buy the components for each
filter for $1. Sandra and Bob
Problem Set Eight Solutions
Chapter 11
3. Suppose the supply curve of boom box rentals on Golden State Park is given by P = 5 + 0.1 Q,
where P is the daily rent per unit in dollars and Q is the number of units rented in hundreds per
day. The demand curve
Problem Set Seven Solutions
Chapter 9
1. Two car manufacturers, Saab and Volvo, have fixed costs of $1 billion and constant marginal
costs of $10,000 per car. If Saab produces 50,000 cars per year and Volvo produces 200,000,
calculate the average fixed co
Problem Set Six Solutions
Chapter 8
3. John Jones owns and manages a caf whose monthly revenue is $5,000. Monthly expenses are:
Labor
$2,000
Food and drink
500
Electricity
100
Vehicle lease
150
Rent
500
Interest on loan for equipment
1,000
a. Calculate Jo
Problem Set Five Solutions
Chapter 7
1. Suppose the weekly demand and supply curves for used DVDs in Lincoln , Nebraska, are
shown in the diagram. Calculate and graph:
a. The weekly consumer surplus.
Answer: Consumer surplus is the triangular area between
Problem Set Four Solutions
Chapter 6
3. The Paducah Slugger Company makes baseball bats out of lumber and receives $10 for each
finished bat. Paducahs only factors of production are lathe operators and a small building with a
lathe. The number of bats per
Problem Set Three Solutions
Chapter 4
4. Is the demand for a particular brand of car, like a Chevrolet, likely to be more or less priceelastic then the demand for all cars? Explain.
Answer: The price elasticity of a good generally increases with the numbe
Problem Set Two Solutions
Chapter 3
2. How would each of the following affect the U.S. market supply for corn? Does the supply
curve shift left or right and why?
a. A new and improved crop rotation technique is discovered.
Answer: The supply curve would s
ECN 712 Fall 2009 Professor Schlee Problem Set IX, Due 11 November
For each problem assume that the monopolist posts a single price (for each good that it sells).
1. A monopolist produces a good according to a constant returns to scale technology with ave
ECN 712 Fall 2009 Professor Schlee Problem Set VIII, Due Tuesday, 3 November
1. An investor has initial wealth of w and divides his wealth between a safe asset with a rate of return of 0, and a risky asset with a rate of return of x + t. The number t is a
ECN 712 Fall 2009 Professor Schlee Problem Set VIIa, Additional Practice
For problems 1 and 2, consider the insurance problem in MWG, exercise 6.C.1, with the following change of notation: W is wealth in the absence of a loss, and L is the magnitude of a
Demand Properties: Summary
Let u represent a continuous preference relation on RL . In what follows assume that u is + continuous and locally nonsatiated. For (p, w) > 0, dene d(p, w) = cfw_x B (p, w) | u(x) u(y ) for all y B (p, w), where B (p, w) = cfw_
The Maximum over a Collection of Convex Functions is Convex
Consider V (t) = max f (x, t)
xC
where t lies in a convex subset T of Rm and C is a subset of Rn , n, m 1. Let a solution exist for every t T .
Theorem If f (x, ) is convex for every x C , then V
Monotone Comparative Statics
Let X and T be subsets of R and let f : X T R. We consider how the set of maximizers of f (x, t) on X vary with the parameter t.
Denition. f satises the strict single-crossing property (SSCP) in (x, t) if for every x , x in X
Welfare in the 2-good quasilinear model (or: concave programming and the invisible hand)
Let = (x1 , m1 ), ., (xI , mI ), q1 , ., qJ ) XI RJ = A denote an allocation, where X = R+ R is each consumers + consumption set. The input allocation (z1 , ., zJ ) i
Pareto Optimality
Let A be a set (of alternatives) and cfw_ Optimal if there is no a A with a Suppose that, for each i = 1, ., I ,
i I i i=1 i
a family of preference relations on A. a A is Pareto a for i = 1, ., I and a j a for some j cfw_1, ., I .
is re
The Arrow-Pratt Theorem
For t = 0, 1, let ut be a C 2 vN-M utility with ut > 0 on R+ . And let L be the set of cumulative distribution functions on R+ with F (0) = 0 and F (z ) = 1 for some number z < u (z ) .1 For t = 0, 1, denote the Arrow-Pratt absolut
Concave Programming
Let f : Rn R for some positive integer n, and let g : Rn Rm for some positive integer m be continuously dierentiable functions. Consider the problem
xRn +
max f (x)
(1)
subject to the constraint that g (x) 0. Let L(x, ) = f (x) g (x).
Choice Under (Objective) Uncertainty
Primitives: X = cfw_x1 , ., xn , a set of outcomes. L = cfw_p Rn | pi = 1 the set of probability distributions (lotteries) on X. + ei , the element of L that assigns probability 1 to xi X , a binary relation on L (with
Pratts Theorem on Risk Aversion and Portfolio Demand 1
For t = 0, 1 let ut be a C 1 vN-M utility with positive rst derivative everywhere. Consider the following problem: t arg max0 where xdF (x) > 0, ut (w + x)dF (x) .
x2 dF (x) > 0 (so F is nondegenerate
Nash Equilibrium: Existence
Let G = (N ; S1 , ., Sn ; u1 , ., un ) be a strategic form game with N = cfw_1, ., n. Theorem G has a Nash Equilibrium if, for every i N , (a) Si is a nonempty, compact, convex subset of Rm for some integer m; (b) ui is quasico
Extensive Form Games Revised item 7
An extensive form game is a collection = (N, A, H, T, , I, (Ui )iN ).
1. N , a nite set of players. 2. A, a (nite) set of actions. 3. H , a (nite) set of histories (sometimes called nodes), with (a) h0 H , where h0 is t