Economic Principles Solutions to Problem Set 5
Question 1 f (= (minf x1 ; x2 g) Since f (kx1 ; kx2 ) = (minf kx1 ; kx2 g) = k minf x1 ; x2 g = k f (x1 ; x2 ), we have that:
< 1 =) f (kx1 ; kx2 ) < kf (x1 ; x2 ) () decreasing returns to scale = 1 =) f (kx1
Economic Principles Problem Set 2
1. Bob consumes ice creams (x1 ) and hamburgers (x2 ) : His utility function is u (x1 ; x2 ) = (x1 ) 2 (x2 ) 2 : Bob' income is $ 100. The price of each hamburger is $ 2. The price of an ice cream s depends on the quantit
Game Theory Problem Set 6
1. (Mixed and behavioral strategies) Consider the following extensive-form game.
1 A 2 L M 1 l r l r R B
(a) Let 1 denote player 1' mixed strategy in which she plays (B; r) with probability 0:4; s (B; l) with probability 0:1; (A;
Economic Principles Solutions to Problem Set 6
Question 1 We know that in the long run equilibrium, price will be equal to the minimum of the AC of the production for a representative .rm, and it is equal to M C as well. AC(q) = a c(q) = + b + cq q q a +c
Economic Principles, Spring 2013 Midterm Solutions
You have two hours to complete this exam. Please answer the following three questions. Be sure to allocate your time in proportion to the points. Always justify your answers by providing a formal proof or
Economic Principles Problem Set 11
1. A monopolist faces a market demand curve given by q (p) = 70 p:
(a) If the monopolist can produce at constant average and marginal costs of AC = M C = 6; what output level will the monopolist choose to maximize pro.ts
Economic Principles Problem Set 8
1. Consider the following Robinson Crusoe economy. Robinson the consumer is endowed with zero units of coconuts, x; and 24 hours of time, h; so that e = (0; 24) : His 1 2 preferences are de.ned over R2 and represented by
Economic Principles Problem Set 10
1. Consider an economy with two consumers. The utility function of consumer i = 1; 2 is equal to u(xi ; li ; xj ) = log(xi + xj ) + li ; where li denotes i' (leisure) time and xi and xj denote the quantity of the public
Economic Principles Problem Set 3
1. (JR 2.8). The consumer buys bundle xi at price pi ; i = 0; 1: Separately for parts (a) to (d), state whether these indicated choices satisfy WARP: (a) p0 = (1; 3); x0 = (4; 2) ; p1 = (3; 5) ; x1 = (3; 1) : (b) p0 = (1;
Economic Principles Problem Set 9
1. Consider a two-consumer, two-good exchange economy. Utility functions and endowments are: uA (x1 ; x2 ) = (x1 x2 )2 and eA = (15; 6) ; B u (x1 ; x2 ) = ln (x1 ) + 2 ln (x2 ) and eB = (3; 8) : (a) Derive the set of Pare
Economic Principles Problem Set 1
1. Let < be represented by u : Rn ! R. Prove that u (x) is strictly quasiconcave if and + only if < is strictly convex. p 2. Consider the following utility functions: u (x1 ; x2 ) = x1 x2 ; and v (x1 ; x2 ) = ln (x1 ) + l
Economic Principles Problem Set 5
1. Suppose a production function has the form y = (min f x1 ; x2 g) ; where > 0; > 0; > 0: Carefully sketch the isoquant map for this technology. Discuss returns to scale as a function of : 2. To produce output y a .rm ne
Economic Principles Solutions to Problem Set 4
Question 1 The investor' problem is s max
n X i=1 i u(w
2[0;1]
(1
) r+w
xi )
Let wi = w (1 ) r+w xi . For simplicity, let' assume that we have an interior solution, then F.O.C. implies that s
n X i=1 iu 0
(wi
Economic Principles Solutions to Problem Set 2
Question 1. Given the conditions, Bob' budget constraint is equal to: s (x1 i.e., 2x1 + 2x2 = 100 if x1 10 x1 + 2x2 = 90 if x1 > 10 Graphically, the budget constraint looks as follows:
50 40 30 20 10 0 0 10 2
Economic Principles Solutions to Problem Set 1
Question 1. Let < be represented by u : Rn ! R. Prove that u (x) is strictly quasiconcave if and only if + < is strictly convex. "If"part: ( "strict convexity of <")" strict quasiconcavity of u" ). n For x; y
Economic Principles Solutions to Problem Set 10
Question 1 (a) In this economy there are two goods, labor h and the public good x. By Walras'law we can set the price of labor w = 1:By constant return to scale, the price of the public good in equilibrium m
Economic Principles Solutions to Problem Set 11
Question 1 (a) The maximization problem of the monopolist is: max
q
(70
q)q
6q
The F.O.C. is 70 2q = 6 =) q = 32 Also, the monopolistic price and pro.ts are p = 70 q = 38 = q (p AC) = 32
(38 5.
6) = 1024
1 1
Economic Principles Solutions to Problem Set 8
Question 1 Let p denote the price of the coconuts and w denote the price of Robinson' time. s Normalize p = 1. p Given the production function, the producer will choose l to maximize p l wl. Hence, the produc
Economic Principles Solutions to Problem Set 9
Question 1 (a) The Pareto-e cient allocations solve to the following problem (where we took the square root of uA and take the logarithm, both monotonic transformations): max ln xA + ln xA 1 2 s.t. ln xB + ln
Economic Principles Solutions to Problem Set 3
Question 1 The WARP requires that, if a bundle x0 is chosen when another bundle x1 is available, then when this new bundle x1 is itself chosen, x0 must not be available. Assuming budget balancedness, wi = pi
Economic Principles, Spring 2013 Final Exam Solutions You have two hours to complete this exam. Please answer the following four questions. Be sure to allocate your time in proportion to the points. Always justify your answers by providing a formal proof
Economic Principles, Spring 2013 Midterm Exam Dino Gerardi April 8th You have two hours to complete this exam. Please answer the following three questions. Be sure to allocate your time in proportion to the points. Always justify your answers by providing
Economic Principles Problem Set 4
1. Consider an investor who has initial wealth w and has to decide how to invest it. There is a riskless asset with rate of return r: The risky asset can have any of the rates of return xi with probability i ; i = 1; :; n
Game Theory Solutions to Problem Set 2
1
(a)
Find all NE
L 3; 4 0; 3 R 2; 6 5; 1
U D
First, note that pure strategy D is strictly dominated by U )we can eliminate D. Then pure strategy L is strictly dominated by R )w e can eliminate L:Hence, the unique NE
Game Theory Problem Set 9
1. Suppose there are n .rms in a Cournot oligopoly. Inverse demand is given by: P (q1 ; : : : ; qn ) = a (q1 + : : : + qn ) :
For simplicity, assume that there are no production costs. Consider the in.nitely repeated game based o
Game Theory Problem Set 10
1. In the following game, nature .rst chooses one of two types of player 1 (in the .gure, the two types are denoted t1 and t2 ). Each type is chosen with equal probability. Player 1 observes her type and decides whether to choos
Game Theory Problem Set 7
1. (The centipede game) In class we showed that in the unique subgame perfect equilibrium of the centipede game (see the .gure below) both players exit in every information set.
1
C E
2
C E
1
C E
2
C E
1
C E
2
C
100,99
E 1,0
0,2
Game Theory, Spring 2013 Midterm Exam Dino Gerardi April 9th You have two hours to complete this exam. Please answer the following three questions. Be sure to allocate your time in proportion to the points. Always justify your answers by providing a forma
Game Theory Problem Set 2
1. Find all Nash equilibria of the following normal-form games. a) L U 3; 4 D 0; 3 b) L R U 4; 5 3; 1 D 4; 0 0; 6 c) L C R U 6; 6 1; 2 3; 3 M 2; 1 4; 7 4; 3 D 3; 4 2; 5 3; 9 2. (Divide the dollar) Players 1 and 2 are bargaining o