Advanced Microeconomic Theory

Transcription

1 Advanced Microeconomic Theory Lecture Notes Sérgio O. Parreiras Fall, 2016

2 Outline Mathematical Toolbox Decision Theory Partial Equilibrium Search Intertemporal Consumption General Equilibrium Financial Markets General Equilibrium with Production Liquidity

3 Mathematical Toolbox How to use the toolbox: 1 Casually read it once so you can classify your understanding of the topics in three categories: mastered, familiar but not mastered, and never seen before. 2 Read it again but, skip the mastered topics. 3 In your second reading, make sure to have pen and paper at hand and, Mathematica open and running. 4 Work the learning-by-doing exercises using pen and paper and, verify using Mathematica if your answers are correct. 5 When you have problems with Mathematica as you will for sure refer to the crash tutorial and Mathematica s help documentation. As last resort, me your.nb file.

4 Mathematical Toolbox Matrices A matrix is just a convenient way of displaying information. A n by m matrix A is composed of n m entires. The entry A ij is displayed in the ith row and jth column. Example of a 3 by 3 matrix, A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33.

5 Mathematical Toolbox Matrices A matrix is just a convenient way of displaying information. A n by m matrix A is composed of n m entires. The entry A ij is displayed in the ith row and jth column. Example of a 3 by 3 matrix, A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33.

6 Mathematical Toolbox: Matrix Multiplication a 11 a a 1p a 21 a a 2p a n1 a n2... a np A : n rows p columns b 11 b b 1q b 21 b b 2q b p1 b p2... b pq B : p rows q columns c 11 c c 1q c 21 c c 2q c n1 c n2... c nq a 21 b 12 a 22 b 22 a 2p b p C = A B : n rows q columns

7 Matrix Multiplication Examples A 1 n = (p 1, p 2,..., p n ) and B n 1 = u(x 1 ) u(x 2 ). u(x n ) C 1 1 = A B = p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n )

8 Matrix Multiplication Examples A 1 n = (p 1, p 2,..., p n ) and B n 1 = u(x 1 ) u(x 2 ). u(x n ) C 1 1 = A B = p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n )

9 Matrix Multiplication Mathematica In Mathematica to enter the matrices, we type: A = ( ) A :={{1, 0, 3}, {5, 4, 7}} and B = B :={{1, 2}, {3, 4}, {7, 0}} shift+enter To multiply the matrices, type: A.B shift+enter. ( A B = ).

10 Mathematical Toolbox Partial Derivatives Often we wish to evaluate the marginal impact of ONE given variable on some function of several variables. M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 We refer to M y as: 1 The marginal change f with respect to y. 2 The partial derivative of f wrt y. 3 The slope of f wrt y.

11 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

12 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

13 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

14 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

15 Partial Derivatives Examples ( ) 1 x + y = x 2 x ( x 2 + 3xy + y 2) = 2x + 3y x x (y log (x)) = y x

16 Partial Derivatives Learning-by-doing exercises Compute the marginal utilities MU X and MU Y for the following utility functions: 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y)

17 Mathemematical Tool Box, The Chain Rule: If h(x) = f(g(x)) then h (x) = f (g(x)) g (x)

18 Chain Rule Learning-by-doing exercises 1 For each of the composite functions below tell us, What are the corresponding f and g and compute h. a) h(x) = 2x b) h(x) = exp( ρ x) c) h(x) = (4 + x σ ) 1 σ 2 Use the Chain Rule to obtain the marginal utilities M X and M Y of the utility function, u(x, y) = 2 3 exp( x) 1 3 exp( y). 3 If k(x) = f(g(h(x))) is a composition of three functions, apply the chain rule twice to compute k (x). 4 Consider f(x, y) and g(x) compute the total derivative of f(x, g(x)) with respect to x using the Chain Rule and partial derivatives.

19 Mathematical Toolbox Taylor s Approximation Consider a function of one variable defined on the real line, f : R R. If f is differentiable, we write the first order Taylor approximation: f(x + h) f(x) f (x) h The approximation works well only if h is small. For a function of two variables and h = (h 1, h 2 ) we have a similar expression: f(x + h 1, y + h 2 ) f(x, y) x f(x, y) h 1 + y f(x, y) h 2

20 Mathematical Toolbox Taylor s Approximation Consider a function of one variable defined on the real line, f : R R. If f is differentiable, we write the first order Taylor approximation: f(x + h) f(x) f (x) h The approximation works well only if h is small. For a function of two variables and h = (h 1, h 2 ) we have a similar expression: f(x + h 1, y + h 2 ) f(x, y) x f(x, y) h 1 + y f(x, y) h 2

21 Mathematical Toolbox Taylor s Approximation Consider a function of one variable defined on the real line, f : R R. If f is differentiable, we write the first order Taylor approximation: f(x + h) f(x) f (x) h The approximation works well only if h is small. For a function of two variables and h = (h 1, h 2 ) we have a similar expression: f(x + h 1, y + h 2 ) f(x, y) x f(x, y) h 1 + y f(x, y) h 2

22 Mathematical Toolbox, Marginal Utility & Taylor s Approximation: U = U(x + x, y + y ) U(x, y) MU x x + MU y y

23 Taylor s Approximation Learning-by-doing exercises Using Taylor s approximation, for each of the utility functions below, compute the change in utility when the consumer moves from consuming the basket (100, 100) to consuming the basket (105, 99). 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y) 4 u(x, y) = 2 3 exp( x) 1 3 exp( y) Hint: What is the value of X? What is the value of Y? What is the value of MU X and MU Y when the basket (100, 100) is consumed?

24 Mathematical Toolbox Implicit Functions Let f(x, y) be a real-valued function of two variables and let g(x) be a real-valued function of one-variable with the following property: If we set y = g(x), f remains constant as we change x. that is, f(x, g(x)) = c for all x, where c is a constant. We refer to the function g as an implicit function since g is implicitly defined by the equation f(x, g(x)) = c.

25 Mathematical Toolbox Implicit Functions Let f(x, y) be a real-valued function of two variables and let g(x) be a real-valued function of one-variable with the following property: If we set y = g(x), f remains constant as we change x. that is, f(x, g(x)) = c for all x, where c is a constant. We refer to the function g as an implicit function since g is implicitly defined by the equation f(x, g(x)) = c.

26 Mathematical Toolbox Implicit Functions Let f(x, y) be a real-valued function of two variables and let g(x) be a real-valued function of one-variable with the following property: If we set y = g(x), f remains constant as we change x. that is, f(x, g(x)) = c for all x, where c is a constant. We refer to the function g as an implicit function since g is implicitly defined by the equation f(x, g(x)) = c.

27 Implicit Functions Learning-by-doing exercises For the utility curves below, find the equation of the indifference curve that gives utility c: 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y) 4 u(x, y) = 2 3 exp( x) 1 3 exp( y) Hint: set u(x, y) = c and solve for y, this is your g function...

28 Mathematical Toolbox Implicit Function Theorem If f(x, g(x)) = c for all x, where c is a constant. f(x, g(x)) Then, g (x) = x f(x, g(x)). y

29 Mathematical Tools The Implicit Function Theorem Proof: Taking the total derivative of f with respect to x, d dx f(x, g(x)) = x f(x, g(x)) x x + y f(x, g(x)) x g(x) = x f(x, g(x)) + y f(x, g(x)) g (x) Since, d f(x, g(x)) dx f(x, g(x)) = 0 g (x) = x. y f(x, g(x)) As f is constant along g(x), we also call g an iso-curve of f. Indifference curves and iso-cost curves are examples of iso-curves that you should be familiar.

30 The Implicit Function Theorem Learning-by-doing exercises For the utility curves below: a) find the marginal rate of substitution; b) in the MRS, replace y by the implicit function g you found in the previous learning-by-doing exercise and simplify the expression; c) compute g (x) for the implicit functions of the previous exercise; d) compare the results you found in items (b) and (c). 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y) 4 u(x, y) = 2 3 exp( x) 1 3 exp( y)

31 Mathematical Toolbox Interior Solutions Maximizing a function of one variable defined on the real line, f : R R. Maximization Problem max f(x) x R (P) First order condition f (x) = 0 (FOC) Second order condition f (x) 0 (SOC) Any point x satisfying FOC and SOC is a candidate for an interior solution.

32 Mathematical Toolbox Interior and Corner Solutions Maximizing a function of one variable defined on an interval, f : [a, b] R. As before, Maximization Problem max f(x) b x a (P) First order condition f (x) = 0 (FOC) Second order condition f (x) 0 (SOC) Any point x satisfying FOC and SOC is a candidate for an interior solution and now, x = a is a candidate for a corner solution if f (a) 0. x = b is a candidate for a corner solution if f (b) 0.

33 Mathematical Toolbox Concavity and Convexity Consider any function f : R k R. Definition: f is concave if and only if, for all α [0, 1], and any two points x, y R k, we have f (α x + (1 α) y) α f(x) + (1 α) f(y). Another definition: We say that f is convex if f is concave.

34 Mathematical ToolBox Global Maxima Proposition. Assume f is concave and also assume that x satisfy the FOC then x is a solution to the maximization problem (i.e. x is a global maximum).

35 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

36 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

37 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

38 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

39 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

40 States of the World Example 1 Decision Problem In a sunny morning, choose whether to carry an umbrella, a parasol or nothing to work. Possible States of The World Ω = {ω 1, ω 3, ω 3 }. ω 1 The afternoon is also sunny. ω 2 The afternoon is cloudy but it does not rain. ω 3 The afternoon is rainy.

41 States of the World Example 1 Decision Problem Buy, sell or take no action regarding Alibaba shares in the NYSE. Possible States of The World Ω = {ω 1, ω 3, ω 3, ω 4 }. ω 1 % BABA > 20%. ω 2 20% > % BABA > 5%. ω 3 5%> % BABA > 0%. ω 4 0%> % BABA>-10%. where above, % BABA is the rate of change of Alibaba s stock price, p 1 p 0 p 0, where p 1 is the future (six months ahead) price and p 0 is the current price.

42 Information Partitions Individual i s knowledge about the states of the world is represented by a partition P of the set Ω. Definition A partition P of Ω is a collection of subsets of Ω such that: 1. If A P and B P and A B then A B =. 2. A P A = P. If two states are in the same element of our knowledge partition, it means we cannot distinguish between these states. But if two states are if different elements of the partition, we can distinguish them.

43 Information Partitions Individual i s knowledge about the states of the world is represented by a partition P of the set Ω. Definition A partition P of Ω is a collection of subsets of Ω such that: 1. If A P and B P and A B then A B =. 2. A P A = P. If two states are in the same element of our knowledge partition, it means we cannot distinguish between these states. But if two states are if different elements of the partition, we can distinguish them.

44 Information partition Weather in the afternoon example Possible States of The World Ω = {ω 1, ω 2, ω 3 }. ω 1 The afternoon is also sunny. ω 2 The afternoon is cloudy but it does not rain. ω 3 The afternoon is rainy. a) P a = {{ω 1 }, {ω 2, ω 3 }}, we know if the afternoon is sunny or not, but if it is not sunny, we cannot tell if it is rainy or cloudy. b) P b = {{ω 1, ω 2, ω 3 }}, we do not know anything about the weather in the afternoon. c) P c = {{ω 1, ω 2 }, {ω 3 }}, we know if the afternoon is rainy or not, but if it is not rainy, we cannot tell if it is sunny or cloudy.

45 Information Partitions Individual i s knowledge about the states of the world is represented by a partition P of the set Ω. Definition A partition P of Ω is a collection of subsets of Ω such that: 1. If A P and B P and A B then A B =. 2. A P A = P. If two states are in the same element of our knowledge partition, it means we cannot distinguish between these states. But if two states are if different elements of the partition, we can distinguish them.

46 Information Partitions Individual i s knowledge about the states of the world is represented by a partition P of the set Ω. Definition A partition P of Ω is a collection of subsets of Ω such that: 1. If A P and B P and A B then A B =. 2. A P A = P. If two states are in the same element of our knowledge partition, it means we cannot distinguish between these states. But if two states are if different elements of the partition, we can distinguish them.

47 Information partition Weather in the afternoon example Possible States of The World Ω = {ω 1, ω 2, ω 3 }. ω 1 The afternoon is also sunny. ω 2 The afternoon is cloudy but it does not rain. ω 3 The afternoon is rainy. a) P a = {{ω 1 }, {ω 2, ω 3 }}, we know if the afternoon is sunny or not, but if it is not sunny, we cannot tell if it is rainy or cloudy. b) P b = {{ω 1, ω 2, ω 3 }}, we do not know anything about the weather in the afternoon. c) P c = {{ω 1, ω 2 }, {ω 3 }}, we know if the afternoon is rainy or not, but if it is not rainy, we cannot tell if it is sunny or cloudy.

48 Information Partitions Cheryl s birthday Albert and Bernard just met Cheryl. When s your birthday? Albert asked Cheryl. Cheryl thought a second and said, I m not going to tell you, but I ll give you some clues. She wrote down a list of 10 dates: May 15, May 16, May 19 June 17, June 18 July 14, July 16 August 14, August 15, August 17 My birthday is one of these, she said. Then Cheryl whispered in Albert s ear the month and only the month of her birthday. To Bernard, she whispered the day, and only the day. Can you figure it out now? she asked Albert. Albert: I don t know when your birthday is, but I know Bernard doesn t know, either.

49 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17

50 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 whispered in Albert s ear the month

51 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 To Bernard, she whispered the day

52 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 whispered in Albert s ear the month To Bernard, she whispered the day

53 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 whispered in Albert s ear the month To Bernard, she whispered the day Albert: I don t know when your birthday is, but I know Bernard doesn t know, either.

54 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 whispered in Albert s ear the month To Bernard, she whispered the day Albert: I don t know when your birthday is, but I know Bernard doesn t know, either. Bernard: I didn t know originally, but now I do.

55 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 whispered in Albert s ear the month To Bernard, she whispered the day Albert: I don t know when your birthday is, but I know Bernard doesn t know, either. Bernard: I didn t know originally, but now I do. Albert: Well, now I know, too!

56 Lotteries Assume we have n states of the world. Definition: A lottery is a list of prizes and probabilities, l = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )), where x k R m is the prize the lottery gives when state k occurs, and p k 0 is the probability that state k occurs.

57 Lotteries Assume we have n states of the world. Definition: A lottery is a list of prizes and probabilities, l = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )), where x k R m is the prize the lottery gives when state k occurs, and p k 0 is the probability that state k occurs.

58 Lotteries In practice, it is very useful to depict lotteries as a graph. x 1 x 2... x n p 1 p 2 p n l

59 Lotteries In practice, it is very useful to depict lotteries as a graph. x 1 x 2... x n p 1 p 2 p n l

60 The Certain Lottery The lottery that gives prize x with probability one (with certainty) is denoted by: δ x = ((x), (1)). 1 x δ x

61 Expectation and Variance The expected value of a lottery l 1 = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) is: E[l 1 ] = p 1 x 1 + p 2 x p n x n = n p i x i ; i=1 and the variance of a lottery l is: Var[l 1 ] =p 1 (x 1 E[l 1 ]) 2 + p 2 (x 2 E[l 1 ]) p n (x n E[l 1 ]) 2 = n = p i (x i E[l 1 ]) 2. i=1

62 Composition of Lotteries Given two lotteries, l 1 = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) and l 2 = (y 1, y 2,..., y m ), (q 1, q 2,..., q n )) and a number α in the interval (0, 1), we can create a compound lottery l that plays l 1 with probability α and l 2 with probability 1 α. l = αl 1 (1 α)l 2 = = ((x 1, x 2, y 1, y 2 ), (αp, α(1 p), (1 α)q, (1 α)(1 q)). x 1 l x 2 y 1

63 Composition of Lotteries The compound lottery l plays l 1 with probability α and l 2 with probability l 2 : l = αl 1 (1 α)l 2 = = ((x 1, x 2, y 1, y 2 ), (αp, α(1 p), (1 α)q, (1 α)(1 q)). p x 1 l α 1 α l 1 l 2 1 p q x 2 y 1 1 q y 2

64 Composition of Lotteries The compound lottery l plays l 1 with probability α and l 2 with probability l 2 : l = αl 1 (1 α)l 2 = = ((x 1, x 2, y 1, y 2 ), (αp, α(1 p), (1 α)q, (1 α)(1 q)). x 1 l α p α (1 p) (1 α) q (1 α) (1 q) x 2 y 1 y 2

65 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a DM l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

66 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a DM l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

67 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a DM l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

68 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a DM l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

69 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a DM l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

70 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a DM l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

71 Preferences Over Lotteries A preference of the DM, DM, over the set of lotteries is just the DM s ranking of lotteries. We wish (for convenience) a numerical score that reflects the DM s ranking.

72 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

73 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

74 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

75 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

76 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

77 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3. If satisfy all of of the above, there exists u : R R such that ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) (y 1, y 2,..., y m ), (q 1, q 2,..., q n )) if and only if n m u(x k ) p k > u(y k ) q k. k=1 k=1

78 Expected Utility We write: U(l 1 ) = u(x 1 ) p u(x n ) p n and refer to U as the expected utility and to u as the: 1 utility for money 2 Bernoulli utility 3 von Neumann-Morgenstern utility (vnm)

79 Expected Utility We write: U(l 1 ) = u(x 1 ) p u(x n ) p n, and refer to U as the expected utility and to u as the: 1 utility for money 2 Bernoulli utility 3 von Neumann-Morgenstern utility (vnm)

80 Expected Utility We write: U(l 1 ) = u(x 1 ) p u(x n ) p n, and refer to U as the expected utility and to u as the: 1 utility for money 2 Bernoulli utility 3 von Neumann-Morgenstern utility (vnm)

81 How to use expected utility 1 List the states of the world: ω 1, ω 2,..., ω n. 2 List states probabilities: p 1, p 2,...,p n. 3 For each possible action: a) List its possible outcomes (prizes): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4 Choose action that delivers the highest expected utility.

82 How to use expected utility 1 List the states of the world: ω 1, ω 2,..., ω n. 2 List states probabilities: p 1, p 2,...,p n. 3 For each possible action: a) List its possible outcomes (prizes): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4 Choose action that delivers the highest expected utility.

83 How to use expected utility 1 List the states of the world: ω 1, ω 2,..., ω n. 2 List states probabilities: p 1, p 2,...,p n. 3 For each possible action: a) List its possible outcomes (prizes): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4 Choose action that delivers the highest expected utility.

84 How to use expected utility 1 List the states of the world: ω 1, ω 2,..., ω n. 2 List states probabilities: p 1, p 2,...,p n. 3 For each possible action: a) List its possible outcomes (prizes): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4 Choose action that delivers the highest expected utility.

85 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 20000x x 2. She has to choose between insurance or no insurance. 1 What are the states? 2 What are the states probabilities? 3 If she does not insure, What are her prizes in each state? 4 If she does not insure, What is her utility in each state? 5 What is the expected utility of not insuring? 6 What is the expected utility of insuring?

86 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 20000x x 2. She has to choose between insurance or no insurance. 1 What are the states? 2 What are the states probabilities? 3 If she does not insure, What are her prizes in each state? 4 If she does not insure, What is her utility in each state? 5 What is the expected utility of not insuring? 6 What is the expected utility of insuring?

87 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 20000x x 2. She has to choose between insurance or no insurance. 1 What are the states? 2 What are the states probabilities? 3 If she does not insure, What are her prizes in each state? 4 If she does not insure, What is her utility in each state? 5 What is the expected utility of not insuring? 6 What is the expected utility of insuring?

88 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 20000x x 2. She has to choose between insurance or no insurance. 1 What are the states? 2 What are the states probabilities? 3 If she does not insure, What are her prizes in each state? 4 If she does not insure, What is her utility in each state? 5 What is the expected utility of not insuring? 6 What is the expected utility of insuring?

89 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 20000x x 2. She has to choose between insurance or no insurance. 1 What are the states? 2 What are the states probabilities? 3 If she does not insure, What are her prizes in each state? 4 If she does not insure, What is her utility in each state? 5 What is the expected utility of not insuring? 6 What is the expected utility of insuring?

90 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 20000x x 2. She has to choose between insurance or no insurance. 1 What are the states? 2 What are the states probabilities? 3 If she does not insure, What are her prizes in each state? 4 If she does not insure, What is her utility in each state? 5 What is the expected utility of not insuring? 6 What is the expected utility of insuring?

91 ECON101 Review Learning-by-doing exercise Definition: Rate of Return If an asset s price today is p 0 and its price tomorrow is p 1 then the asset s rate of return is r = p 1 p 0. With x dollars, how many units of the asset can you buy today? What is the revenue you obtain by selling these units tomorrow?

92 Expected Utility Learning-by-doing Exercises For each of the following settings, describe: the states, the probabilities, the prizes and, compute the expected utility. Assume the utility for money is u(x) = x and the initial wealth is w. 1 A monetary loss L, where w > L > 0 happens with probability 0 < p < 1. The insurance premium and the deductible are P > 0 and D > 0. The individual buys insurance. 2 A fraction 0 < α < 1 of the initial wealth is invested in bonds with a rate of return r 1, the remaining is invested in a risky asset that has return R H with probability p and R L with probability 1 p, where R H > r > R L 0.

93 Extracting u from

94 Extracting u from

95 Extracting u from

96 Extracting u from

97 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 102), ( 1 2, 1 2 ) and δ 101 = ((101), (1)) We have U(l 1 ) = u(100) u(102) 1 2 and U(δ 101 ) = u(101) 1.

98 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 102), ( 1 2, 1 2 ) and δ 101 = ((101), (1)) We have U(l 1 ) = u(100) u(102) 1 2 and U(δ 101 ) = u(101) 1.

99 Risk Aversion U(l 1 ) U(δ 101 ) = [u(102) u(101) (u(101) u(100))] 1 2 U(l 1 ) = u(100) u(102) 1 2 and U(δ 101 ) = u(101).

100 Risk Aversion U(l 1 ) U(δ 101 ) = u(102) u(101) u(101) u(100) 1 }{{}}{{} 2 Mu(101) Mu(100) U(l 1 ) = u(100) u(102) 1 2 and U(δ 101 ) = u(101).

101 Risk Aversion U(l 1 ) U(δ 101 ) = u(102) u(101) u(101) u(100) 1 }{{}}{{} 2 Mu(101) Mu(100) U(l 1 ) > U(δ 101 ) Mu(101) > Mu(100). U(l 1 ) = u(100) u(102) 1 2 and U(δ 101 ) = u(101).

102 Risk Aversion U(l 1 ) U(δ 101 ) = u(102) u(101) u(101) u(100) 1 }{{}}{{} 2 Mu(101) Mu(100) U(l 1 ) > U(δ 101 ) Mu(101) > Mu(100). U(l 1 ) = U(δ 101 ) Mu(101) = Mu(100). U(l 1 ) < U(δ 101 ) Mu(101) < Mu(100). U(l 1 ) = u(100) u(102) 1 2 and U(δ 101 ) = u(101).

103 Risk Aversion u u(102) u(100) x U(l 1 ) E[l 1 ] U(δ 101 ) = u(101)

104 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U(l 1 ) E[l 1 ] U(δ 101 ) = u(101)

105 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U(l 1 ) E[l 1 ] U(δ 101 ) = u(101)

106 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U(l 1 ) E[l 1 ] U(δ 101 ) = u(101)

107 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U(l 1 ) E[l 1 ] U(δ 101 ) = u(101)

108 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U(X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U(X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U(X) = u(e[x]) for all X

109 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U(X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U(X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U(X) = u(e[x]) for all X

110 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U(X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U(X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U(X) = u(e[x]) for all X

111 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U(X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U(X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U(X) = u(e[x]) for all X

112 Measuring the Degree of Risk-Aversion The Arrow-Pratt or Absolute Measure of Risk Aversion Definition The Arrow-Pratt absolute measure of risk-aversion of an agent with VN-M utility u at wealth level w is: ρ u (w) = u (w) u (w). If for two individual with VN-M utilities u and ũ we have that ρ u (w) > ρũ(w) for all wealth levels w then we say that the agent with utility u is more risk-averse than the agent with utility ũ.

113 Measuring the Degree of Risk-Aversion The Arrow-Pratt or Absolute Measure of Risk Aversion Definition The Arrow-Pratt absolute measure of risk-aversion of an agent with VN-M utility u at wealth level w is: ρ u (w) = u (w) u (w). If for two individual with VN-M utilities u and ũ we have that ρ u (w) > ρũ(w) for all wealth levels w then we say that the agent with utility u is more risk-averse than the agent with utility ũ.

114 Measuring the Degree of Risk-Aversion The Relative Measure of Risk-Aversion We are not covering this material, please skip this slide... Definition The relative absolute measure of risk-aversion of an agent with VN-M utility u at wealth level w is: r u (w) = u (w) w u. (w)

115 The Coal Industry A motivating example Click to read the July of 2015, NY Times article, Coal Miners Struggle to Survive in an Industry Battered by Layoffs and Bankruptcy. Since then, the trend described in the article has continued, Arch Coal Files for Chapter 11 Bankruptcy Protection. China has also experienced a similar trend (although for slightly different reasons), Mass Layoffs in China s Coal Country Threaten Unrest.

116 Long-run (partial) equilibrium Consider an industry (e.g. energy) where the cost function is c(q) = q and demand function Q = 100 2p. 1 What is the long-run price? 2 What is the long-run aggregate quantity? 3 How many firms are in the market? The marginal and average costs are MC(q) = 2q and AC(q) = q + 100/q. So zero-profit, p = AC, and profit maximization, p = MC, imply AC = MC so q = 10 and p = 20. Aggregate demand is then Q = = 60, so there are N = Q/q = 60/10 = 6 firms.

117 Long-run (partial) equilibrium Consider an industry (e.g. energy) where the cost function is c(q) = q and demand function Q = 100 2p. 1 What is the long-run price? 2 What is the long-run aggregate quantity? 3 How many firms are in the market? The marginal and average costs are MC(q) = 2q and AC(q) = q + 100/q. So zero-profit, p = AC, and profit maximization, p = MC, imply AC = MC so q = 10 and p = 20. Aggregate demand is then Q = = 60, so there are N = Q/q = 60/10 = 6 firms.

118 Long-run (partial) equilibrium Consider an industry (e.g. energy) where the cost function is c(q) = q and demand function Q = 100 2p. 1 What is the long-run price? 2 What is the long-run aggregate quantity? 3 How many firms are in the market? The marginal and average costs are MC(q) = 2q and AC(q) = q + 100/q. So zero-profit, p = AC, and profit maximization, p = MC, imply AC = MC so q = 10 and p = 20. Aggregate demand is then Q = = 60, so there are N = Q/q = 60/10 = 6 firms.

119 Long-run (partial) equilibrium Suppose now there is technological progress (e.g. fracking) that allows some firms to produce (energy) cheaper (e.g. natural gas instead of coal). The new cost function is ĉ(q) = q 2 / and so MC = q. Assume there is only one firm using the new technology and 6 using the old one. 1 What is the equilibrium price? 2 What are the firms profits? If the price is p, an old firm supplies q = p/2 and a new firm supplies q = p. Total supply is then Q = 6p/2 + p = 4p. Equating it with total demand, Q = 100 2p, give us p = 100/6 < 20. Profits are π = 100/6 (50/6) (50/2) and π = (100/6) 2 (100/6) 2 /2 80.

120 Long-run (partial) equilibrium Suppose now there is technological progress (e.g. fracking) that allows some firms to produce (energy) cheaper (e.g. natural gas instead of coal). The new cost function is ĉ(q) = q 2 / and so MC = q. Assume there is only one firm using the new technology and 6 using the old one. 1 What is the equilibrium price? 2 What are the firms profits? If the price is p, an old firm supplies q = p/2 and a new firm supplies q = p. Total supply is then Q = 6p/2 + p = 4p. Equating it with total demand, Q = 100 2p, give us p = 100/6 < 20. Profits are π = 100/6 (50/6) (50/2) and π = (100/6) 2 (100/6) 2 /2 80.

121 Long-run (partial) equilibrium Suppose now there is technological progress (e.g. fracking) that allows some firms to produce (energy) cheaper (e.g. natural gas instead of coal). The new cost function is ĉ(q) = q 2 / and so MC = q. Assume there is only one firm using the new technology and 6 using the old one. 1 What is the equilibrium price? 2 What are the firms profits? If the price is p, an old firm supplies q = p/2 and a new firm supplies q = p. Total supply is then Q = 6p/2 + p = 4p. Equating it with total demand, Q = 100 2p, give us p = 100/6 < 20. Profits are π = 100/6 (50/6) (50/2) and π = (100/6) 2 (100/6) 2 /2 80.

122 Long-run (partial) equilibrium Suppose now there is technological progress (e.g. fracking) that allows some firms to produce (energy) cheaper (e.g. natural gas instead of coal). The new cost function is ĉ(q) = q 2 / and so MC = q. Assume there is only one firm using the new technology and 6 using the old one. 1 What is the equilibrium price? 2 What are the firms profits? If the price is p, an old firm supplies q = p/2 and a new firm supplies q = p. Total supply is then Q = 6p/2 + p = 4p. Equating it with total demand, Q = 100 2p, give us p = 100/6 < 20. Profits are π = 100/6 (50/6) (50/2) and π = (100/6) 2 (100/6) 2 /2 80.

123 Long-run (partial) equilibrium Suppose now there is technological progress (e.g. fracking) that allows some firms to produce (energy) cheaper (e.g. natural gas instead of coal). The new cost function is ĉ(q) = q 2 / and so MC = q. Assume there is only one firm using the new technology and 6 using the old one. 1 What is the equilibrium price? 2 What are the firms profits? If the price is p, an old firm supplies q = p/2 and a new firm supplies q = p. Total supply is then Q = 6p/2 + p = 4p. Equating it with total demand, Q = 100 2p, give us p = 100/6 < 20. Profits are π = 100/6 (50/6) (50/2) and π = (100/6) 2 (100/6) 2 /2 80.

124 Long-run (partial) equilibrium Suppose now there is technological progress (e.g. fracking) that allows some firms to produce (energy) cheaper (e.g. natural gas instead of coal). The new cost function is ĉ(q) = q 2 / and so MC = q. Assume there is only one firm using the new technology and 6 using the old one. 1 What is the equilibrium price? 2 What are the firms profits? If the price is p, an old firm supplies q = p/2 and a new firm supplies q = p. Total supply is then Q = 6p/2 + p = 4p. Equating it with total demand, Q = 100 2p, give us p = 100/6 < 20. Profits are π = 100/6 (50/6) (50/2) and π = (100/6) 2 (100/6) 2 /2 80.

125 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

126 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

127 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

128 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

129 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

130 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

131 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

132 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

133 Beyond partial equilibrium Labor Market After the old technology firms are displayed by the new ones, What happens with the labor force? To answer this, we have to model both technologies and so, we assume: 1 Interest rate, r = 1. 2 Old firms need 100 units of capital to produce k = New firms need 80 units of capital to produce k = The wage rate is w. 5 Old production function is q = f(k, l) = l if k 100 and zero otherwise. 6 New production function is q = f(k, l) = 2l if k 80 and zero otherwise. Cost functions are: c(q) = w q and ĉ(q) = w 2 q

134 Beyond partial equilibrium Labor Market The demand for labor, for the old and new technology firms are: d(w) = What about the labor supply? ( p 2 ( p ) 2 and d(w) = /2. 2w) w

135 Beyond partial equilibrium Labor Market The demand for labor, for the old and new technology firms are: d(w) = What about the labor supply? ( p 2 ( p ) 2 and d(w) = /2. 2w) w

136 Labor Supply a first look Assume individuals have utility for two goods: leisure and a consumption good, l and x: u(l, x) = ln(l) + ln(x). Also assume that: individuals are endowed with h hours that can either be sold at the labor market or used for leisure and; individuals only source of income is labor. Let p x be the price of the consumption good and w be the wage rate. The individuals budget constraint is: p x x w (h l).

137 Labor Supply a first look Solving for the optimal basket, MRS Mu l Mu x = w p x w l = p x x Thus, p x x = w (h l) l(w, p x ) = h/2 and x(w, p x ) = So individual labor supply is inelastic, l h l(w, p x ) = h/2. w 2p x h.

138 Labor Supply a first look Solving for the optimal basket, MRS Mu l Mu x = w p x w l = p x x Thus, p x x = w (h l) l(w, p x ) = h/2 and x(w, p x ) = So individual labor supply is inelastic, l h l(w, p x ) = h/2. w 2p x h.