EC422 Mathematical Economics 2

Transcription

1 EC422 Mathematical Economics 2 Chaiyuth Punyasavatsut Chaiyuth Punyasavatust 1

2 Course materials and evaluation Texts: Dixit, A.K ; Sydsaeter et al. Grading: 40,30,30. OK or not. Resources: ftp://econ.tu.ac.th/class/archan/c HAIYUTH/ Chaiyuth Punyasavatust 2

3 Objectives Dynamic economics is about explaining economic behaviors through time. Explain the time paths of economic variables. Optimization concept. The choice is the use of resources in different time periods. Maximizing a multiperiod objective function of variables in many periods, subject to budget or resource constraints on these variables through time. Chaiyuth Punyasavatust 3

4 Objectives Find optimal time path for every choice variable. Three major approaches: Calculation of variation, maximum principle or optimal control theory (Hamiltonian), and dynamic programming (Bellman equation). And also method of Lagrange Chaiyuth Punyasavatust 4

5 Example: Economic Growth choice between current consumption and future consumption. By reducing consumption today, will lead to more resources or capital stock that can be used to produce consumption goods for future. What is the optimum consumption decision in each period. Chaiyuth Punyasavatust 5

6 Example: Business Cycles Market equilibrium in a dynamic setting Explaining economic fluctuations from technology shocks Chaiyuth Punyasavatust 6

7 Example: Dynamic Games Oligopoly pricing Two players try to solve the standard optimization problem where the state variable depends on control variables chosen by both players. Nash equilibrium vs subgame perfect Stackelberg game Chaiyuth Punyasavatust 7

8 Common ingredients 1. Initial point and terminal point 2. Admissible paths from initial to terminal 3. Path values 4. Objective to max or min path value and get the optimal path. Chaiyuth Punyasavatust 8

9 state Z A 0 T stage Chaiyuth Punyasavatust 9

10 Lecture 1 Preliminary Mathematics Mathematics is a language. Chaiyuth Punyasavatust 10

11 Convex Sets 1. We often assume this or convexity to guarantee that our analysis is tractable mathematically and the results are well-behaved or clearcut. 2. A set is convex if for any two points in the set, all convex combinations of those two points are also points in the same set. Chaiyuth Punyasavatust 11

12 Convex Sets S is a convex set if for all x 1 and x 2 belong to S, we have t x 1 + (1--t) x 2 S, for all t 0 t 1. Equivalently, a set is convex iff we can connect any two points in the set by a straight line that lie entirely within this set. 3. For example, consider two points in a real line, R. Chaiyuth Punyasavatust 12

13 Convex Sets 5. Example of S in R 2. Not a convex set when it has a dense or holes, ie. Heart, doughnut. Chaiyuth Punyasavatust 13

14 Closed Sets 1. Consider the simplest case, a close interval, [a, b], in the real line is a closed set. The sets on both sides of this interval (open interval) are open sets. The union of open sets is an open set. Thus, [a, b] c is an open set. Chaiyuth Punyasavatust 14

15 Closed Sets 2. We call [a, b] a closed set if its complement is an open set. 3. A set is closed if it contains all of the points in its boundary. Chaiyuth Punyasavatust 15

16 Bounded Sets 1. Every open interval on the real line, (a, b) R is a bounded set. 2. In R 2, an open ball is a disk containing set of points inside (excluding the circumference). 3. In R 3, an open ball is the set of points inside the sphere of radius ε. Chaiyuth Punyasavatust 16

17 Compact Sets 1. A set S in R n is called compact if it is closed and bounded. 2. Any open interval in R is not a compact set. (since not closed) 3. An open ball in R n is not compact, similarly. 4 R n is not compact since it is not bounded. Chaiyuth Punyasavatust 17

18 Real-valued functions f is a real-value function if it maps elements of its domain into the real line. Ex. Y = ax1 + bx2 Ex. U = x1*x2 Chaiyuth Punyasavatust 18

19 Level set This allows us to study functions of three variables by looking upon sets in the two dimensional plane. 1. A level set is the set of all elements in the domain of a function that map into the same number in the range. Chaiyuth Punyasavatust 19

20 Level set 2. L( y=10) is a level set of the real-valued function f : D R iff L( y=10) = {x x D, f(x) =10 }. Ex. An isoquant is a set that all input bundles producing exactly 10 units of output. 3. Note that since f is a function-- that is it assigns a single number in the range therefore, two difference level sets will never cross each other. Chaiyuth Punyasavatust 20

21 Level Set Draw a picture Chaiyuth Punyasavatust 21

22 Level set 4. Draw level sets in R 2 and define the superior set, S(y=10) {x x D, f(x) 10}, for level y=10. Ex. An input requirement set is a se of all input bundles that produce at least y units of output. 5. When f(x) is increasing, then S(y=10) will always lie on and above the level set L(y=10) Chaiyuth Punyasavatust 22

23 Level set 6. Note that (a) f is increasing if f(x 0 ) f(x 1 ), for all x 0 x 1. (notice vector notation here). 7. Think of the level sets when f is decreasing. Chaiyuth Punyasavatust 23

24 Concave functions. 1. In economics, we often see concave real-valued functions when domains are convex sets. 2. f is a concave function iff for every pair of points on its graph, the chord joining them lies on or below the graph. Here, we look at the value of the function evaluated at convex combinations of any two points. Chaiyuth Punyasavatust 24

25 Concave Functions f(x t ) t. f(x 0 ) + (1-t). f(x 1 ), RHS=chord Chaiyuth Punyasavatust 25

26 Concave Functions f(x t ) x 0 x t x 1 Chaiyuth Punyasavatust 26

27 Concave functions. 3. Another equivalent definition: f is a concave function iff the set of points on and below (i.e. beneath) the graph is a convex set. 4. A concave function allows for linear segments. To rule out this, we require a strictly concave function. 5. Strictly normally is equivalent to get rid of equality sign in definition. Chaiyuth Punyasavatust 27

28 Concave functions. 6. So, f is a strictly concave function iff the chords joining any pairs of points must lie below the graph. 7. Nice properties of the concave functions are (1) the critical points are always maxima; (2) sum of concave functions is concave; Chaiyuth Punyasavatust 28

29 Concave functions. (3) their level sets have just the right shapes; they bound convex subsets from below. For (2), we need it for the grand utility or social welfare function. For (3), it has nice interpretations, diminishing marginal rate of substitution, and mixing goods make you happier. Chaiyuth Punyasavatust 29

30 Concave functions. 8. A monotonic transformation of a concave function needs not be concave. In fact, any monotonic transformation of a concave function is a quasiconcave function. Think of f(x) = x, and g(x) = x 2. Chaiyuth Punyasavatust 30

31 g is a monotonic transformation of I if g is strictly increasing function of I. Example. U(x,y)= xy. Then, 3xy+2, (xy) 2, (xy) 2 +2, ln (xy) are its monotone transformation. We use this property for an ordinal concept of utility. Chaiyuth Punyasavatust 31

32 Quasiconcave functions 1. f is a quasiconcave function iff the superior set is a convex set. 2. The superior set is the set containing points in domain that gives the function a value a specific value of y. So, it is the area on and above the level set of a given value of y. Chaiyuth Punyasavatust 32

33 Quasiconcave functions Chaiyuth Punyasavatust 33

34 Quasiconcave functions Ex. F(x, y) = min { ax, by} with a, b > 0. y Slope=a/b x Chaiyuth Punyasavatust 34

35 Quasiconcave functions 3. Another definition is to look at the value of the function: the value of the function at points formed by convex combination must be greater or equal the lowest value of the functions at any two points. That is, if f(x 1 ) < f(x 2 ), then f(x t ) f(x 1 ), for all t between 0 and 1, and x t t x1 + (1-t)x 2. f(x t ) Min [ f(x 1 ), f(x 2 ) ], for all t [0,1]. Read the smaller of.. Chaiyuth Punyasavatust 35

36 Quasiconcave functions f(x) f(x 2) f(x t) f(x 1) x x 1 x 2 Chaiyuth Punyasavatust 36

37 Quasiconcave functions 4. Noting that if the level set has some linear portion, it is still a convex set. So, if we want to get rid of this linear segment of the level curve, we will require a strictly quasiconcave function. 5. To remember this, if we assume that the utility function is strictly quasiconcave, then their indifference curves have no linear segments. Chaiyuth Punyasavatust 37

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39 Quasiconcave functions 6. Some properties of quasiconcave functions: (1) sum of quasiconcave functions is not necessary quasiconcave; (2) a critical point need not be a maximum. For (1), think of z=x 3 -x, we know that both x 3 and x are both monotone in R, so they are quasiconcave, but z, a sum of quasiconcave functions is neither quasiconcave nor quasiconvex. Chaiyuth Punyasavatust 39

40 Concavity and quasiconcavity 1. A concave function is always quasiconcave. So, does the strictly function. 2. The reverse is not true. I.e. the bell curve, graph of x3, a step function 3. From (1), we know that superior sets of a quasiconcave function are convex. 4. The linear graph is both quasiconcave and quasiconvex. Chaiyuth Punyasavatust 40

41 Concavity and quasiconcavity 5. So far, we should feel that we only need a function to be only a quasiconcave, if we need only a nice level curve that is convex to the origin, and a strictly quasiconcave for a nicer level curve. Chaiyuth Punyasavatust 41

42 Checking Concavity 1. R 1 : f is concave iff f " 0, for all x. 2. R 2 : f is concave iff the Hessian matrix is negative semidefinite for all x in domain. The simplest way to check this is to verify that for all x, f 11 0 and f Note that f 11 is just a change in slope of the function in the direction of x1, keeping x2 constant. And for f 12 is just a curvature of the function when x1 and x2 change at the same time. This rule of thumb does not allow for both zero values of f 11 and f 22. Chaiyuth Punyasavatust 42

43 Checking Concavity 2.1 The precise way to check for a concave function for two variables is to verify that (a) f 11 0; and (b) f 11 f 22 -f 12 f Conditions (a) and (b) are equivalently stated as the principal minors of the Hessian matrix always alternate in sign, starting with negative. 2.3 Principal minors are just the determinants of submatrices evaluated at the point x, as we move down the principal diagonal of the Hessian. Chaiyuth Punyasavatust 43

44 Checking Concavity f 11, f 11 f 12 f 21 f 22 Ex. f(x)=x 2-4 X X 1 X 2 -X 2 2 X1=3/7; x2= 8/7 H = -8 3, 3-2 f 11 = -8 H = 16-9=7 > 0 Chaiyuth Punyasavatust 44

45 Checking Concavity 3. So, the Hessian matrix tells us about the curvature of the function evaluated at certain points. 4. For a strictly concave function, we require the Hessian matrix to be negative definite for all x in domain set. That is, we need f 11 < 0 and f 22 < 0. Chaiyuth Punyasavatust 45

46 Checking Concavity 5. Example: f(x 1, x 2 )= x x We can verify that f 11 0 and f 22 0 for all x in domain. Thus, f is a concave function, and is also quasiconcave. 6. How about f(x 1, x 2 ) = x 1 x 2. Here we have f 11 = 0 and f 22 = 0 and the determinant of the Hessian matrix (second order principal minor) is 0. In fact, this function is not neither concave nor convex. Chaiyuth Punyasavatust 46

47 Homogenous functions 1. f is homogenous of degree k if f(tx) = t k f(x), for all t > When k =1, f is also called linear homogenous. α β 3. f(x) = A xx 1 2 is homogenous of degree α + β. This function is also known as Cobb-Douglas function. 4. If f is homogenous of degree k, its partial derivatives are homogenous of degree k-1. Chaiyuth Punyasavatust 47

48 Homogenous functions 5. Euler s theorem: f(x) is homogenous of degree k iff n fx () k f(x) = x for all x. x i= 1 6. When k=1, we can write the linear homogenous function in terms of its partial derivatives. 7. The most useful one is when f is the production function using K and L: f(k, L) = AK α L 1 α = MPK* K + MPL * L. i i Chaiyuth Punyasavatust 48

49 Useful theorems 1. If f is quasiconcave and linearly homogenous, then f is concave. 2. Every Cobb-Douglas function of two variables is quasiconcave. 3. CES function is quasiconcave since it is a monotonic transformation of a concave function. 4. A Cobb-Douglas function is concave iff it is CRTS or DRTS. Chaiyuth Punyasavatust 49

50 Nice examples of functions used in economics 1. f(x 1, x 2 ) = x 0.5 x Chaiyuth Punyasavatust 50

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52 Nice examples of functions used in economics 2. f(x 1, x 2 ) = x 0.3 x Chaiyuth Punyasavatust 52

53 Nice examples of functions used in economics 3. f (x 1, x 2 ) = min { x 1, 2x 2 } Chaiyuth Punyasavatust 53

54 Nice examples of functions used in economics 4. f(x 1, x 2 ) = x 1 + 2x 2 Chaiyuth Punyasavatust 54

55 Chaiyuth Punyasavatust 55