4) Univariate and multivariate functions

Transcription

1 30C00300 Mathematical Methods for Economists (6 cr) 4) Univariate and multivariate functions Simon & Blume chapters: 13, 15 Slides originally by: Timo Kuosmanen Slides amended by: Anna Lukkarinen Lecture held by: Anna Lukkarinen 1

2 How much more rope? 1m 1m 1m 1m 1m 1m 1m 1m

3 Outline 1. Function as a mapping 2. Inverse function 3. Implicit function and correspondence 4. Microeconomic content Production function Utility function Cost, revenue and profit functions 3

4 Function Definition (S&B, Ch. 13.1): A function from a set A to a set B is a rule that assigns to each object in A, one and only one object in B. In this case, we write f: A B. Set A is called the domain of f. Set B is called the target or target space. y=f(x) is the image (or range) of x under f. Note: it is incorrect to write f(x) to denote the function itself. It is the value of function f at point x. 4

5 Function In other words A function is a rule that assigns a unique object to each object in the function s domain Input Function Output

6 Function - Basic definitions y = f(x) y Depends on x Dependent variable Endogenous variable Image or range Set of all output elements x Fixed outside model Independent variable Exogenous variable Domain Set of all input elements

7 Polynomial functions f x = a n x n + a n 1 x n a 1 x 1 + a 0 (a n 0) Polynomial functions Linear functions Cubic functions Quadratic functions Etc.

8 Exponential functions f x = Aa x (a > 0, a 1) Source of picture: Matti Karvonen, lecture notes for Mathematics and Statistics for Managers, 2008

9 Firm s production function Production function f: R + k R + indicates the maximum output that can be produced with given input vector x. Definition: f ( x) max y ( x, y) T Note: production function represents Eff(T). Example: Cobb-Douglas production function k k i i1 f x x x x x k i ( x)... 9

10 Consumer s utility function Utility function u can be used for modeling choices. Utility function indicates the level of satisfaction obtained from consumption of commodity basket x = (x 1, x 2,, x m ) Example: Cobb-Douglas utility function m u : R R, u x x x x m ( x)... m i1 x i i m 10

11 Graph of a function The graph of a function can help to visualize functions of one or two variables. The graph is a set that contains pairs of points consisting of the elements of domain A and the corresponding values of f: ( x, f ( x) x A 11

12 Outline 1. Function as a mapping 2. Inverse function 3. Implicit function and correspondence 4. Microeconomic content Production function Utility function Cost, revenue and profit functions 12

13 Inverse function If f: A B is an injection, the inverse function f 1 : B A exists. Note: The domain of the inverse function is the target of the original function, and vice versa. Given equation y = f(x), the inverse function of f is obtained by solving x. Example: f(x) = 2x + 3 f 1 (y) = ½(y 3). 13

14 Examples of inverse functions The inverse of the power function f(x) = x n is the nth root f(x) -1 = x 1/n. The inverse of the exponential function f(x) = exp(x) is the logarithm function f(x) -1 = lnx. Examples in economics: Demand function: Inverse demand: q = D(p) p = D -1 (q) 14

15 Exponential and logarithmic functions Source of picture: Matti Karvonen, lecture notes for Mathematics and Statistics for Managers, 2008

16 Increasing and decreasing functions In R 2, if the graph of a function rises (drops) from left to right on an interval I, it is increasing (decreasing) on I Function f is increasing on [a,b] Function g is decreasing on [a,b] Function h is neither increasing nor decreasing on R Modified from: Matti Karvonen, lecture notes for Mathematics and Statistics for Managers, 2008

17 Increasing and decreasing functions y = f(x) Increasing Decreasing x 1 < x 2 f(x 1 ) f(x 2 ) x 1 < x 2 f(x 1 ) f(x 2 ) Strictly increasing x 1 < x 2 f(x 1 ) < f(x 2 ) Strictly decreasing x 1 < x 2 f(x 1 ) > f(x 2 ) y (1,1) (-2,2) y (-2, - 2) x (1,-1) x

18 Inverse function Note: Not all functions are invertible: the inverse function does not necessarily exist. To be invertible, a real valued function f must be strictly monotonic increasing or strictly monotonic decreasing in its domain. 18

19 Outline 1. Function as a mapping 2. Inverse function 3. Implicit function and correspondence 4. Microeconomic content Production function Utility function Cost, revenue and profit functions 19

20 Implicit function Explicit form: y = F x 1,, x n y is an explicit function of the x i s Implicit form: G x 1,, x n, y = 0 If the equation determines a corresponding value y for each (x i,, x i ), it defines y is an implicit function of the x i s 20

21 Correspondence A correspondence F from set A to set B is a rule that maps each x in A into a subset F(x) of B A correspondence is different from a function in that a given domain is mapped into a set (not a single object as in a function) 21

22 Outline 1. Function as a mapping 2. Inverse function 3. Implicit function and correspondence 4. Microeconomic content Production function Utility function Cost, revenue and profit functions 22

23 Cost function k Given input prices w R and technology T the minimum, cost of producing output y is given by the cost function km C : R R, Note: w, y, and T are exogenously given. C( w, y) min w x ( x, y) T x 23

24 Revenue function m Given output prices p R, and technology T, the maximum revenue obtainable with inputs x is given by the revenue function km R : R R, Here x, p, and T are taken as given. R( x, p) max p y ( x, y) T y 24

25 Profit function Given input and output prices the maximum profit is given by the profit function km : R R, xy, k m w R, p R, ( w, p) max p y w x ( x, y) T Note: prices p and w are exogenously given, quantities y and x are optimized endogenously. 25

26 Next time Wed 23 March Mathematical programming 26