IMP 007 Introductory math course 5. Optimization Antonio Farán Vallespín Toniaran@hotmail.com
Derivatives Why are derivatives so important in economics? Derivatives inorm us o the eect o changes o the independent variable on the dependent variable. e.g. what happens with GDP i salaries increase? and with demand? Derivatives are a tool or solving optimisation problems. Humans are assumed to behave optimising their utility.
Economics is a science o choice: How do individuals choose their consumption bundles? How do individuals choose among what to consume today and tomorrow? How do irms choose how many units should they produce? How do irms choose how many workers should they hire? How do you choose how many hours should you study and how many hours should you spend in the disco? 3
We assume: Agents have an objective unction. (e.g. utility, costs, proits ) Agents can choose the value o some variables argument o the objective unction: Choice variables Agents attempt either to: Maximize: choose the values o the choice variables that yield the maximum possible value o the objective unction. (e.g. maximize proit or utility), or to: Minimize: choose the values o the choice variables that yield the minimum possible value o the objective unction. (e.g. minimize total cost) 4
Optimization: Given an objective unction rom set A to the real numbers, it is sought: I minimizing: an element x 0 in A such that ( x 0 ) ( x) or all x in A. Then we call x 0 a minimum. I maximizing: an element x 0 in A such that ( x 0 ) ( x) or all x in A. Then we call x 0 a maximum. : A R 5
Relative versus absolute extremum An extremum (a maximum or a minimum) can be: Absolute (or global): i it is maximum or minimum in all the range o the unction. Relative (or local): i it is only maximum or minimum in the immediate neighbourhood o the point only. SEE EXAMPLES 6
What does this have to do with derivatives???? Increasing and decreasing unctions Stationary points Convexity and concavity 7
Increasing unctions:. Optimization A unction is strictly increasing in an interval when or any two points in this interval, x and (x+h) it can be veriied that: x x h ( x) ( x h) We can express this by using the dierence quotient: y x ( x h) h ( x) I this true or any value o h in the interval, then it is also true or values o h approaching zero. Thereore, the derivative o any point at an interval where the unction is increasing must be positive. Necessary condition. 0 8
Decreasing unctions:. Optimization A unction is strictly decreasing in an interval when or any two points in this interval, x and (x+h) it can be veriied that: x x h ( x) ( x h) We can express this by using the dierence quotient: y x ( x h) h ( x) I this true or any value o h in the interval, then it is also true or values o h approaching zero. Thereore, the derivative o any point at an interval where the unction is decreasing must be negative. Necessary condition. 0 9
Summarizing:. Optimization I the derivative at one point is positive: The unction is increasing at that point. I the derivative at one point is negative: The unction is decreasing at that point. But, what happens when the derivative is neither positive nor negative? this is to say, when it is equal to zero. 10
Case when derivative is zero: What does it mean that the derivative is zero? For being zero it must be that (x+h)=(x). It implies that at this point the unction is neither increasing nor decreasing. Thus we call these points where =0: stationary points. 11
What can a stationary point be? A local maximum: I the neighbouring points have a smaller (x). This also implies that the unction is: Increasing to the right o this point. >0 Decreasing to the let o this point. <0 A local minimum: I the neighbouring points have a larger (x). This also implies that the unction is: Decreasing to the right o this point: <0 Increasing to the let o this point: >0 A point o inlection I the unction is increasing or decreasing on both sides. See Exercise 1. 1
How can be know whether a stationary point is a local maximum, a local minimum or a point o inlection? a) Evaluating whether the unction is increasing or decreasing to the let and to the right. b) Evaluating the second derivative. 13
The second derivative: It is obtained by taking the derivative o the derivative I the irst derivative tells us about the rate o change o the dependent variable, the second derivative tells us about the rate o change o the rate o change The second derivative is denoted by: (x) d y d dy 14
Second derivative being positive means that the irst derivative is increasing: I the irst derivative was positive, then the unction increases at increasing rates. I the irst derivative was negative, then the unction decreases at increasing rate. Second derivative being negative means that the irst derivative is decreasing: I the irst derivative was positive, then the unction increases at decreasing rates. I the irst derivative was negative, then the unction decreases at decreasing rates. 15
Assuming an ininitesimal increase in the independent variable rom a point x=x 0 (x 0 )>0 means the value o the unction tends to increase. (x 0 )<0 means the value o the unction tends to decrease. (x 0 )>0 means the slope o the curve tends to increase. (x 0 )<0 means the slope o the curve tends to decrease. 16
Dierentiability:. Optimization I second derivative exists or all values o an interval, then the unction is twice dierentiable in that interval. I (x) is continuous in an interval, then (x) is said to be twice continuously dierentiable. 17
Curvature o a unction: Second derivative also inorm us o the curvature o a unction. Inormally, curvature: how the curve bends. A curve can be: Convex: i shape is a valley, a U, happy. Concave: i shape is a hill, an inverse U, sad. 18
Concave: I we pick any two pair o points M and N on its curve. and join them by a straight line. the line segment MN must lie entirely below the curve, except points M and N. Convex: I we pick any two pair o points M and N on its curve. and join them by a straight line. the line segment MN must lie entirely above the curve, except points M and N. 19
Concave: (x 0 )<0. Optimization Start with positive slope and it reduces becoming negative ater a maximum. Convex: (x 0 )>0 Start with negative slope and it increases becoming positive ater a minimum. I (x 0 )=0 (x 0 )>0 Inlection point rom concavity to convexity. (x 0 )<0 Inlection point rom convexity to concavity 0
Necessary and suicient conditions: First-order condition: =0: necessary condition or a point x=x 0 to be a local extremum. But =0 is not suicient condition or a point x=x 0 to be a local extremum. Second order condition, ater =0: >0 and <0 suicient condition or being a local maximum or minimum But it is not necessary, you can have extremum with =0. See y=x 4 at x=0. 1
>0 in an interval: unction strictly increasing in the interval. <0 in an interval: unction strictly decreasing in the interval. =0: stationary >0, local minimum <0, local maximum =0, inlection point >0 concave-convex <0 convex-concave See Exercise and 3.
Objective unctions with more than one choice variable: z=(x 1,x,,x n ) At a extremum, it must be that dz=0 or ininitesimal values o 1,,., n. Since we know that: dz 11 33 Then, the necessary condition (irst-order) or extremum is that: 1 3 0 3
Second order condition: Ater the irst-order condition is met, A stationary value will be a maximum i: the second total derivative o z (d z) is positive deinite. A stationary value will be a minimum i: The second total derivative o z (d z) 4
What is d z? d Substituting dz by dz. Optimization ( dz) ( dz) ( dz) z d dz) 1 x1 x x3 ( 11 33 3 We obtain: d z 1 31 1 11 1 3 1 3 1 3 1 3 33 13 1 3 3 3 5
How do we know i that is positive or negative deinite? We use the Hessian: 11 1 31 Whose succesive principal minors are: H1 11 H 1 3 13 3 33 11 1 H H3 H 1 6
Negative d z deinite i: H1 0; H 0; H 3 0 Positive d z deinite i: H1 0; H 0; H 3 0 In using this condition we must evaluate all the principal minors at the stationary points where 1 3 0 7
Advance o matrix algebra: How to calculate the determinant? Given the matrix: a11 a1 a13 a1 a a3 a31 a3 a33 The determinant det(a) or is: A a 13 a 11 a a a a 31 33 a 11 a 13 a 3 a 1 a 3 a A 3 a 33 a 31 a 1 a A1 a 11 A a11 a a1 a1 1 a 1 a 3 8
BIBLIOGRAPHY:. Optimization Alpha C. Chiang (1984) Fundamental Methods o Mathematical Economics Third edition. McGraw-Hill, Inc. Ch. 9,11,1 9