Quantitative Subject. Quantitative Methods. Calculus in Economics. AGEC485 January 23 nd, 2008

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1 Quantitative Subect AGEC485 January 3 nd, 008 Yanhong Jin Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 1 Quantitative Methods Calculus Statistics Regression analysis Linear programming Risk and decision making under risk Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 Calculus in Economics Optimization (Minimize vs. maximize) First-order condition to find the possible optimal solution Second-order condition to verify whether it is min or max Elasticity Own price elasticity inelastic vs. elastic elasticity along a linear demand curve Constant elasticity demand Cross price elasticity (complements vs. substitutes) Income elasticity (normal goods vs. inferior goods) Marginal analysis Marginal revenue Marginal cost Marginal profit Marginal productivity Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

2 Maximization Problems: max f(x) Solution: Derive First Order Condition (FOC): f (x)=0 Check Second Order Condition (SOC): f (x)<0 Local vs. global: If more than one point satisfy both FOC and SOC, evaluate the obective function at each point to identify the maximum. Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 4 Minimization Problems: Min f(x) Solution: Derive First Order Condition (FOC): f (x)=0 Check Second Order Condition (SOC): f (x)>0 Local vs. global: If more than one point satisfy both FOC and SOC, evaluate the obective function at each point to identify the minimum. Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 5 Unconstrained Multivariate Optimization Max FOC: y = g( x, ( ) 0 Min y = g FOC: y ( x, y = g ( x, x x = g ( x, x x y y = g ( x, = g ( x, z z z z SOC: SOC: w w yy ( y, < 0 yy ( y, > 0 w ( y, > 0 zz ( y, < 0 w ( y, w ( y, w ( y, > w ( y, z ) w ( y, z ) w ( y, z ) yy zz yz > yy zz yz w zz ( ) 0 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 6

3 Constrained Optimization: using Lagrangian method Maximize y = f(x 1, x, x 3,, x n ) s.t. g(x 1, x, x 3,, x n ) < b Solution: Set up Lagrangian: L( x1, x,..., xn, λ) = f ( x1, x,..., xn) + λ( b g( x1, x,..., xn) ) FOC: Lx 1( x1, x,..., xn, λ) = 0... L ( x1, x,..., xn, λ) = 0 xn Lλ ( x1, x,..., xn, λ) = b g( x1, x,..., xn) = 0 Interpretation of Lagrangian multiplier: shadow value of constrained resource. Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 7 Elasticity Price elasticity >-1 inelastic % change of Q Q / p <-1 elastic ε p = = % change of p Q / p =1 unitary Cross price elasticity >0 substitutes % change of Q Q / p 0 ε po = = <0 complements % change of other product price Q / p0 Income elasticity >0 normal goods % change of Q Q / I ε I = = <0 inferior goods % change of I Q / I Interpretations of elasticity Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 8 Completely Inelastic Demand Curve: = 0 Price per unit ($) Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

4 Completely Elastic Demand Curve: = -8 - Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Price Elasticity of Demand Varies Along a Linear Demand Curve Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Price Elasticity and Revenue Maximization Revenue: R=P*Q, Marginal revenue MR: dr dp Q dp Q / p 1 MR = = p + Q = p(1 + ) = p(1 + ) = p(1 + ) dq dq p dq dq / dp ep The change in revenue if the price goes down: Elastic demand: increases Inelastic demand: decrease Unitary demand: unchanged Linear demand case: P=a-b*Q Price elasticity: p ep = b Q Marginal revenue: MR=a-*b*Q Total revenue is maximized where MR=0 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

5 Optimal Pricing for Monopoly Optimal pricing: MR=MC MR P 1 1 = 1 + = MC P = MC / 1 + EP EP Interpretation: the more elastic demand, the lower the price a monopoly can charge Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Marginal Analysis Marginal revenue MR Marginal cost MC Marginal profit MP Marginal productivity MPP Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Sampling Sample (Data) Descriptive Statistics Population Regression Population parameters Inference Sample parameters T test, F test, confidence interval OLS: Assumptions, properties of OLS estimates, interpretations of estimates Measures of central tendency: mean, median, mode Measures of variability or dispersion: range, variance, standard deviation, coefficient of variation Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

6 Two Types of Random Variables Continuous variables Discrete variables Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Descriptive Statistics Measures of central tendency Mean (average) Median Mode Measures of dispersion/variability Range Variance Standard deviation Coefficient of variation Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Normal Distribution Bell-shaped Standard normal distribution: mean=0, stdev=1 Property of normal distributions If x has a normal distribution with mean u and stdev sigma, z=(x-u)/sigma has a standard normal distribution. The sum of two independent normally distributed variables is also normally distributed with mean=u1+u, stdev=sqrt(sigma1^+sigma^). Central Limit Theorem: Consider a random sample of n OBSs from a population with mean u and stdev sigma. When n is sufficiently large, the sampling distribution of sample mean will be approximately a normal distribution with mean u and stdev is sigma/sqrt(n). Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

7 The Simple Regression Model Dependent Variable Left-Hand Side Variable Explained Variable Regressand Response Variable Independent Variable Right-Hand Side Variable Explanatory Variable Regressor Control Variable y = β 0 + β 1 x + u Coefficients β 0 : Intercepts β 1 : Slope Error Term Disturbance Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Goal of OLS y y 4 y 3 y Choose β 0 and β 1 to minimize these sum of squared prediction errors..} û û. 3 { û 4. { yˆ = ˆ β + ˆ β1x 0 y 1. } û 1 x 1 x x 3 x 4 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 0 x Minimizing Sum of Least Squares n n min ( uˆ i ) = min ( yi ˆ β xi ) i= 1 Simple linear regressions: ˆβ 1 n ( xi x)( yi y) i= 1 = n ( xi x) ˆ i= 1 β = y βˆ 1x 0 i= 1 Multiple linear regressions: 1 ˆ β = ( x ' x) x' y where x=(x 1, x, x k ). Note: In the simple regressions, the slope estimate is the sample covariance between x and y divided by the sample variance of x. Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

8 Assumptions Assumption 1: E(u)=0: The average value of u in the population is 0. Assumption : Zero conditional mean: E(u x) = E(u) = 0, implying E(y x) = β 0 + β 1 x Assumption 3: Independent error terms: Each observed u i is independent of all other u, E[u i u ] = 0 for all i Assumption 4: Var(u x) = σ Homoskedasticity (vs. heteroskedasticity) Assumption 5: u~norm(0, σ ) Normality Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 Assumption 4: Homoskdasticity Var(u x) = σ f(y x) x 1 x. x 3 y. x f(y x). x 1 x y x 3.. x Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 3 Gauss Markov Assumptions for cross-section regressions Gauss Markov Theorem: The OLS estimator is the best linear unbiased estimator (BLUE). BLUE: Assumptions: Linear in parameters Random Sample Zero conditional mean No perfect collinearity Homoskedasticity Best: OLS estimators have the smallest sum of squared errors among all linear estimators; Linear in the dependent variable Y; Unbiased: OLS estimator represents the true value on the average; Estimator. Questions: (a) Which assumption you do not need to get unbiasness of OLS estimators? (b) What s the mean of best, linear, and unbiasness, respectively? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

9 Goodness-of-Fit: Coefficient of Determination ŷ yˆ = ˆ β ˆ x 0 + β1 (x i, y i ): sample OBS yˆ = ˆ β + ˆ β x : regression line 1 0 y i : true value ŷ y ŷ y yi yˆ : fitted value : average value : explained variation : Unexplained variation Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 5 Goodness-of-fit: some terminologies We can think of each observation as being made up of an explained part, and an unexplained part, y = yˆ + uˆ We then define the following : i ( yi y) ( yˆ i y) uˆ i is i i is is the residual sum of squares (SSR) Thus, SST = SSE + SSR the total sum of squares (SST) the explained sum of squares (SSE) Goodness-of-fit: How well does the simple regression line fit the sample data? Calculate R = SSE/SST = 1 SSR/SST Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 6 R and Adusted R R ( yˆ i y) Explained sample variability i= 1 SSE SSR R = = = = 1 Total sample variability yi y n ( ) SST SST n i= 1 Adusted R SSR /( n k 1) R == 1 = 1 σ SST / n SST / n Questions: (a) Is adusted R always better than R? (b) Why do we care about the adusted R? (c) What s the relationship between R and adusted R? (d) What s appears in the F-Statistics in the regression results? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

10 The t Test Form hypothesis One-sided hypothesis Two-sided hypothesis Calculate t statistic: for example, t statistic for Final the critical value, c ˆ β ( β ) β : t ˆ β se ˆ Given a significance level, α, we look up the corresponding percentile in a t distribution with n k 1 degree of freedom and call this c, the critical value Apply reection rule to determine whether reect the null hypothesis ˆ Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 8 One-Sided Alternatives y i = β 0 + β 1 x i1 + + β k x ik + u i H 0 : β = 0 H 1 : β > 0 Fail to reect (1 α) 0 Critical value c: the (1 α)th percentile in a t-dist with n k 1 DF. ˆ β T-statistic: t ˆ β se( ˆ β ) Results: Reect H 0 if t-statistic>c; Fail to reect H 0 if t-statistic<c c reect α Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 9 One-Sided Alternatives (cont ) α y i = β 0 + β 1 x i1 + + β k x ik + u i H 0 : β = 0 H 1 : β < 0 reect (1 α) -c 0 Fail to reect Critical value c: the (1 α)th percentile in a t-dist with n k 1 DF. ˆ β T-statistic: t ˆ β se( ˆ β ) Results: Reect H 0 if t-statistic<-c; Fail to reect H 0 if t-statistic>c Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

11 Two-Sided Alternatives reect α/ y i = β 0 + β 1 X i1 + + β k X ik + u i H 0 : β = 0 H 1 : β 0 fail to reect -c (1 α) 0 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU c reect α/ Critical value: the (1 α/)th percentile in a t-dist with n k 1 DF. ˆ β T-statistic: t ˆ β se( ˆ β ) Results: Reect H 0 if t-statistic >c; Fail to reect H 0 if t-statistic <c Confidence Intervals Use the same critical value as was used for a twosided t test to construct a confidence interval. A (1 - α) % confidence interval is defined as ( ˆ β ), ˆ α β ± c se where c is the 1- percentile in a t distribution n k 1 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 3 The F Tests The F tests are used to test multiple linear restrictions ( SSRr SSRur) / q F = SSRur /( n k 1) SSR r is the sum of squared residuals from the restricted model SSR ur is the sum of squared residuals from the unrestricted model. q=df r -df ur is the degrees of freedom of numerator (no. of independent variables dropped) n-k-1=df ur is the degrees of freedom of denominator. F is always positive Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

The F Tests (cont ) f(f) fail to reect Reect H 0 at α significance level if F > c (1 α) 0 c α reect F Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU 51 34 F-test

12 The F Tests (cont ) f(f) fail to reect Reect H 0 at α significance level if F > c (1 α) 0 c α reect F Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU F-test for the overall significance A special case of exclusion restrictions is to test H 0 : β 1 = β = = β k = 0 Since the R from a model with only an intercept will be zero, the F statistic is simply F = R k ( 1 R ) ( n k 1) Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Be Careful about Interpretations of Estimation Results Regression line y=a 0 +(a 1 )(x 1 )+a (x )+a 3 (x 1 ) Ceteris paribus analysis (controlling for all other factors) What s the change in y if x 1 goes up by one unit? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

13 Linear Programming The main assumption: the obective and constrain functions are linear. Applications: resource allocation, investment, profit maximization, input allocation To set up a linear programming problem: Obective function Resource constraints Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Linear Programming (cont ) Interpret outputs: The value of the obective function Resource allocations: binding vs. non-binding resources Shadow price: an additional value increase in the obective function if one more unit of this resource is available. Sensitivity analysis What happens to the optimal solution if one variable changes? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Risk and Decision Making Under Risk Degree of risk Decision making under risk Risk attitudes Risk aversion and risk premium Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

14 Minimizing Variance or Standard Deviation The payoffs of all events: x 1, x,, x N The probability of each event: p 1, p,, p N N Expected value of x: EV ( x) = x1 p1 + x p xn pn = xi pi = ( x1 EV ) p1+ ( x EV ) p ( xn EV ) p = xi Variance: N ( i = 1 Standard deviation: square root of variance σ Coefficient of variance: CV = σ EV i= 1 N EV ) p i Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Risk Measurement Absolute Risk: Overall dispersion of possible payoffs Measurement: variance, standard deviation The smaller variance or standard deviation, the lower the absolute risk. Relative Risk Variation in possible returns compared with the expected payoff amount Measurement: coefficient of Variation (CV), CV = The lower the CV, the lower the relative risk. σ EV Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Decision Making Under Uncertainty Criteria: Maximize expected value Minimize variance or standard deviation Coefficient of variance Maximin criterion Maximax criterion Incorporate risk attitudes: certain equivalent Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

15 Maximize Expected Value Event (State of Economy) Recession Normal Boom Prob Profit Pro. A Pro. B $4,000 $0 $5,000 $6,000 $5,000 $1,000 Questions: Which proect you will choose based on this criteria? What are your concerns? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Minimizing Variance/Standard Deviation Event (State of Economy) Recession Normal Boom Prob Profit pro. A pro. B $4,000 $0 $5,000 $5,000 $6,000 $ Questions: Which proect you will choose based on this criteria? What is ignored using this criterion? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Coefficient of Variance Expected value Variance Coefficient of Variance A $5, B $5, Questions: Which proect you will choose based on this criterion? What is ignored? Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

16 Maxmin and Maxmax Criterion Event (State of Economy) Recession Normal Boom Questions: Which proect you will choose? Based on Maxmin? Based on Maxmax? What is ignored? Profit pro. A pro. B $4,000 $0 $5,000 $5,000 $6,000 $1.000 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Risk Attitudes Scenario: A person has two choices, a sure thing and a risky option, and both yields the same expected value. Risk averse: take the sure thing Risk neutral: indifferent between two choices Risk loving: take the risky option Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Fundamental Reasons to Risk Attitudes Risk averter: diminishing MU Risk neutral: constant MU Risk lover: increasing MU Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

17 Risk Premium Initial wealth=40 which results in U(40)=10 Utility (U) Buy a vase (a)if it is a Ming vase,u(70)=140 (b)if it is an imitation, U(10)=70 (c)0.5u(70)+0.5u(10)=105 Questions: (a)buy the vase? U(70)=140 U(40)=10 0.5U(10)+0.5U(70) =U(6)=105 U(10)=70 (b)risk premium? Wealth ($) Risk premium Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Incorporating Risk Attitudes: Certain Equivalent Initial wealth $1000, and you face two options: (a) Keep your initial wealth and do nothing; (b) Invest: receive $10,000 if succeed; lose 1000 if fail. After assessing these two options, you find yourself indifferent between two. Analysis: Certain Equivalent EquivalentCertainSum α = Certain Equivalent factor: ExpectedRiskSUm Risk averse: α<1 Risk neural: α=1 Risk loving: α>1 Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU Summary Calculus Statistics Regression Linear programming Risk and decision making under risk Quantitative Analysis of Agribusiness and Agricultural Economics Yanhong Jin at TAMU

CHAPTER 3. Quantitative Demand Analysis

CHAPTER 3. Quantitative Demand Analysis CHAPTER 3 Quantitative Demand Analysis Copyright 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Outline