Ch. 3 - Quantitative Demand Analysis PDF

# Ch. 3 - Quantitative Demand Analysis ## Elasticity Elasticity measures the responsiveness (how much it will change) of one variable to changes in another variable. **Example:** How much will your grade change if you study more hours? * **Formula:** $E_{G,S} = \frac{\% \Delta G}{\% \Delta S}...

# Ch. 3 - Quantitative Demand Analysis ## Elasticity Elasticity measures the responsiveness (how much it will change) of one variable to changes in another variable. **Example:** How much will your grade change if you study more hours? * **Formula:** $E_{G,S} = \frac{\% \Delta G}{\% \Delta S} = \frac{\frac{\Delta G}{G}}{\frac{\Delta S}{S}} = \frac{\Delta G}{G} \times \frac{S}{\Delta S} = \frac{\Delta G}{\Delta S} \times \frac{S}{G}$ or $E_{G,S} = \frac{\Delta G}{S} \times \frac{S}{\Delta G}$ * $\frac{\Delta G}{\Delta S}$ is the slope of the functional relation between G and S. * Elasticity does not depend on the units in which we measure the variables. ## Is It Negative or Positive? * **When E > 0:** An increase in one variable increases the other. * **When E < 0:** An increase in one variable decreases the other. ## What Is Its Absolute Value? * **When |E| > 1:** A small change in one variable will cause a big change in the other. We say demand is elastic. * **When |E| < 1:** We will need a big change in one variable to cause a small change in the other. We say demand is inelastic. * **When |E| = 1:** When a change in one variable causes the same magnitude in change of the other variable. We say demand is unitary elastic. ## Own Price Elasticity of Demand Measures the responsiveness of quantity demanded of a product when its price changes. * **Formula:** $E_{Qx,Px} = \frac{\% \Delta Qx}{\% \Delta Px}$ or $E_{Qx,Px} = \frac{\frac{\partial Qd }{\partial Px}}{\frac{\partial Px}{\partial Qx}} \times \frac{Px}{Qx}$ **Question:** Given the law of Demand, is the Own Price Elasticity of Demand negative or positive? ## Interpreting Elasticities * **Example 1:** $E_{Qx,Px} = -4$ * **Example 2:** $E_{Qx,Px} = 12$ * **Example 3:** Based on the following graph, calculate $E_{Qx,Px}$ if it lowered price from \$12 to \$10. **(Insert image of graph here)** ## Elasticity and Total Revenue **Total Revenue Test:** * **When |$E_{QxPx}$| > 1:** An increase in price leads to a decrease in total revenue. * **When |$E_{QxPx}$| < 1:** An increase in price leads to an increase in total revenue. * **When |$E_{QxPx}$| = 1:** Total revenue is maximized. **Note:** Even though the slope of a linear demand function is always the same, the elasticity varies along the demand curve. * **Example 4:** Back to the previous graph, now calculate $E_{Qx,Px}$ if it lowered price from \$6 to \$4. **(Insert image of graph here)** **(Insert image of graph here)** ## Extreme Cases * **Perfectly Elastic:** $|E_{Qx,Px}| = \infty$ "Any change in P will make consumers change their consumption." **(Insert image of graph here)** * **Perfectly Inelastic** $|E_{Qx,Px}| = 0$ **(Insert image of graph here)** ## What Affects the Own Price Elasticity? * Available Substitutes * Time * Expenditure share ## Marginal Revenue and the Own Price Elasticity of Demand * **General Relationship Between MR and Elasticity:** $MR = P \left [ \frac{1 + E}{E} \right ]$ * **When −∞ < E < −1:** Demand is elastic, and MR > 0. * **When E = −1:** Demand is unitary elastic, and MR = 0. * **When −1 < E < 0:** Demand is inelastic, and MR < 0. **Example:** Suppose the equilibrium price in the market is \$24, and the price elasticity of demand for the linear demand function at the market equilibrium is -1.5. Calculate MR, and interpret the result. ## Cross-Price Elasticity Measures the responsiveness of the demand for a good (X) given changes in the price of a related good (Y). * **Formula:** $E_{Qx,Py} = \frac{\% \Delta Qx}{\% \Delta Py} = \frac{\frac{ \partial Qd }{\partial Py}}{\frac{ \partial Py }{\partial Qx}} \times \frac{Py}{Qx}$ **Example:** Netflix and Hulu; Pepsi and Coke; etc. * **Note:** If products are substitutes, $E_{QxPY}$ > 0 * **Note:** If products are complements, $E_{QxPY}$ < 0 ## Total Revenue of Two Products Suppose that a firm that sells two products (X and Y). * **Total Revenue:** $R_x + R_y = P_xQ_x + P_yQ_y$ **Question:** How do we measure the impact of a percent change in $P_x$ on Total Revenues? * **Formula:** $\Delta R = [R_x(1+ E_{Qx,Px}) + R_y E_{Qy,Px}] \times \% \Delta P_x$ **Example 1:** Suppose a restaurant earns \$4,000 per week in revenues from hamburger sales (X) and \$2,000 per week from soda sales (Y). If the own price elasticity for burgers is $E_{Qx,Px} = -1.5$ and the cross-price elasticity of demand between sodas and hamburgers is $E_{Qy,Px} = -4$ what would happen to the firm’s total revenues if it reduced the price of hamburgers by 1 percent? **Example 2:** You are the manager of a firm that receives revenues of \$20,000 per year from product X and \$80,000 per year from product Y. The own price elasticity of demand for product X is -3, and the cross-price elasticity of demand between product Y and X is -1.6. How much will your firm’s total revenues (revenues from both products) change if you increase the price of good X by 2 percent? **Example 3:** You are a manager in charge of monitoring cash flow at a major publisher. Paper books comprise 40 percent of your revenues, which grow about 2 percent annually. You recently received a preliminary report that suggests the growth rate in e-book reading has leveled off, and that the cross-price elasticity of demand between paper books and e-books us -0.3. In 2019, your company earned about \$600 million from sales of e-books and about \$400 million from sales of paper books. If your data analytics team estimates the own price elasticity of demand for paper books is -2, how will a 4 percent decrease in the price of paper books affect your overall revenues from both paper books and e-books sales? ## Income Elasticity Measure the responsiveness of consumer demand in given a change in income. * **Formula:** $E_{Qx,M} = \frac{\% \Delta Qx}{\% \Delta M} = \frac{\frac{ \partial Qd }{\partial M}}{\frac{ \partial M }{\partial Qx}} \times \frac{M}{Qx}$ * **Normal good:** $E_{Qx,M}$ > 0 * **Inferior good:** $E_{Qx,M}$ < 0 **Example:** Let's interpret the following table: | Good | Income Elasticity | |:--------------------------|:-------------------| | Organic Potatoes | 2.26 | | Meat | 0.11 | | Beer | -0.83 | ## Own Advertising Elasticity Own Advertising Elasticity measures how much demand of a product changes when we change the amount of advertising. * **Formula:** $E_{Qx,Ax} = \frac{\% \Delta Qx}{\% \Delta Ax}$ ## Other Elasticities Any other factor that can affect the demand for a product. ## Elasticities From a Demand Function * **Linear Demand Function:** $Q_x^d = a_o + a_xP_x + a_yP_y +a_mM + a_hH$ where H is any other factor that can affect demand. To find the elasticity, we take the derivative of the demand function with respect to the variable of interest. * **Own Price Elasticity:** $E_{Qx,Px} = \frac{\partial Qd}{\partial Px} \times \frac{Px}{Qx} = a_x \frac{Px}{Qx}$ * **Cross-Price Elasticity:** $E_{Qx,Py} = a_y \frac{Py}{Qx}$ * **Income Elasticity** $E_{Qx,M} = a_m \frac{M}{Qx}$ ## Nonlinear Demand Functions $Q_x^d = cP_x^{\beta_x}P_y^{\beta_y}M^{\beta_M}H^{\beta_H}$ If we take the natural log of this equation, we end up with a log-linear demand function: $\ln Q_x^d = \beta_o + \beta_x\ln P_x + \beta_y\ln P_y + \beta_M\ln M + \beta_H\ln H$ * **Own Price Elasticity:** $E_{Qx,Px} = \beta_x$ * **Cross-Price Elasticity:** $E_{Qx,Py} = \beta_y$ * **Income Elasticity:** $E_{Qx,M} = \beta_M$ **Note:** Because it is a log-linear function, coefficients are interpreted as elasticities! **Example:** Suppose a research report the demand for a product as: $\ln (Q) = 7 - 1.5\ln(P_x) + 5\ln(P_y) - 0.5\ln(M) - \ln(A)$ Where $P_x$ = \$15, $P_y$ = \$6, M = \$40,000, and A = \$350. Determine the cross-price elasticity of demand between good X and good Y, and state whether these two goods are complements or substitutes.