PowerPoint Presentations for Principles of Macroeconomics Sixth Canadian Edition by Mankiw/Kneebone/McKenzie Adapted for the Sixth Canadian Edition by Marc Prud’homme University of Ottawa

APPENDIX: THE MATHEMATICS OF MARKET EQUILIBRIUM Chapter 4 Copyright © 2014 by Nelson Education Ltd.4A-2

Appendix  In this appendix, simple mathematical methods are used to help solve algebraically for a market’s equilibrium price and quantity using supply and demand curves.  In Figure 4.8, we saw how the equilibrium price and quantity for a good are determined by the intersection of the supply and demand curves.  Although they don’t have to be, for simplicity, these curves are often drawn as linear (the “curves” are actually straight lines!). Copyright © 2014 by Nelson Education Ltd.4A-3

Figure 4.8 Copyright © 2014 by Nelson Education Ltd.4A-4

Appendix  The general equation for a linear demand curve is as follows:  Q D is the quantity demanded.  P is the price.  The letters a and b are referred to as demand parameters.  The parameter a can be viewed as incorporating all of the things other than the own price of the good that affect demand.  The parameter b reflects the sensitivity of demand to changes in its own price. Copyright © 2014 by Nelson Education Ltd.4A-5

Appendix  For a linear demand curve, we can determine its intercept with the price axis (the y -intercept) by setting Q D = 0 and solving the demand equation for P.  Solving for P gives P = a / b.  The intercept with the quantity axis (the x -intercept) is determined by setting P = 0.  Solving for Q D gives Q D = a. Copyright © 2014 by Nelson Education Ltd.4A-6

Appendix Copyright © 2014 by Nelson Education Ltd.  Figure 4A.1 plots the demand curve for a general linear demand curve given by the equation Q D = a - bP, identifying the x - and y -intercepts determined previously. 4A-7

Appendix  We saw in the appendix to Chapter 2 that the slope of a linear demand curve is equal to the “rise over the run” as we move along the line.  The “rise” is the change in price measured along the y -axis as we move from one point on the demand curve to another  The “run” is the change in quantity demanded measured along the x -axis.  So the slope of the demand curve is measured as, as we move from one point on the demand curve to another. Copyright © 2014 by Nelson Education Ltd.4A-8

Appendix Copyright © 2014 by Nelson Education Ltd. Using two points on the demand curve to derive the slope: Gathering and cancelling: RunRise 4A-9

Appendix Copyright © 2014 by Nelson Education Ltd. The “rise” over the “run”: 4A-10

Appendix  The general equation for a linear supply curve is as follows:  Q S is the quantity supplied.  P is the price.  The letters c and d are referred to as supply parameters.  The parameter c can be viewed as incorporating all of the things other than the own price of the good that affect supply.  The parameter d reflects the sensitivity of supply to changes in its own price. Copyright © 2014 by Nelson Education Ltd.4A-11

Appendix  For a linear supply curve, we can determine its intercept with the price axis (the y -intercept) by setting Q S = 0 and solving the demand equation for P.  Solving for P gives P = - c / d.  The intercept with the quantity axis (the x -intercept) is determined by setting P = 0.  Solving for Q S gives Q S = c. Copyright © 2014 by Nelson Education Ltd.4A-12

Appendix Copyright © 2014 by Nelson Education Ltd.  Figure 4A.1 plots the supply curve for a general linear supply curve given by the equation Q S = c + dP, identifying the x - and y -intercepts determined previously.  Because supply curves are upward sloping, they can intersect the x - or y -axis at either a positive or negative number, so the supply parameter c can be either positive or negative (although d is always positive). 4A-13

Appendix Copyright © 2014 by Nelson Education Ltd. Using two points on the supply curve to derive the slope : Gathering and cancelling: Run Rise 4A-14

Appendix Copyright © 2014 by Nelson Education Ltd. The “rise” over the “run”: 4A-15

Appendix Copyright © 2014 by Nelson Education Ltd. Equilibrium price is found by setting: And solving for P: 4A-16

Appendix Copyright © 2014 by Nelson Education Ltd.  To determine the equilibrium quantity in the market, substitute the equilibrium price into the equation for quantity demanded and do some simple algebra to get: Or: 4A-17

Appendix  For example:  The demand schedule for a good is given by:  The supply schedule is given by:  a = 20, b = 2, c = - 10, and d = 4. Copyright © 2014 by Nelson Education Ltd.4A-18

Appendix  The equilibrium price is  The equilibrium quantity is Copyright © 2014 by Nelson Education Ltd.4A-19

Appendix  Alternatively:  Set :  Solve for P : Copyright © 2014 by Nelson Education Ltd.4A-20

THE END Chapter 4 Appendix Copyright © 2014 by Nelson Education Ltd.4A-21