ECO101 Tutorial 02

• ECO101 - Week 02 Elasticity.pdf
• 1 - (02 Elasticity, p.9)
  • Demand is given by $D=20-2p$. Calculate the own price elasticity of demand in the range of prices $p=1$ to $p=3$. Show this calculation on a demand curve diagram.
  • $0=20-2p$
  • $2p=20$
  • $p=10$
  • For $\mathrm{\Delta }Q,\mathrm{\Delta }p$ show work.
  • ${\eta }_{xx}^d=\frac{\frac{\mathrm{\Delta }Q}{\overline{Q}}}{\frac{\mathrm{\Delta }p}{\overline{p}}}$
  • ${\eta }_{xx}^d=\frac{\frac{14-18}{16}}{\frac{2}{2}}$
  • ${\eta }_{xx}^d=-\frac{4}{16}$
  • ${\eta }_{xx}^d=|\frac{1}{4}|$
• 2 - (02 Elasticity, p.9)
  • Prices are determined by demand $D=80-2p$ and supply $S=2p-20$. Calculate the price at which demand is unit elastic. Provide a Supply and Demand Diagram to illustrate this calculation.
  • Supply
    • $0=2p-20$
    • $2p=20$
    • $p=10$
  • Demand
    • $D=80-2p$
    • $0=80-2p$
    • $2p=80$
    • $p=40$
  • When is ${\eta }_{xx}^d=1$?
  • So ${\eta }_{xx}^d=\frac{\frac{\mathrm{\Delta }Q}{\overline{Q}}}{\frac{\mathrm{\Delta }p}{\overline{p}}}$
  • Midpoint method:
    • Take your demand intercept:
    • $p=40$
    • $\frac{40}{2}=20$
    • So at $p=20$, we have unit elasticity.
    • And take the Q-int's half point.
    • $\frac{80}{2}=40$
  • Another way to calculate $\eta$ is to differentiate the functions given:
    • ${\eta }_{xx}^d=\frac{dD}{dp}\cdot \frac{p}{D}$
    • ${\eta }_{xx}^d=-2\cdot \frac{p}{80-2p}$
    • ${\eta }_{xx}^d=\left|\frac{-2p}{80-2p}\right|$
    • $1=\left|\frac{-2p}{80-2p}\right|$
    • $80-2p=2p$
    • $80=4p$
    • $20=p$

    ---

    • $D=80-2(20)$
    • $D=40$
  • Equilibrium
    • $D=S$
    • $80-2p=2p-20$
    • $100-2p=2p$
    • $100=4p$
    • $25=p$
    • $D=80-2(25)$
    • $D=30$
  • Unit Elastic happens only at the midpoint of the demand curve.
• 3 - (02 Elasticity, p.10)
  • Prices are determined by demand $D=80-p$ and supply $S=p-20$. Expenditure is at a maximum when price is equal to xxxxx. Provide a diagram.
  • $S=p-20$
  • $0=p-20$
  • $20=p$
  • Equilibrium
    • $D=S$
    • $80-p=p-20$
    • $100=2p$
    • $50=p$
    • $D=80-p$
    • $D=80-50$
    • $D=30$
  • Maximum Expenditure is also at Unit Elasticity.
    • So you can do the same method, finding the midpoint of the demand curve on both intercepts to find the unit elastic point.
  • Or solve for $\eta :1=\frac{\overline{p}}{\overline{q}}\cdot \left|\frac{1}{slope}\right|$
  • To maximize expenditure:
    • $E=p\cdot Q_d$
    • $E=p\cdot (80-p)$
    • $E=80p-p^2$
    • $E=-p^2+80p$
    • $\frac{dE}{dp}=-2p+80$
    • $0=-2p+80$
    • $-80=-2p$
    • $40=p$
    • So max expenditure is at $p=40$
    • $D=80-40$
    • $D=40$
• 4 - (02 Elasticity, p.10)
  • The demand for good $X$ depends on the price of good $Y$: $D_x=5-p_x+p_y$. The supply for good $X$ is $S_x=p_x$. The price of good $Y$ changes from $p_y=1$ to $p_y=3$. Calculate the cross-price elasticity of demand for good $X$ at the original equilibrium price. Provide a Supply and Demand Diagram to illustrate this calculation.
  • ${\eta }_{xy}^d=\frac{\frac{\mathrm{\Delta }Q}{\overline{Q}}}{\frac{\mathrm{\Delta }p}{\overline{p}}}$
  • $D_x^1=6-p_x$
    • $0=6-p_x$
    • $6=p_x$
    • $D_x^1=6-0=0$
    • Eq:
      • $p_x=6-p_x$
      • $2p_x=6$
      • $p_x=3$
      • $D_x^1=6-3$
      • $D_x^1=3$
  • $D_x^2=8-p_x$
    • $(0,8),(8,0)$
    • Eq:
      • $p_x=4$
  • ${\eta }_{xy}^d=\frac{\frac{1}{3.5}}{\frac{1}{3.5}}$
  • ${\eta }_{xy}^d=1$
  • ${\eta }_{xy}^d=\frac{\frac{5-3}{4}}{\frac{3-1}{2}}=\frac{1}{2}$