ECO101 Lecture 07

ECO101 - Week 07 Household Demand.pdf

Household Demand

• Utility Maximization
• Tastes are given by exponents of the level curve
• $f(x,y)=\sqrt{x}\sqrt{y}$
• Marginal Rate of Substitution is $\frac{y}{x}$
• Figure 2-8
• Explain why indifference curves are convex,
  • Due to diminishing marginal utility
• Budget Lines
  • Properties
    • Down Slope
    • Vertical Intercept is when $x=0$, then $y=\frac{m}{P_y}$
    • Horizontal Intercept when $y=0$, then $x=\frac{m}{P_x}$
  • $p_{x}x+p_{y}y=m$
  • $OCOS{T}_x=\frac{p_x}{p_y}=2$
  • Opportunity cost of $x$ to $y$ is $2$
  • For each $y$ you give up $2x$
  • Highest Affordable Indifference Curve
    • You're maximizing utility subject to what you're able to afford
    • This works for all indifference curves
    • $MRS=OCOS{T}_x$
    • $\frac{y}{x}=\frac{8}{4}$
    • $y=2x$ is our budget line
    • $8x+2(2x)=2y$
    • $x=2$
    • $y=4$
• Other indifference curves, where they're not continuous, smooth and convex
  • Perfect Substitutes
    • Completely interchangeable
    • Ex:
      • Pepsi Coke
      • Yellow or White tennis ball
    • Marginal Rate of Substitution of $1$
    • $f(x,y)=x+y$
      • $m=24$
      • $p_x=8$
      • $p_y=4$
    • $MRS=\frac{M{U}_x}{M{U}_y}=\frac{1}{1}=1$
      • Willing to give up
    • $OCOS{T}_x=\frac{8}{4}=2$
      • Have to give up
    • Then Utility Maximization happens at the 'willing to' curve
  • Perfect Complements
    • $f(x,y)=min(x,y)$
    • $y=x$
    • $8x+4y=2y$
    • $x=2$
    • $y=2$
    • $y=2$
  • Addictive Goods
    • $f(x,y)=x^2+y^2$
    • Unit Circle
  • Economic Bads
    • Something that lowers your utility
    • $f(x,y)=(16-x{)}^{\frac{1}{2}}{y}^{\frac{1}{2}}$
    • $m=32$
    • $p_y=4$
    • $p_x=-8$
    • Budget line is $4y-8x=32$

Statics of Choice

• Change:
  • Income Effect
    • Income
  • Substitution Effect
    • Price of $Y$
    • Price of $X$
• Cross Price Effects
  • $f(x,y)=\sqrt{x}\sqrt{y}$
  • $m=12$
  • $p_x=2$
  • $p_y^o=2$
  • $p_y^{\prime }=1$
  • Budget line is
  • $B{L}_0:2x+2y=12$
  • $B{L}_1:2x+1y=12$
  • Horizontal intercept is unchanged because maximum purchasing power for good $X$ is unaffected when the price of good $Y$ changes, when someone doesn't buy the other good
  • Vertical intercept goes up because it's cheaper for good $X$.
  • Slope is steeper because the OCOST of good $X$ is $1$, now it's $2$
  • A bundle with complements
  • If they're substitutes
    • If good $y$ is cheaper, more good $y$ and less good $x$
• Income Effect
  • We increase income, OCOST is the same
• Rental Plans

  • (07 Household Demand, p.7)

  • Change price and income
  • a
    • $BL:-0.2x+Y=120-20$
    • Tangent:
      • $y=0.2x$
    • Sub:
  • b
    • $BL:0.4x+y=120-40$
  • Choose Bundle A
    • It has the highest utility

Income Effect And Substitution Effect

• Own price effect
• Reduce $p_x$
• Substitution Effect
  • $SE(A\to B)$
  • $OCOS{T}_X$ goes down then buy more of good $X$
• Income Effect
  • $IE(B\to C)$
  • $p_w$ perceived wealth
  • Real Income
  • Depends on the type of the good
  • Normal
    • Buy more
  • Inferior
    • Buy less
• $$TE(A\to C)=\{\begin{array}{llll}{C}_1& \text{Normal}& IE\text{ Reinforces }SE& \text{More Good X}\\ C_2& \text{Inferior}& SE\text{ Dominates }IE& \text{More Good X}\\ C_3& \text{Inferior}& IE\text{ Dominates }SE& \text{Less Good X}\end{array}$$
• Decrease in the price of good $X$, will mean that you buy more of good $X$ if it's normal or inferior.
• Only decreases in the price of good $X$ and decrease in demands for $X$ are Giffen Goods
• Example:

  • (07 Household Demand, p.8)

  • $BL:2x+y=8$
  • Tangent $y=2x$
    • $A(2,4)$
  • $BL:0.5x+y=8$
  • Tangent $2y=x$
    • $C(8,4)$
  • Utility Condition
    • My happiness at $B$ needs to be the same as happiness as $A$
    • That's how we find point $B$
    • $\sqrt{x}\sqrt{y}=\sqrt{2}\sqrt{4}$
    • $xy=8$
    • Tangent: $2y=x$
    • $2y^2=8$
    • $y^2=4$
    • $y=2$
    • $x=4$
  • $e_b^{\prime }=0.5(4)+1(2)=4$
  • $BL:0.5x+2y=4$
  • Most this person would be willing to pay:
    • Not a member we get point $A$
    • Member, we get point $C$
    • Not a member, utility is $\sqrt{8}$
    • Max amount this person would pay is $\mathrm{\$}4$
    • Because a $\mathrm{\$}4$ membership fee, the utility at $B$ is just as good as utility at $C$
• Example:

  • (07 Household Demand, p.9)

  • Comparing alternatives
  • Toronto jeans are $80$
  • $BL:100x+2y=800$
  • $P_x:x+2y=640$
  • $p_{x}x+2y=640$
  • $xy=800$
  • $2y=p_{x}x$
  • $2y+2y=640$
  • $y=160$
  • $x(160)=800$
  • $x=5$
    • $5$ pairs of pants in buffalo
  • $2(160)=p_x(5)$
  • $p_x=64$
  • They can't be more expensive than $\mathrm{\$}64$ or else it doesn't cancel out the trip costs to shop in buffalo.

Demand Curves Embody Utility Maximizing Behaviour

• $B{L}_1:4x+2y=24$
• To find the demand curve formula
• Keep $p_x,p_y,m$ as variables
• Maximize utility
• Tangency:
  • $\frac{y}{x}=\frac{p_x}{p_y}$
  • $y=\frac{p_x}{p_y}x$
  • $p_{x}x+p_{y}y=m$
• Now sub in
  • $p_{x}x+p_y\left(\frac{p_x}{p_y}x\right)=m$
  • $p_{x}x\cancel{+p_y}\left(\frac{p_x}{\cancel{p_y}}x\right)=m$
  • $2p_{x}x=m$
  • $x=\frac{m}{2p_x}$
  • $D_x=\frac{m}{2p_x}$
• Sub in again
• How do changes in $P_y$ affect the position of the demand curve
  • Decrease in substitute, demand curve shifts left
  • Decrease in complement - Shift right
  • Price of good y is $2\to 4$
  • Nothing would happen, good $y$'s price doesn't appear in demand for $x$
• Avoiding a tax:
  • $B{L}_0$ is $\frac{1800}{20+25}$
  • how much are they willing to pay, to avoid paying the tax
  • Solution
    • Membership fee