ECO101 Week 8 Review

ECO101 Lecture: ECO101 Lecture 08

ECO101 - Week 08 Household Supply.pdf

• a: theory of labour supply
  • 1:
    • a:
      • Find the budget line
      • $f=A+wh$
      • $h+z=24$
      • $h=24-z$
      • $f=A+w(24-z)$
      • $f=A+24w-wz$
    • b:
      • Budget line at $w=1$ and $A=16$
      • $f(z)=-z+40$
      • If you take no leisure per day, you can make 40 a day.
      • If you take 24 hours of leisure per day, you can make 16 a day which is $A$, asset income
      • After that, it violates our constraint curve of $h+z=24$.
    • c:
      • If we have two inputs of leisure and food as: $u(z,f)=\sqrt{z}\sqrt{f}$
      • We get $MRS=\frac{\frac{\partial u}{\partial z}}{\frac{\partial u}{\partial f}}=\frac{M{U}_z}{M{U}_f}=\frac{f}{z}$
  • 2:
    • $u(z,f)=\sqrt{z}\sqrt{f}$
    • $w=1$ per hour
    • $A=16$
    • Budget Line:
      • $f(z)=A+24w-wz$
      • $f(z)=16+24-z$
      • $f(z)=-z+40$
    • Indifference Curve:
      • $MRS=OCOS{T}_Z$
      • $\frac{f}{z}=\frac{1}{1}$
      • $f=z$
      • $-z+40$
• b: Comparing employment alternatives
  • 3:
    • $u(z,f)=\sqrt{z}\sqrt{f}$
    • $w_0=4$
    • $A_0=0$
    • $w_1=1$
    • $A_1=16$
    • Case 1: $w_0,A_0$
      • $f(z)=A+24w-wz$
      • $f(z)=24(4)-(4)z$
      • $f(z)=96-4z$
      • $MRS=OCOS{T}_Z$
      • $\frac{f}{z}=w$
      • $f=wz$
      • $f=4z$
      • $4z=96-4z$
      • $8z=96$
        $z=12$
      • At $(12,48)$ we maximize utility
      • $\sqrt{12}\sqrt{48}=24$
    • Case 2: $w_1,A_1$
      • $f(z)=A+24w-wz$
      • $f(z)=40-z$
      • $MRS=OCOS{T}_Z$
      • $\frac{f}{z}=w$
      • $f=wz$
      • $f=z$
      • $z=40-z$
        $2z=40$
        $z=20⟹f=20$
      • $(20,20)$ we maximize utility
      • $\sqrt{20}\sqrt{20}=20$
    • The first option maximizes utility.
• c: Income Effect and Substitution Effect
  • 4:
    • $u(f,z)=fz$
    • $w_0=2.25$
    • $w_1=4$
    • $A=36$
    • Case 1: $w_0$
      • $f(z)=A+24w-wz$
      • $f(z)=36+24(2.25)-2.25z$
      • $f(z)=90-2.25z$
      • $MRS=OCOS{T}_z$
      • $\frac{f}{z}=w$
      • $f=wz$
      • $f=2.25z$
      • $2.25z=90-2.25z$
      • $4.5z=90$
      • $z=20⟹f=45$
      • Utility: $u=900$
    • Case 2: $w_1$
      • $f(z)=A+24w-wz$
      • $f(z)=36+24(4)-4z$
      • $f(z)=132-4z$
      • $MRS=OCOS{T}_z$
      • $f=wz$
      • $f=4z$
      • $8z=132$
      • $z=16.5⟹f=66$
      • Utility: $u=1089$
    • b:
      • What is the max amount she would pay to join the union?
      • $p_f$ as fee
      • $f(z)=132-4z$
      • $MRS=OCOS{T}_z$
      • Income is $132-p_f$
      • $fz=900$
      • $f=4z$
      • $4z^2=900$
      • $z^2=225$
      • $z=15⟹f=60$
      • $f+wz=132-p_f$
      • $60+4(15)=132-p_f$
      • $120=132-p_f$
      • $12=p_f$
      • $12$ dollars
• d: The Labour Supply Curve
  • 5:
    • Derive labour supply curve for $u(z,f)=\sqrt{z}\sqrt{f}$
    • $A=36$
    • $f(z)=A+24w-wz$
    • $f(z)=36+24w-wz$
    • $\frac{f}{z}=w$
    • $wz=36+24w-wz$
    • $2wz=36+24w$
    • $z=\frac{36}{2w}+12$
    • $h+z=24$
    • $z=24-h$
    • $24-h=\frac{36}{2w}+12$
    • $-h=\frac{36}{2w}-12$
    • $h=-\frac{36}{2w}+12$
    • $0=-\frac{36}{2w}+12$
    • $12=\frac{36}{2w}$
    • $24w=36$
    • $w=1.5$
    • Reservation Wage
    • It may bend backwards because eventually you get diminishing marginal utility. It's not worth it to work as much when you already have so much
• e: Households are savers in capital markets

  • (08 Household Supply, p.6)

  • Mavis who has preferences u = F11/2F21/2 will earn m1 = 240 in period 1 and will be retired m2 = 0 in period 2. The interest rate is 25%. Provide an Indifference Curve diagram to illustrate her optimal choices for consumption. A mutual fund offers savers an interest rate of 80% but there is a fee to invest in this fund. Show on your diagram the maximum fee that Mavis would be willing to pay

  • $u(f_1,f_2)=\sqrt{f_1}\sqrt{f_2}$
  • $m_1=240$ in period 1
  • $m_2=0$ in period 2
  • $r_0=0.25$
  • $r_1=0.8$
  • Bundle A
    • Tangency Condition
      • $MRS=OCOS{T}_{f_2}$
      • $\frac{f_2}{f_1}=1+r$
      • $\frac{f_2}{f_1}=1.25$
      • $f_2=1.25f_1$
    • Budget Line
      • $f_1+\frac{1}{1+r}{f}_2=m_1$
      • $f_1+\frac{1}{1.25}{f}_2=240$
      • $f_1+0.8f_2=240$
      • $f_1+0.8(1.25f_1)=240$
      • $2f_1=240$
      • $f_1=120⟹f_2=150$
    • $u(120,150)=\sqrt{18000}$
  • Bundle C
    • Tangency Condition
      • $MRS=OCOS{T}_{f_2}$
      • $\frac{f_2}{f_1}=1+r$
      • $\frac{f_2}{f_1}=1.25$
      • $f_2=1.8f_1$
    • Utility
      • $f_1{f}_2=18000$
      • $1.8f_1^2=18000$
      • $f_1^2=10000$
      • $f_1=100⟹f_2=180$
      • $u(100,180)=\sqrt{18000}$
    • Cost
      • $f_1+\frac{1}{1+r}{f}_2=240$
      • $100+\frac{1}{1.8}180=240$
      • $200=240$
      • $40$ is the maximum fee
  • Graph:

        left=-10; right=300;
        top=350; bottom=-10;
        ---
        xy=18000
        x+0.8y=240
        (100,180)|label:C
        (120,150)|label:A
        y=1.25x|dashed
        y=1.8x|dashed
    

ECO101 Tutorial: ECO101 Tutorial 08

• 1
  • Utility Maximization
  • $u(f_1,f_2)=\sqrt{f_1}\sqrt{f_2}$
  • $m_0=80$
  • $m_1=50$
  • $r=0.25$
  • How much will she save
  • Bundle A
    • Tangency Condition
      • $\frac{f_2}{f_1}=1+r$
      • $f_2=1.25f_1$
    • Budget Line
      • $f_1+\frac{1}{1+r}{f}_2=m$
      • $f_1+0.8f_2=120$
      • $2f_1=120$
      • $f_1=60⟹f_2=75$
    • Utility
      • $u(60,75)=\sqrt{4500}$
  • Bundle C
    • Same as A
    • Savings
    • $m_0-f_1$
    • $80-60=20$ dollars saved
• 2
  • $u(z,f)=\sqrt{z}\sqrt{f}$
  • $z+h=24$
  • $f(z)=A+24w-wz$
  • $f(z)=60+24(5)-5z$
  • $f(z)=180-5z$
  • Bundle A
    • Tangency Condition
      • $MRS=OCOS{T}_z$
      • $\frac{f}{z}=w$
      • $f=wz$
      • $f=5z$
    • Budget Line
      • $f(z)=180-5z$
      • $5z=180-5z$
      • $10z=180$
      • $z=18⟹f=90$
    • Utility
      • $u(18,90)=\sqrt{1620}$
  • Spend a maximum of 90 on food
  • Graph:

        left=-10; right=100;
        top=200; bottom=-10;
        ---
        f(x)=180-5x
        xy=1620
        (18,90)|label:A
    

• 3
  • Utility Maximizing
  • $u(c_1,c_2)\sqrt{c_1}\sqrt{c_2}$
  • $m_0=60$
  • $m_1=40$
  • How many dollars will she save at $r=0.25$
  • Bundle A
    • Tangency Condition
      • $\frac{c_2}{c_1}=1+r$
      • $c_2=1.25c_1$
    • Budget Line
      • $c_1+\frac{1}{1+r}{c}_2=m_0+\frac{1}{1+r}{m}_1$
      • $c_1+0.8c_2=92$
      • $2c_1=92$
      • $c_1=46⟹c_2=57.5$
    • Utility
      • $u(46,57.5)=\sqrt{2645}$
  • Bundle C
    • Same as A
  • Bundle B
    • $c_1{c}_2=2645$
    • $1.25c_1^2=2645$
    • $c_1^2=2116$
    • $c_1=46⟹c_2=57.5$
  • Savings: $s=m_0-c_1$
    • $s=60-46=14$ dollars

        left=-10; right=100;
        top=130; bottom=-10;
        ---
        x+0.8y=92
        xy=2645
    	(46,57.5)|label:A
    	y=1.25x|dashed
    

• 4
  • Maximize
  • $u(z,f)=\sqrt{z}\sqrt{f}$
  • $z+h=24$
  • $A=6$
  • $w=0.5$
  • Tangency Condition
    • $MRS=OCOS{T}_z$
    • $\frac{f}{z}=w$
    • $f=wz$
    • $f=0.5z$
  • Budget Line
    • $f(z)=A+24w-wz$
    • $f(z)=6+24(0.5)-(0.5)z$
    • $f(z)=18-0.5z$
    • $z=18⟹f=9$
  • $z=18⟹h=6$
  • Utility $u(18,9)=\sqrt{162}$

        left=-0.5; right=40;
        top=20; bottom=-0.5;
        ---
        f(x)=18-0.5x
        xy=162
        (18,9)|label:A
        x<=24
    

ECO101 - Week 08 Household Supply.pdf

• 1
  • Utility maximizing
  • $u(f_1,f_2)=\sqrt{f_1}\sqrt{f_2}$
  • $m_1=80$
  • $m_2=50$
  • $r=0.25$
  • Bundle A: $m_1$
    • Tangency
      • $\frac{f_2}{f_1}=1+r$
      • $f_2=1.25f_1$
    • Budget Line
      • $f_1+\frac{1}{1+r}{f}_2=m_1+\frac{1}{1+r}{m}_2$
      • $f_1+0.8f_2=120$
      • $2f_1=120$
      • $f_1=60⟹f_2=75$
    • Utility
      • $u(60,75)=\sqrt{4500}$
  • Bundle C: $m_2$
    • $S=m_0-f_1$
    • $S=80-60=20$ dollars saved

ECO101 Applications: ECO101 Applications 8