# Lecture 5: Budget Lines

### Labour Supply

• ﻿$U=\delta Z$﻿
• B- Leisure per day
• H- Hours of work /day
• F- Dollars on food/day
• T- 24 hours per day
• W- Wage per hour
• A- Asset income per day
• ﻿$F=A+24W-WZ$﻿ ﻿$H=24-Z$﻿ ﻿$A=12$﻿ ﻿$W=1$﻿ ﻿$F=12+24-Z$﻿ (﻿$Z=0$﻿ means work 24 hours)

﻿$U=\delta Z$﻿ ﻿$MRS=\frac{MU_{3}}{MUf}= \frac{F}{Z}$﻿ ﻿$\rightarrow$﻿ How much willing to give up?

﻿$V-int=A+24W$﻿

﻿$H-int=24+\frac{A}{W}$﻿ ﻿$\rightarrow$﻿ Time value ﻿$(TV)$﻿ of A, How much ﻿$A$﻿ is worth in hours of leisure

(In kind transfer)

• ﻿$CD$﻿ -Convex ﻿$IC_{s}$﻿ , caused by diminishing ﻿$MR$﻿ .

﻿$TAN: MRS=O-Cost$﻿

﻿$\frac{E}{Z}=W$﻿ ﻿$F=WZ$﻿

If ﻿$A=12$﻿ ﻿$W=1$﻿

﻿$TAN: F=Z\left. \right \}Z=18, H=6, F=18$﻿

﻿$BL: F=36-Z\left. \right \}U=18^{2}$﻿ ### Derive Labour Supply Curve • Suppose- ﻿$W\uparrow$﻿ to ﻿$4$﻿

﻿$ES$﻿ - Employee Surplus

﻿$ES=\int_{1}^{4}12-\frac{6}{w} dw=12w-6 In w\int_{1}^{4}$﻿

﻿$=48-12-6Inn=27.68$﻿ Approx ﻿$ES$﻿ goes ﻿$A-C$﻿ but need ﻿$A-B$﻿ ﻿$W=U$﻿ Bundle B (Hicksian)

﻿$U_{a}=U_{B}-18^{2}$﻿ ﻿$FZ=18^{2}$﻿ ﻿$UZ^{2}=18^{2}$﻿ ﻿$H=24-9=15$﻿

﻿$F=UZ$﻿ ﻿$Z=\frac{18}{2}=9$﻿ ﻿$F=36$﻿

﻿$CV=?$﻿ ﻿$F=A+WH-CV$﻿

﻿$=12+U?(15)-CV$﻿

﻿$36=72-CV$﻿

﻿$CV=\ 36$﻿ - Says if we take ﻿$\ 36$﻿ and pay you ﻿$\ 4/$﻿ hour, you will be as well off.

﻿$CBL: F=12+24(U)-36-UZ$﻿

﻿$F=72-UZ$﻿ ﻿$($﻿ intercepts ﻿$(18,72)$﻿

﻿$CV=\ 36\rightarrow$﻿ Approx understates ### Compensated Labour Supply

﻿$TAN: F=wZ$﻿ ﻿$wZ^{2=18^{2}}$﻿

﻿$F_{Z}=18^{2}$﻿ ﻿$Z=\frac{18}{Uw}=18w^{\frac{1}{2}}$﻿

`﻿$H_{c}=24-Z=24-18w^{\frac{-1}{2}}$﻿

Exact ﻿$ES$﻿ ﻿$=\int_{1}^{4}24-18w^{\frac{-1}{2}}dw=24w-36w{\frac{-1}{2}}\int_{1}^{4}$﻿

﻿$=36=CV$﻿

• C is left of B because leisure is a norm good where ﻿$SE$﻿ Dominates ﻿$IE$﻿ .

﻿$W\uparrow$﻿ ﻿$SE$﻿ ﻿$O-Cost\uparrow$﻿ ﻿$Z\downarrow$﻿ ﻿$H\uparrow$﻿

﻿$IE$﻿ ﻿$RW\uparrow$﻿ ﻿$Z\uparrow$﻿ ﻿$H\downarrow$﻿

﻿$TE$﻿ ﻿$C_{1}$﻿ inf ﻿$SER$﻿ ﻿$IE$﻿ ﻿$Z\downarrow$﻿ ﻿$H\uparrow$﻿

﻿$C_{2}$﻿ norm ﻿$SE$﻿ ﻿$D$﻿ ﻿$IE$﻿ ﻿$Z\downarrow$﻿ ﻿$H\uparrow$﻿ - Always for Cob Douglas

﻿$C_{3}$﻿ nor, ﻿$IE$﻿ ﻿$D$﻿ ﻿$SE$﻿ ﻿$Z\uparrow$﻿ ﻿$H\downarrow$﻿ ### Commuting

﻿$U=\delta Z$﻿ ﻿$A=144$﻿ ﻿$W^{2}=16$﻿ ﻿$W^{1}=9$﻿

﻿$BL^{o}=F=144+16(24)-16Z$﻿

﻿$=16Z$﻿

﻿$Z=16.5$﻿ • Find ﻿$CV=\ 36\rightarrow$﻿ means ﻿$4$﻿ hours must be saved ﻿$(4-9)$﻿ ﻿$\frac{36}{9}$﻿
• Find ﻿$EV=\ 48\rightarrow$﻿ willing to spend up to 3 hours ﻿$\frac{48}{16}$﻿

### Endowments

﻿$x=2$﻿ ﻿$Y=4$﻿ ﻿$Y=U$﻿ ﻿$4=XY$﻿

﻿$Px^{0}=4$﻿ ﻿$Py=2$﻿

﻿$BL: P_{x}X+P_{y}Y=Px(2)+Py(4)$﻿

﻿$4y+2y=4(2)+2(4)=16\rightarrow$﻿ Represents ﻿$\ 16$﻿ income

﻿$4x+2y=12$﻿ ﻿$MRS=\frac{Y}{X}$﻿

﻿$TAN: Y=2X$﻿ ﻿$a.) x=2, y=4, U=8$﻿

﻿$8x=16$﻿

• Set up chose portfolio ﻿$X=X^{2}$﻿ , ﻿$Y=Y^{4}$﻿ What if ﻿$Px'=8$﻿

﻿$M = 8(2) +2(4) = 24$﻿

﻿$8x + 2y = 24$﻿

﻿$y=4x$﻿

﻿$x =$﻿ ﻿$\frac{3}{2}$﻿ ﻿$y=6$﻿ ﻿$u=9$﻿

What if ﻿$Px' =1$﻿ & ﻿$Px=8$﻿ never happened?

﻿$M= 1(2) +2(4)=10$﻿ (using endowment values)

U-int: ﻿$s$﻿ H-int ﻿$=10$﻿ In the endowment model, if you can trade to advantage of a price change, you can always make yourself better off.

﻿$*$﻿Slutsky SE ﻿$P_x = 4$﻿ to ﻿$P_x = 8$﻿ (Draws CBL through ﻿$A$﻿)

Slutsky IE ﻿$= 0$﻿. Change in endowment offsets the change in cost of buying the good.

5) Time & Money Prices:

﻿$f_x = 15$﻿min ﻿$P_x = 20$﻿ ﻿$w=20$﻿

﻿$f_y = 30$﻿min ﻿$P_y = 40$﻿ ﻿$H=120$﻿ ﻿$H + f_xY + f_yY = 24 \rightarrow$﻿ time working & time spending can't exceed 24 hours

﻿$H = 24 - f_xX - f_yY$﻿

﻿$P_xX + PyY = A + 24W - wtxX - wtyY$﻿ ﻿$[20 + \frac{1}{4} 20]x + [40 +\frac{1}{2} 20]y=120+240 \times 2 = 600$﻿

﻿$25x + 50y = 600$﻿ ﻿$\Rightarrow$﻿ Budget Line

TAN: ﻿$\frac{y}{x} = \frac{25}{50} = x = 2y$﻿

﻿$100 y = 600$﻿ ﻿$y=6$﻿ ﻿$x=12$﻿

﻿$H=24-\frac{1}{4}(12)-\frac{1}{2}(6) = 18$﻿ Hours of work

### What concepts can be tested?

1. How much would you be willing to pay to buy ﻿$x$﻿ faster?
2. Time vs Wage: when ﻿$w$﻿ is ﻿$\downarrow$﻿, you can't afford to care about time.
3. Does "tap" increase purchase?

Intertemporal - Two period model

﻿$M_1 = 84$﻿ ﻿$F_1 =$﻿ ﻿$r=25 \%$﻿ ﻿$(PV = 84+ \frac{45}{1/25} = 120$﻿﻿$)$﻿

﻿$M_2 = 45$﻿ ﻿$F_2=$﻿ ﻿$(FV = 84(1.25)+45 = 150)$﻿ Slope: 1.25 0-cost of consumption today ﻿$u = F_1F_2$﻿ ﻿$MRS = \frac{MU_1}{MU_2}$﻿ if ﻿$u$﻿ ﻿$MRS = \frac{f_2}{f_1}$﻿

TAN: ﻿$F_2 = 1.25 F_1$﻿ ﻿$\Rightarrow F_1 =60$﻿

BL: ﻿$F_1+ \frac{F_2}{1.25}=120$﻿ ﻿$\Rightarrow F_2=75$﻿

Saving: ﻿$84-60 = \ 24$﻿

Return on Saving: ﻿$75-45=30$﻿

What if ﻿$r=80 \%$﻿? o-cost ﻿$1.8$﻿ ﻿$u=60$﻿

so ﻿$F_2 = 1/8 F_1 \Rightarrow F_1=54.5$﻿ save: ﻿$84-54.5=29.5$﻿

﻿$F_2 = 1/8 F_1 + \frac{F_2}{1.8}=109 \Rightarrow F_2=98.1$﻿ ﻿$r \uparrow$﻿, saver saves more

Power

﻿$r \uparrow$﻿ Saver ﻿$(w \uparrow)$﻿

﻿$SE$﻿ ﻿$A-B$﻿ ocost ﻿$\uparrow \ F_1 \downarrow$﻿

﻿$IE$﻿ ﻿$B-C$﻿ Pwealth﻿$\uparrow \ F_1 \downarrow$﻿ inf

﻿$F_1 \uparrow$﻿ norm

﻿$IE$﻿ ﻿$c_1$﻿ inf ﻿$SE$﻿ ﻿$R$﻿ ﻿$IE$﻿ ﻿$F_1 \downarrow \ S \uparrow$﻿

﻿$c_2$﻿ norm ﻿$SE$﻿ ﻿$D$﻿ ﻿$IE$﻿ ﻿$F_1 \downarrow \ S \uparrow$﻿

﻿$c_3$﻿ norm ﻿$IE$﻿ ﻿$D$﻿ ﻿$SE$﻿ ﻿$F_1 \uparrow \ S \downarrow$﻿

﻿$S = M_1-F_1$﻿

Controlled

Borrower ﻿$(P \times \uparrow)$﻿

﻿$SE$﻿ ﻿$A-B$﻿ ocost ﻿$\uparrow$﻿ ﻿$F_1 \downarrow$﻿

﻿$IE$﻿ ﻿$B-C$﻿ Pwealth ﻿$\downarrow$﻿ ﻿$F_1 \downarrow$﻿ norm

﻿$TE$﻿ ﻿$c_1$﻿ norm ﻿$SE$﻿ ﻿$R$﻿ ﻿$IE$﻿ ﻿$F \downarrow \ B \downarrow$﻿

﻿$c_2$﻿ inf ﻿$SE$﻿ ﻿$D$﻿ ﻿$IE$﻿ ﻿$F \downarrow \ B \downarrow$﻿

﻿$c_3$﻿ inf ﻿$IE$﻿ ﻿$D$﻿ ﻿$SE$﻿ ﻿$F_1 \downarrow \ B \uparrow$﻿

A borrower will borrow less unless they have a large income effect

Bundle B (Hicksian)

﻿$F_1F_2 = 1.25 \cdot 60^2 \Rightarrow 1.8 F_1 \ ^2 = 1.25 \cdot 60^2$﻿ ﻿$F_1 = \frac{60}{1.2}=50$﻿

﻿$F_2 = 1.8 F_1 \Rightarrow \frac{1.8}{1.25}=1.44 \sqrt{1.44}=1.2$﻿ ﻿$F_2=90$﻿ ﻿$PV_B \Rightarrow F_1 + \frac{F_2}{1.8}=100$﻿ U-int: ﻿$180$﻿ Derive Supply Curves:

TAN: ﻿$F_2 = (1+r)F_1$﻿

BL: ﻿$F_1 + \frac{F_2}{1+r} = M_1 + \frac{M_2}{1+r}$﻿

Sub: ﻿$F_1 = \frac{1}{2} (M_i+ \frac{M_2}{1+r}) \Rightarrow$﻿ Consume ﻿$\frac{1}{2}$﻿ of present value income

﻿$S=M_1-F$﻿﻿$= \frac{1}{2} M_1 - \frac{1}{2} \frac{M_2}{1+r}$﻿

if ﻿$m_1 =84$﻿ ﻿$m_2 =45$﻿

﻿$S = 42 - \frac{22.5}{1+r}$﻿ ﻿$S_c$﻿: ﻿$F_2 = (1+r)F_1$﻿

﻿$u = F_1F_2 = 1.25 \times 60^2$﻿

sub: ﻿$(1+r)F_1 \ ^2 = 1.25 \times 60^2$﻿

﻿$F_1 = (1.25)^{1/2}(1+r)^{-1/2} \times 60$﻿

﻿$S_c$﻿﻿$= 60(1.25)^{1/2}(1+r) ^{-1/2}$﻿

﻿$CV = \int^{1.8}_{1.25} 60(1.25)^{1/2}(1+r)^{-1/2} \sigma (1+r)$﻿

﻿$=120(1.25)^{1/2}(1+r)^{1/2} | ^{1.8}_{1.25}$﻿

﻿$=16.2$﻿

Human Capital (more B-line) going to school

﻿$\ _2$﻿ ﻿$R=42 + 12 E^ {1/2}$﻿ ﻿$\ _1 E -$﻿Tuition Human capital production has diminishing ﻿$MP$﻿

﻿$F_1 + \frac{F_2}{Hr}=(M_1 - E)+ \frac{M_2 +R}{1+r}$﻿

2 Stages: 1) Max ﻿$PV$﻿

2) Mac ﻿$u$﻿ (subject to ﻿$B$﻿ constraint)

• Slope of ﻿$BL =$﻿ o-cost ﻿$=$﻿ ﻿$1+r$﻿
• Slope of HC (Human Capital) ﻿$MP \rightarrow$﻿ Return on education "﻿$1.75 -60$﻿," "﻿$1.00$﻿" don't go Return is compared to interest rate. ﻿$R = 42+ 12 E ^{1/2}$﻿

﻿$\frac{\sigma R}{\sigma E} = \sigma E ^ {-1/2}$﻿

﻿$\mu_1 + \frac {\mu_2 + R}{1+r} -E = PV$﻿

﻿$\sigma E ^{-1/2} =1.2 \rightarrow 5 = E ^{1/2}$﻿

﻿$E=25$﻿

﻿$R=42+60=102$﻿