Lecture 5: Budget Lines

Labour Supply

  • U=δZU=\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+24WWZF=A+24W-WZ H=24ZH=24-Z

A=12A=12 W=1W=1 F=12+24ZF=12+24-Z (Z=0Z=0 means work 24 hours)

U=δZU=\delta Z MRS=MU3MUf=FZMRS=\frac{MU_{3}}{MUf}= \frac{F}{Z} \rightarrow  How much willing to give up?

Vint=A+24WV-int=A+24W

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

(In kind transfer)

  • CDCD -Convex ICsIC_{s} , caused by diminishing MRMR .

TAN:MRS=OCostTAN: MRS=O-Cost

EZ=W\frac{E}{Z}=W F=WZF=WZ

If A=12A=12 W=1W=1

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

BL:F=36Z}U=182BL: F=36-Z\left. \right \}U=18^{2}

Derive Labour Supply Curve

  • Suppose- WW\uparrow to 44

ESES - Employee Surplus

ES=14126wdw=12w6Inw14ES=\int_{1}^{4}12-\frac{6}{w} dw=12w-6 In w\int_{1}^{4}

=48126Inn=27.68=48-12-6Inn=27.68 Approx ESES goes ACA-C but need ABA-B


W=UW=U  Bundle B (Hicksian)

Ua=UB182U_{a}=U_{B}-18^{2} FZ=182FZ=18^{2} UZ2=182UZ^{2}=18^{2} H=249=15H=24-9=15

F=UZF=UZ Z=182=9Z=\frac{18}{2}=9 F=36F=36

CV=?CV=?  F=A+WHCVF=A+WH-CV

=12+U?(15)CV=12+U?(15)-CV

36=72CV36=72-CV

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

CBL:F=12+24(U)36UZCBL: F=12+24(U)-36-UZ

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

CV=$36CV=\$ 36\rightarrow  Approx understates

Compensated Labour Supply

TAN:F=wZTAN: F=wZ wZ2=182wZ^{2=18^{2}}

FZ=182F_{Z}=18^{2} Z=18Uw=18w12Z=\frac{18}{Uw}=18w^{\frac{1}{2}}

`Hc=24Z=2418w12H_{c}=24-Z=24-18w^{\frac{-1}{2}}

Exact ESES =142418w12dw=24w36w1214=\int_{1}^{4}24-18w^{\frac{-1}{2}}dw=24w-36w{\frac{-1}{2}}\int_{1}^{4}

=36=CV=36=CV

  • C is left of B because leisure is a norm good where SESE Dominates IEIE .

WW\uparrow  SESE OCostO-Cost\uparrow ZZ\downarrow  HH\uparrow

IEIE  RWRW\uparrow ZZ\uparrow HH\downarrow 

TETE C1C_{1} inf SERSER IEIE  ZZ\downarrow HH\uparrow

C2C_{2} norm SESE  DD IEIE ZZ\downarrow HH\uparrow - Always for Cob Douglas

C3C_{3} nor, IEIE  DD SESE Z Z\uparrow HH\downarrow 

Commuting

U=δZU=\delta Z A=144A=144 W2=16W^{2}=16 W1=9W^{1}=9

BLo=F=144+16(24)16ZBL^{o}=F=144+16(24)-16Z

=16Z=16Z

Z=16.5Z=16.5

  • Find CV=$36CV=\$ 36\rightarrow  means 44 hours must be saved (49)(4-9) 369\frac{36}{9}
  • Find EV=$48EV=\$ 48\rightarrow  willing to spend up to 3 hours 4816\frac{48}{16}

Endowments

x=2x=2 Y=4Y=4 Y=UY=U 4=XY4=XY

Px0=4Px^{0}=4 Py=2Py=2

BL:PxX+PyY=Px(2)+Py(4)BL: P_{x}X+P_{y}Y=Px(2)+Py(4)

4y+2y=4(2)+2(4)=164y+2y=4(2)+2(4)=16\rightarrow  Represents $16\$ 16 income

4x+2y=124x+2y=12 MRS=YXMRS=\frac{Y}{X}

TAN:Y=2XTAN: Y=2X a.)x=2,y=4,U=8a.) x=2, y=4, U=8

8x=168x=16

  • Set up chose portfolio X=X2X=X^{2} , Y=Y4Y=Y^{4}

What if Px=8Px'=8

M=8(2)+2(4)=24M = 8(2) +2(4) = 24

8x+2y=248x + 2y = 24

y=4xy=4x

x=x =  32\frac{3}{2}  y=6y=6 u=9u=9


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

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

U-int: ss H-int =10=10

In the endowment model, if you can trade to advantage of a price change, you can always make yourself better off.

*Slutsky SE Px=4P_x = 4 to Px=8P_x = 8 (Draws CBL through AA)

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


5) Time & Money Prices:

fx=15f_x = 15min Px=20P_x = 20 w=20w=20

fy=30f_y = 30 min Py=40P_y = 40 H=120H=120

H+fxY+fyY=24H + f_xY + f_yY = 24 \rightarrow  time working & time spending can't exceed 24 hours

H=24fxXfyYH = 24 - f_xX - f_yY

PxX+PyY=A+24WwtxXwtyYP_xX + PyY = A + 24W - wtxX - wtyY

[20+1420]x+[40+1220]y=120+240×2=600[20 + \frac{1}{4} 20]x + [40 +\frac{1}{2} 20]y=120+240 \times 2 = 600

25x+50y=60025x + 50y = 600 \Rightarrow Budget Line

TAN: yx=2550=x=2y\frac{y}{x} = \frac{25}{50} = x = 2y

100y=600100 y = 600 y=6y=6 x=12x=12

H=2414(12)12(6)=18H=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 xx faster?
  2. Time vs Wage: when ww is \downarrow, you can't afford to care about time.
  3. Does "tap" increase purchase?


Intertemporal - Two period model

M1=84M_1 = 84  F1=F_1 = r=25%r=25 \% (PV=84+451/25=120(PV = 84+ \frac{45}{1/25} = 120))

M2=45M_2 = 45 F2=F_2= (FV=84(1.25)+45=150)(FV = 84(1.25)+45 = 150)

Slope: 1.25 0-cost of consumption today

u=F1F2u = F_1F_2 MRS=MU1MU2MRS = \frac{MU_1}{MU_2} if uu MRS=f2f1MRS = \frac{f_2}{f_1}

TAN: F2=1.25F1F_2 = 1.25 F_1 F1=60\Rightarrow F_1 =60

BL: F1+F21.25=120F_1+ \frac{F_2}{1.25}=120 F2=75\Rightarrow F_2=75

Saving: 8460=$2484-60 = \$ 24

Return on Saving: 7545=3075-45=30

What if r=80%r=80 \%? o-cost 1.81.8 u=60u=60

so F2=1/8F1F1=54.5F_2 = 1/8 F_1 \Rightarrow F_1=54.5  save: 8454.5=29.584-54.5=29.5

F2=1/8F1+F21.8=109F2=98.1F_2 = 1/8 F_1 + \frac{F_2}{1.8}=109 \Rightarrow F_2=98.1  rr \uparrow, saver saves more


Power

rr \uparrow Saver (w)(w \uparrow)

SESE ABA-B ocost  F1\uparrow \ F_1 \downarrow

IEIE BCB-C Pwealth F1\uparrow \ F_1 \downarrow  inf

F1F_1 \uparrow norm

IEIE c1c_1 inf SESE RR IEIE F1 SF_1 \downarrow \ S \uparrow

c2c_2 norm SESE DD IEIE F1 SF_1 \downarrow \ S \uparrow

c3c_3 norm IEIE DD SESE F1 SF_1 \uparrow \ S \downarrow

S=M1F1S = M_1-F_1


Controlled

Borrower (P×)(P \times \uparrow)

SESE ABA-B ocost \uparrow F1F_1 \downarrow

IEIE BCB-C Pwealth \downarrow F1F_1 \downarrow norm

TETE c1c_1 norm SESE RR IEIE F BF \downarrow \ B \downarrow

c2c_2 inf SESE DD IEIE F BF \downarrow \ B \downarrow

c3c_3 inf IEIE DD SESE F1 BF_1 \downarrow \ B \uparrow

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


Bundle B (Hicksian)

F1F2=1.256021.8F1 2=1.25602F_1F_2 = 1.25 \cdot 60^2 \Rightarrow 1.8 F_1 \ ^2 = 1.25 \cdot 60^2 F1=601.2=50F_1 = \frac{60}{1.2}=50

F2=1.8F11.81.25=1.441.44=1.2F_2 = 1.8 F_1 \Rightarrow \frac{1.8}{1.25}=1.44 \sqrt{1.44}=1.2 F2=90F_2=90

PVBF1+F21.8=100PV_B \Rightarrow F_1 + \frac{F_2}{1.8}=100 U-int: 180180

Derive Supply Curves:

TAN: F2=(1+r)F1F_2 = (1+r)F_1

BL: F1+F21+r=M1+M21+rF_1 + \frac{F_2}{1+r} = M_1 + \frac{M_2}{1+r}

Sub: F1=12(Mi+M21+r)F_1 = \frac{1}{2} (M_i+ \frac{M_2}{1+r}) \Rightarrow Consume 12\frac{1}{2}  of present value income

S=M1FS=M_1-F =12M112M21+r= \frac{1}{2} M_1 - \frac{1}{2} \frac{M_2}{1+r}

if m1=84m_1 =84 m2=45m_2 =45

S=4222.51+rS = 42 - \frac{22.5}{1+r}


ScS_c: F2=(1+r)F1F_2 = (1+r)F_1

u=F1F2=1.25×602u = F_1F_2 = 1.25 \times 60^2

sub: (1+r)F1 2=1.25×602(1+r)F_1 \ ^2 = 1.25 \times 60^2

F1=(1.25)1/2(1+r)1/2×60F_1 = (1.25)^{1/2}(1+r)^{-1/2} \times 60

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

CV=1.251.860(1.25)1/2(1+r)1/2σ(1+r)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)1/21.251.8=120(1.25)^{1/2}(1+r)^{1/2} | ^{1.8}_{1.25}

=16.2=16.2


Human Capital (more B-line) going to school

$2\$ _2  R=42+12E1/2R=42 + 12 E^ {1/2} $1E\$ _1 E -Tuition

Human capital production has diminishing MPMP

F1+F2Hr=(M1E)+M2+R1+rF_1 + \frac{F_2}{Hr}=(M_1 - E)+ \frac{M_2 +R}{1+r}

2 Stages: 1) Max PVPV

2) Mac uu (subject to BB constraint)


  • Slope of BL=BL = o-cost == 1+r1+r
  • Slope of HC (Human Capital) MPMP \rightarrow Return on education "1.75601.75 -60," "1.001.00" don't go Return is compared to interest rate.

R=42+12E1/2R = 42+ 12 E ^{1/2}

σRσE=σE1/2\frac{\sigma R}{\sigma E} = \sigma E ^ {-1/2}

μ1+μ2+R1+rE=PV\mu_1 + \frac {\mu_2 + R}{1+r} -E = PV

σE1/2=1.25=E1/2\sigma E ^{-1/2} =1.2 \rightarrow 5 = E ^{1/2}

E=25E=25

R=42+60=102R=42+60=102



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