Lecture 13: Market Failure, Voting, Edgeworth Box, and Efficiency in Production

Market Failure

In general when Q + Q^n and thee is an e loss

1) Public Goods - Non excludable, non rival (don't get used up), tend to be expensive (library). No one has incentive to pay due to non-excludability - Q=0Q=0. Markets fail. Non-rival-vertical sums of demand-add

Up prices willing to pay: P1=120205P_1= 120 -205

P2=40105P_2 = 40 -105

C=1052+605C=105^2 +605

4 is reservation quantity - Don't buy more than 4 at any price. Kinck out the reservation QQ of person with lowest QQ.

MCMC cuts 000 left to 444. Plug 4 into McMc. At Q=4Q=4 demand is:

pp=16030540  =160-305 -40 McMc  is 205+60205 +60 which is higher than 40.

160305=205+60s20, c=160160-305=205 +60 \Rightarrow s -20, \ c=160 \Rightarrow Who should pay?


If s=2, p1=80, p2=20s=2, \ p_1 = 80, \ p_2 =20

Person 1 =80%= 80 \%  of cost

Person 2 =20%= 20 \%  of cost $32\$ 32


Example

PA=102QP_A = 10 -2Q

PB=20QP_B = 20 -Q

PC=302QP_C = 30 -2Q

Mc=20Mc=20


Voting

i) Majority Rule - Who would win under MR?

ii) Dictatorship - Choose for everyone

iii) Borda - First Choice = 3 votes. Second = 2 votes Third = 1 votes used in San Fransisco to select mayor, Grammy

A=17A= 17 B=20B=20 C=17C= 17 \Rightarrow Be everyone's second choice

iv) Pairwise - Playoff between 2 and the wind fights the winner.

 "set up determines outcome"


2, Common Goods - Rival, non-excludable. Trouble of congestions. Each person thinks only of themselves. Go vs Car.

Jimmy Bay street bigger

r(rev)=1002n(beggers)r (rev) = 100 - 2n (beggers)

R=rn100n2n2R = rn -100n -2n^2

Optimal \Rightarrow R=1004n=0R' = 100 - 4n = 0 

n=25n = 25 


Edgeworth Box - A & EA \ \& \ E eA(4x, 2y)e_A \overset{x}{(4}, \ \overset{y}{2})  eE(2, 4)) e_E (2, \ 4))

Dimention is aggregate supply

PPR - Pareto Preferred Region - points better than e. Anything between in difference curves.

CC- contract curves. Points that are best- where you can't do better.

MRSA=MRSBMRS_A = MRS_B why? if MRSA=3 & MRSV1,MRS_A = 3 \ \& \ MRS_V -1,  person A will give 3y3y to 1x1x Person B trades: for 1. Both can be better of 6y6y trade.

Core is set of possible equilibria. Points are better & best. eC inside PPR. If both indifference are convert equilibrium is a single price.

if UA=XAYAU_A = X_A Y_A  MRSYAXAMRS \frac{Y_A}{X_A} and UE=YEYEU_E = Y_EY_E  MRS=YEXEMR_S = \frac{Y_E}{X_E}

P=PxPyP = \frac{P_x}{P_y}  TANATAN_A = YA=PxAY_A = Px_A

BLA=PxX+PyY=Px(4)+Py(2)÷PyBL_A = P_xX+P_yY = P_x(4) + P_y(2) \div P_y

=PxPyx+PyPyy=Pxy4+Pyy2= \frac{Px}{Py} x + \frac{Py}{Py}y= \frac{Px}{y} 4 + \frac {Py}{y} 2

BLApX+Y=p4+2TANAY=Px}\left.\begin{matrix} BL_A & pX+Y=p4+2 \\ TAN_A & Y=P_x \end{matrix}\right\} xE=2p+42px_E = \frac {2p+4}{2p}

xE+xA=64P+22P=6x_E +x_A = 6 \rightarrow \frac{4P+2}{2P} =6  \rightarrow 6p+c=12pp=16p+c = 12 p \Rightarrow p = 1

xA=3, xE=3, yA=3, yE=3 uA=9 uE=9x_A =3, \ x_E = 3, \ y_A = 3, \ y_E = 3 \ u_A = 9 \ u_E = 9 \Rightarrow  Trade == better

Efficient if MRSA=MRSEMRS_A = MR_{SE} \rightarrow  No remaining gains from trade


Efficiency in Production

Efficient if TRSx=TRSyTRS_x = TRS_y

TRS=MPLMPKTRS = \frac{MP_L}{MP_K} Isoquants are tangent

At 0, TRSx>TRSy0, \ TRS_x > TRS_y


Consistency in xx, y, \ y Mix

A, B & C are all efficient but which is best?

PPBoundary slope MRT=MCxMCy MRT = \frac{MCx}{MCy} 

Marginal rate of transformation how much yy must five for nxnx

Concave because decreasing returns to scale

MCsMCs are upward sloping

At A, MRTA, \ MRT small because MCxMCx is small, MCyMCy is large.

At C, MRTC, \ MRT large. At BB MRT=MRsMRT =MRs \Rightarrow MRTMRT must MRSMRS willing

DD  is below the curves because it is not efficient


If MRS>MRTMRS >MRT, move right

If MRS<MRTMRS < MRT, move left


1)MRSA=MRSE1) MRS_A =MRS_E  2)TRSA=TRSY2) TRS_A =TRS_Y 3)MRT=MRS3) MRT =MRS

\Rightarrow  For an economy to be efficient


Achieve

1) Very smart economist makes all decisions

Central Planning - Russia, China. Only work on a small scale.

2) Markets take care. MRSA=PxPy MRS_A = \frac{P_x}{P_y} MRSE=PxPyMRS_E = \frac{P_x}{P_y} both people set tangency, to same prices. TRSx=wrTRS_x = \frac{w}{r}  TRSy=wrTRS_y = \frac{w}{r}  if firms face same wages an interest, efficiency is automatic. MRT=MCxMCyG1MRT = \frac{MC_x}{MC_y} G_1 max Px=MCyxP_x = MCy \rightarrow x  Py=MCyyP_y = MCy \rightarrow y 

MCxMCy=PxPy\therefore \frac{MCx}{MCy} = \frac{Px}{Py} Profix max firms, firms with competitive prices are efficient.


But Markets Fail!

Public & common goods & monopoly \rightarrow  government intervention

Monopoly

n=1n=1 x1x_1 yy is comp.

MRy=MCxMRy = MCx MRx<PxMRx < P_x so MRT=McxMcy<PxPyMRT = \frac{Mcx}{Mcy} < \frac{Px}{Py}

BB is better but monopoly won't go there. Markets fail.


Externalities

Should be MRT=MSCxMSCyMRT = \frac{MSCx}{MSCy}

MSCxMSCy>PxPy\frac{MSCx}{MSCy} > \frac{P_x}{P_y} Because social cost is higher than price!

1) Solution

2) Merger

3) Coase Theorem

Assign property rights. Jia Jia has a right to clean air. Bailey will pay Jia Jia for the smoking damage. Bargaining must be possible.

++ve ++ve: Apples & bees

++ve -ve: He is bad for you but you are good for him

-ve -ve: Bad for each other


New York Fries & Wendy's

NYF: G1y=(40+2x)yy2G_{1_y} = (40 +2x)y-y^2 δπyy=40+2x2y=0\frac{\delta \pi y}{y} = 40 + 2x-2y = 0

W: G1y=(80y)xx2G_{1_y} = (80 -y)x-x^2 δG1xxx=80y2x=0\frac{\delta G_{1_x} x}{x} = 80 - y-2x= 0

G1x=10×552=25G_{1_x} = 10 \times 5 -5^2 = 25

G1y=50×70702=1400G_{1_y} = 50 \times 70 -70^2 = -1400

\Rightarrow  Private solution

G1=(40+2x)yy2+(80y)xx2G_1 = (40+2x)y-y^2+(80-y)x-x^2 δG1x=2y+80y2x=0\frac{\delta G_1}{x}= 2y+80−y−2x=0

x=2003y=1603}\left.\begin{matrix} x = \frac{200}{3} \\ y = \frac{160}{3} \end{matrix}\right\} Ideal! δG1y=40+2x2yx=0\frac{\delta G_1}{y} = 40 +2x -2y - x = 0 

G1xsub=(80y)xx2+5xG1xTax=(40+2x)yy2ty}δg1xx=80y2x+5=0δG1yy=40+2x2yt=0 \left.\begin{matrix} G_{1_x}^{sub} = (80-y)x - x^2+5x \\ G_{1_x}^{Tax} = (40+2x)y - y^2-ty \end{matrix}\right\} \begin{matrix} \frac{\delta g_{1_x}}{x} = 80-y -2x + 5 =0 \\ \frac{\delta G_{1_y}}{y} = 40 +2x-2y-t=0 \end{matrix}

S=39203S= \frac{3920}{3} T=38003T= \frac{3800}{3}


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