# 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 - ﻿$Q=0$﻿. Markets fail. Non-rival-vertical sums of demand-add

Up prices willing to pay: ﻿$P_1= 120 -205$﻿

﻿$P_2 = 40 -105$﻿

﻿$C=105^2 +605$﻿ 4 is reservation quantity - Don't buy more than 4 at any price. Kinck out the reservation ﻿$Q$﻿ of person with lowest ﻿$Q$﻿.

﻿$MC$﻿ cuts 0﻿$0$﻿ left to 4﻿$4$﻿. Plug 4 into ﻿$Mc$﻿. At ﻿$Q=4$﻿ demand is:

﻿$p$﻿﻿$﻿ =160-305 -40$﻿ ﻿$Mc$﻿ is ﻿$205 +60$﻿ which is higher than 40.

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

If ﻿$s=2, \ p_1 = 80, \ p_2 =20$﻿

Person 1 ﻿$= 80 \%$﻿ of cost

Person 2 ﻿$= 20 \%$﻿ of cost ﻿$\ 32$﻿

#### Example

﻿$P_A = 10 -2Q$﻿

﻿$P_B = 20 -Q$﻿

﻿$P_C = 30 -2Q$﻿

﻿$Mc=20$﻿

#### 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

﻿$A= 17$﻿ ﻿$B=20$﻿ ﻿$C= 17 \Rightarrow$﻿ 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 ﻿$r (rev) = 100 - 2n (beggers)$﻿

﻿$R = rn -100n -2n^2$﻿

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

﻿$n = 25$﻿

Edgeworth Box - ﻿$A \ \& \ E$﻿ ﻿$e_A \overset{x}{(4}, \ \overset{y}{2})$﻿ ﻿$e_E (2, \ 4))$﻿ 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.

﻿$MRS_A = MRS_B$﻿ why? if ﻿$MRS_A = 3 \ \& \ MRS_V -1,$﻿ person A will give ﻿$3y$﻿ to ﻿$1x$﻿ Person B trades: for 1. Both can be better of ﻿$6y$﻿ 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 ﻿$U_A = X_A Y_A$﻿ ﻿$MRS \frac{Y_A}{X_A}$﻿ and ﻿$U_E = Y_EY_E$﻿ ﻿$MR_S = \frac{Y_E}{X_E}$﻿

﻿$P = \frac{P_x}{P_y}$﻿ ﻿$TAN_A$﻿ = ﻿$Y_A = Px_A$﻿

﻿$BL_A = P_xX+P_yY = P_x(4) + P_y(2) \div P_y$﻿

﻿$= \frac{Px}{Py} x + \frac{Py}{Py}y= \frac{Px}{y} 4 + \frac {Py}{y} 2$﻿

﻿$\left.\begin{matrix} BL_A & pX+Y=p4+2 \\ TAN_A & Y=P_x \end{matrix}\right\}$﻿ ﻿$x_E = \frac {2p+4}{2p}$﻿

﻿$x_E +x_A = 6 \rightarrow \frac{4P+2}{2P} =6$﻿ ﻿$\rightarrow$﻿ ﻿$6p+c = 12 p \Rightarrow p = 1$﻿

﻿$x_A =3, \ x_E = 3, \ y_A = 3, \ y_E = 3 \ u_A = 9 \ u_E = 9 \Rightarrow$﻿ Trade ﻿$=$﻿ better

Efficient if ﻿$MRS_A = MR_{SE} \rightarrow$﻿ No remaining gains from trade

### Efficiency in Production Efficient if ﻿$TRS_x = TRS_y$﻿

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

At ﻿$0, \ TRS_x > TRS_y$﻿

### Consistency in ﻿$x$﻿﻿$, \ y$﻿ Mix

A, B & C are all efficient but which is best? PPBoundary slope ﻿$﻿MRT = \frac{MCx}{MCy}$﻿

Marginal rate of transformation how much ﻿$y$﻿ must five for ﻿$nx$﻿

Concave because decreasing returns to scale

﻿$MCs$﻿ are upward sloping

At ﻿$A, \ MRT$﻿ small because ﻿$MCx$﻿ is small, ﻿$MCy$﻿ is large.

At ﻿$C, \ MRT$﻿ large. At ﻿$B$﻿ ﻿$MRT =MRs \Rightarrow$﻿ ﻿$MRT$﻿ must ﻿$MRS$﻿ willing

﻿$D$﻿ is below the curves because it is not efficient

If ﻿$MRS >MRT$﻿, move right

If ﻿$MRS < MRT$﻿, move left

﻿$1) MRS_A =MRS_E$﻿ ﻿$2) TRS_A =TRS_Y$﻿ ﻿$3) MRT =MRS$﻿

﻿$\Rightarrow$﻿ 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. ﻿$MRS_A = \frac{P_x}{P_y}$﻿ ﻿$MRS_E = \frac{P_x}{P_y}$﻿ both people set tangency, to same prices. ﻿$TRS_x = \frac{w}{r}$﻿ ﻿$TRS_y = \frac{w}{r}$﻿ if firms face same wages an interest, efficiency is automatic. ﻿$MRT = \frac{MC_x}{MC_y} G_1$﻿ max ﻿$P_x = MCy \rightarrow x$﻿ ﻿$P_y = MCy \rightarrow y$﻿

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

### But Markets Fail!

Public & common goods & monopoly ﻿$\rightarrow$﻿ government intervention

#### Monopoly

﻿$n=1$﻿ ﻿$x_1$﻿ ﻿$y$﻿ is comp.

﻿$MRy = MCx$﻿ ﻿$MRx < P_x$﻿ so ﻿$MRT = \frac{Mcx}{Mcy} < \frac{Px}{Py}$﻿ ﻿$B$﻿ is better but monopoly won't go there. Markets fail.

### Externalities

Should be ﻿$MRT = \frac{MSCx}{MSCy}$﻿

﻿$\frac{MSCx}{MSCy} > \frac{P_x}{P_y}$﻿ 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: ﻿$G_{1_y} = (40 +2x)y-y^2$﻿ ﻿$\frac{\delta \pi y}{y} = 40 + 2x-2y = 0$﻿

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

﻿$G_{1_x} = 10 \times 5 -5^2 = 25$﻿

﻿$G_{1_y} = 50 \times 70 -70^2 = -1400$﻿

﻿$\Rightarrow$﻿ Private solution

﻿$G_1 = (40+2x)y-y^2+(80-y)x-x^2$﻿ ﻿$\frac{\delta G_1}{x}= 2y+80−y−2x=0$﻿

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

﻿$\left.\begin{matrix} G_{1_x}^{sub} = (80-y)x - x^2+5x \\ G_{1_x}^{Tax} = (40+2x)y - y^2-ty \end{matrix}\right\} \begin{matrix} \frac{\delta g_{1_x}}{x} = 80-y -2x + 5 =0 \\ \frac{\delta G_{1_y}}{y} = 40 +2x-2y-t=0 \end{matrix}$﻿

﻿$S= \frac{3920}{3}$﻿ ﻿$T= \frac{3800}{3}$﻿