Welfare Analysis
of Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 22

The Derivation I Wanted To Do Last Time

Is this the “right" price and quantity?

If you were an omniscient social planner, could you do "better"
than the price the market "chooses"?

Welfare Analysis:
Consumer and Producer Surplus

Last time, we calculated the equilibrium price and quantity.

The Social Planner's Problem

Suppose you were in charge of the economy.

How would you answer the fundamental economic questions about a particular good?

How to produce it?

Want to produce any given quantity Q
at the lowest possible cost

Who gets to consume it?

How much to produce?

Want to distribute any given quantity Q
to the people who value it the most

Want to choose the quantity Q*
to maximize total surplus 
(benefit to consumers minus costs of production)

Example: One Consumer, One Firm

FIRM

CONSUMER

Quasilinear utility function:

u(x_1,x_2) = 10x_1 - {1 \over 2}x_1^2 + x_2

Good 2 is "dollars spent on other goods"

p_2 = 1\text{, 1 util = 1 dollar}

Total benefit (in dollars)
from \(x_1\) units of good 1:

v(x_1) = 10x_1 - {1 \over 2}x_1^2

Total cost function:

c(q) = q + {1 \over 4}q^2

Note: variable costs only

GROSS CONSUMER'S SURPLUS

(total benefit, in dollars)

v(x_1) = 10x_1 - {1 \over 2}x_1^2

Marginal benefit,
in dollars per unit:

v'(x_1) = 10 - x_1

(also MRS, marginal willingness to pay)

TOTAL VARIABLE COST

(dollars)

c(q) = q + {1 \over 4}q^2

Marginal cost,
in dollars per unit:

c'(q) = 1 + {1 \over 2}q

What is the optimal quantity \(Q^*\) to produce and consume?

FIRM

CONSUMER

Total benefit:

v(Q) = 10Q - {1 \over 2}Q^2

Total cost:

c(Q) = Q + {1 \over 4}Q^2

Total welfare: 

Marginal welfare from producing another unit:

W(Q) = 10Q - {1 \over 2}Q^2 - [Q + \tfrac{1}{4}Q^2]
W'(Q) = 10 - Q - [1 + \tfrac{1}{2}Q]
=0
10 - Q = 1 + \tfrac{1}{2}Q
Q^* = 6

TOTAL WELFARE

(dollars)

Marginal welfare,
in dollars per unit:

Total benefit to consumers minus total cost to firms

Marginal benefit to consumers minus marginal cost to firms

How do competitive markets
solve this problem?

FIRM

CONSUMER

Maximize net consumer surplus

v(x_1) - e(x_1) = 10x_1 - {1 \over 2}x_1^2 - p_1x_1

Maximize profits

r(q) - c(q) = pq - \left[q + {1 \over 4}q^2\right]

FIRM

CONSUMER

Net benefit from buying \(Q\) units at price \(P\):

TB(Q) - P \times Q

Net benefit from selling \(Q\) units at price \(P\):

P \times Q - TC(Q)

Total welfare: 

Marginal welfare from producing another unit:

TB(Q) - P \times Q
P \times Q - TC(Q)
W(Q) =
+
MB(Q) - P
P - MC(Q)
W'(Q) =
+

profit-maximizing
firms set P = MC

utility-maximizing consumers set P = MB

as long as consumers and firms face the same price, markets set MB = MC and maximize total welfare!

What happens if not everyone is identical?

FIRMS: SUBWAY AND TOGO'S

CONSUMERS: ADAM AND EVE

A = number of sandwiches for Adam

v^A(A) = 8 \ln A
v^E(E) = 4 \ln E

S = number of sandwiches produced by Subway

E = number of sandwiches for Eve

T = number of sandwiches produced by Togo's

c^S(S) = S^2
c^T(T) = 2T^2

What happens in market equilibrium?

FIRMS: SUBWAY AND TOGO'S

CONSUMERS: ADAM AND EVE

A = number of sandwiches for Adam

v^A(A) = 8 \ln A
v^E(E) = 4 \ln E

S = number of sandwiches produced by Subway

E = number of sandwiches for Eve

T = number of sandwiches produced by Togo's

c^S(S) = S^2
c^T(T) = 2T^2

What happens in market equilibrium?

{8 \over A} = P
{4 \over E} = P
2S = P
4T = P
S+T
A + E
=

Firm Optimization:
Every firm produces up until the point where their marginal cost equals the market price (MC = P)

Market clearing: supply equals demand.

Consumer Optimization:
Every consumer buys up until the point where their marginal benefit equals the market price (MB = P)

Welfare Analysis

FIRMS: SUBWAY AND TOGO'S

CONSUMERS: ADAM AND EVE

A = number of sandwiches for Adam

v^A(A) = 8 \ln A
v^E(E) = 4 \ln E

S = number of sandwiches produced by Subway

E = number of sandwiches for Eve

T = number of sandwiches produced by Togo's

c^S(S) = S^2
c^T(T) = 2T^2

How can we choose A, E, S, and T to maximize total benefit minus total cost

subject to the constraint that the total amount produced is the total amount consumed?

v^A(A) = 8 \ln A
v^E(E) = 4 \ln E
c^S(S) = S^2
c^T(T) = 2T^2

How can we choose A, E, S, and T to maximize total benefit minus total cost

subject to the constraint that the total amount produced is the total amount consumed?

(S^2 + 2T^2)
(8 \ln A + 4 \ln E)
-
(S+T)
(A + E)
=
=0
-

Rewrite the constraint so that it's something equal to zero...

...and set up the Lagrangian!

v^A(A) = 8 \ln A
v^E(E) = 4 \ln E
c^S(S) = S^2
c^T(T) = 2T^2

How can we choose A, E, S, and T to maximize total benefit minus total cost

subject to the constraint that the total amount produced is the total amount consumed?

(S^2 + 2T^2)
(8 \ln A + 4 \ln E)
-
(S+T)
(A + E)
-
+ \lambda[
]
\mathcal{L}(A,E,S,T,\lambda)=
{\partial \mathcal{L} \over \partial A} = 0
{\partial \mathcal{L} \over \partial E} = 0

First order conditions:

\Rightarrow
{8 \over A} = \lambda
\Rightarrow
{4 \over E} = \lambda
{\partial \mathcal{L} \over \partial S} = 0
{\partial \mathcal{L} \over \partial E} = 0
\Rightarrow
2S = \lambda
\Rightarrow
4T = \lambda
{\partial \mathcal{L} \over \partial \lambda} = 0
\Rightarrow
(S+T)
(A + E)
=
{8 \over A} = \lambda
{4 \over E} = \lambda
2S = \lambda
4T = \lambda
S+T
A + E
=

What does an omniscient "social planner" do?

Productive Efficiency:
Allocate production so that each firm has the same marginal cost of making the last unit.

Don't overproduce or underproduce.

Allocative Efficiency:
Allocate consumption so that each person gets the same marginal utility from the last unit.

Ensure that the last unit consumed brings the same
marginal benefit to each consumer as the marginal cost it requires to produce.

{8 \over A} = P
{4 \over E} = P
2S = P
4T = P
S+T
A + E
=

What occurs in market equilibrium?

Firm Optimization:
Every firm produces up until the point where their marginal cost equals the market price (MC = P)

Market clearing: supply equals demand.

Consumer Optimization:
Every consumer buys up until the point where their marginal benefit equals the market price (MB = P)

Consumers and producers all face the same market price

"Individual ambition serves the common good." - Adam Smith

If there is a single price in the market that all consumers pay, and all producers receive, and all consumers and producers are “price takers,” then:

Every consumer sets MB = P:

  • Everyone’s MB from the last unit bought is the same.
  • Cannot increase total benefit by reallocating the good from one consumer to another

Every firm set MC = P:

  • Every firm’s MC from the last unit produced is the same.

  • Cannot reduce total costs by reallocating production from one firm to another

The MB of the last unit consumed by some person
equals the MC of the last unit produced by some firm

Effect of Taxes

  • Last time, we saw how taxes affected the equilibrium quantity, as well as the price paid by consumers and the price received by firms.
  • How does this affect overall welfare?

Consumers set \(P_C = MB\)

Firms set \(P_F = MC\)

\(P_C\)

\(MB\)

\(>\)

\(>\)

Since

\(P_F \)

\(MC \)

,

.

Why are Taxes Inefficient/Distortionary?

  • Consumers no longer set their MB equal to
    the same thing firms are setting their MC to.
  • Therefore, the quantity produced no longer sets MB of consumers equal to the MB of firms.