Welfare Analysis of Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 15

Today's Agenda

What can go wrong?

Model 2: two consumers, two firms

Model 1: one consumer, one firm

The "social planner's problem"

Market Power

Externalities

Taxes

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)

Model 1: 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:

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 \(P\):

v(Q) - P \times Q

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

P \times Q - c(Q)

Total welfare: 

Marginal welfare from producing another unit:

Model 2: Two Consumers, Two Firms

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?

"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

What can go wrong?

Copy of Econ 50 | Fall 2021 | 15 | Welfare Analysis of Equilibrium

By Chris Makler

Copy of Econ 50 | Fall 2021 | 15 | Welfare Analysis of Equilibrium

Bringing supply and demand together

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