1. Eco 3311 Lecture 12 One Period Closed Economy Model - Equilibrium
(Williamson, Chapter 5)

2. Outline Representative Consumer and Firm, Government
Competitive Equilibrium in One Period Model
Pareto Optimality

3. Motivation we considered consumers and firms separately
we looked at the comparative statics for their decisions when wage changes, but we treated wages as exogenous
we thus did not consider how labor supply and labor demand interact in labor market to determine the equilibrium wage
we will do that now and proceed with a general equilibrium analysis of the economy as whole

4. Learning Objectives
understand the difference between of exogenous and endogenous variables
understand the concept of a competitive general equilibrium
learn what Pareto efficiency is
understand the advantages and limitations to using Pareto efficiency as criteria for optimal allocation

5. Where Are We Representative Consumer and Firm, Government
Competitive Equilibrium in One Period Model
Pareto Optimality

6. Representative Consumer and Firm
suppose that we have two models
Model A: a complex model with multiple heterogeneous firms (that differ in capital, productivity, …) and multiple heterogeneous households (that differ in productivity, income, wealth, …)
Model B: a simple model with one representative consumer and one representative firm
in some cases Model A and Model B can yield same results for aggregate variables (prices, aggregate output, aggregate consumption, aggregate savings,…)
in these cases, if we are only interested in aggregate variables, we only need to understand the decisions of that one consumer and that one firm

7. Representative Consumer and Firm
representative consumer acts as a stand-in for all consumers in the economy
representative firm acts as a stand-in for all producers in the economy
think of the representative consumer and the representative firm as “average consumer” and “average firm”

8. Review – Consumer’s Problem
consumer’s utility maximization problem
or, if we assume a particular utility function

9. Review – Consumer’s Problem
optimal unconstrained choice: given wage w optimal point is H, and consumer chooses leisure
labor supply is

10. Review – Firm’s Problem
firm’s profit maximization problem
for example, if we assume a symmetric Cobb-Douglas production function

11. Review – Firm’s Problem
marginal product of labor
optimal choice for labor demand and labor demand curve

12. Government in general, government constraint is where
T are taxes: lump-sum, tax on income, tax on consumption, property tax, estate tax, …
G are expenditures on goods and services: national defense, police, highways, education,…
TR are transfers: Social Security, Medicare/Medicaid, unemployment insurance, welfare programs, …
INT are interest payments on outstanding debt
in the one period model expenditures on goods and services are the only outlays, government budget constraint thus becomes
we take government expenditures to be exogenous, taxes are endogenous

13. What is G good for?
we need to decide how to treat G in the model - in what way it affects consumer and the firm
to simplify the analysis, textbooks usually at first treat government expenditure as “pure waste”
G then uses up resources but does not affect consumers or firms in any other way – it does not provide any benefits

14. What is G good for?
More realistic treatment: G enters

15. Where Are We Representative Consumer and Firm, Government
Competitive Equilibrium in One Period Model
Pareto Optimality

16. Equilibrium
in physics an equilibrium is a state in which all objects are at rest, i.e. net forces acting on the objects are zero
in economics an equilibrium is a state in which
(i)
(ii)

17. Equilibrium in One Period Model
Ad (1): no economic agent has incentives to change behavior:
taking prices as given consumers maximize utility, firms maximize profits, government’s budget constraint is satisfied
Ad (2): behavior of all agents is mutually consistent:
prices which consumers and firms take as given in their problems clear all the markets, that is in all markets
sum of individual demands of all agents
= sum of individual supply across all agents
or equivalently
aggregate supply = aggregate demand

18. Equilibrium in One Period Model
aggregate consumption and aggregate labor supply are determined by consumption and labor supply of the representative consumer
C=c Ns =ns
aggregate production and aggregate labor demand are determined by production and labor demand of the representative firm
Y=y Nd =nd
the equilibrium conditions that markets clear are thus
c+G=y nd =ns

19. Equilibrium in One Period Model

20. Equilibrium in One Period Model
we defined the real wage, real dividends, real lump sum tax as
all the flows can thus be expressed in terms of the flows of final consumption good
𝑤= 𝑤 𝑛 𝑝 𝜋= 𝜋 𝑛 𝑝 𝑡= 𝑡 𝑛 𝑝

21. Equilibrium in One Period Model

22. Exogenous and Endogenous Variables
exogenous variables:
endogenous variables:
experiments using models: alter an exogenous variable and observe what happens to the endogenous variables
exogeneity and endogeneity are properties of a model, thus a variable can be exogenous in one model but endogenous in another model

23. Equilibrium in One Period Model
when we analyzed the consumer’s and the firm’s problems separately we did not consider market interactions
wage was exogenous in consumer’s problem and also in firm’s problem
we now put together consumer, firm, and the government and let them interact in the markets
this leads to general equilibrium:

24. Equilibrium in One Period Model
equilibrium labor: N* is determined in labor market equilibrium output: Y*=zF(k,N*) equilibrium consumption: C*=Y*-G
labor market
production function
real wage w
output Y
labor N
labor N 24

25. Equilibrium in One Period Model
we can also show the competitive equilibrium in a diagram which combines production possibility frontier (depicting firm’s technology) and indifference curves (depicting consumer’s preferences)

26. Production Possibilities Frontier (PPF)
Production Possibilities Frontier (PPF): set of bundles that are technologically possible

27. Competitive Equilibrium

28. Where Are We Representative Consumer and Firm, Government
Competitive Equilibrium in One Period Model
Pareto Optimality

29. Competitive Equilibrium
To determine whether government should intervene we have to be able to judge whether the free market results in the best allocation of resources possible.
Question "Which allocation is best?" is similar to question "What kind of bear is best?"

30. Pareto Optimality/Pareto Efficiency
An allocation is Pareto optimal/Pareto efficient if it is impossible to rearrange production and reallocate goods in a way that
someone is made better off
no one else is made worse off
Vilfredo Pareto ( )

31. Pareto Optimality/Pareto Efficiency
Notice: Pareto optimality has to do with efficiency (are we making the pie as big as possible?) but has nothing to do with inequality (how are we sharing the pie?)
If one person consumes 99% of what the economy produces and the rest of the population only remaining 1% the allocation can still be Pareto optimal
Pareto optimality thus should not be confused with social optimality which would take inequality into account for which a “social welfare function” concept is needed

32. Finding the Pareto Optimum
to find Pareto Optimal allocation we introduce a social planner
social planner’s goal is to maximize consumers’ well being (utility)
to achieve the highest utility for consumer social planner decides about:
firm’s production level and labor input
consumer’s work and leisure

33. Finding the Pareto Optimum
social planner’s problem
or if we combine the constraints

34. Finding the Pareto Optimum
in the one period model the competitive equilibrium and the Pareto optimum are identical

35. First Welfare Theorem (FWT)
The unregulated market (laissez-faire) allocation is Pareto efficient if markets are perfectly competitive and there are no missing markets (i.e. if there are no market failures)
Related to Adam Smith’s idea of an invisible hand:
Free market interaction of self-interested consumers and firms produces an outcome that is efficient, a benevolent invisible hand guides the actions of selfish agents

36. In General FWT Does Not Hold
Market Failures:

37. Second Welfare Theorem (SWT)
Any Pareto-efficient allocation can be implemented using lump-sum wealth redistribution (transfers) and then letting the market take over.
note: transfers have to be lump-sum and the government needs to have perfect information on consumers' preferences as well as the production possibilities of firms

38. What did we learn?
representative consumer - the “average” consumer, stands in for all consumers in economy
representative firm - the “average” firm, stands in for all firms in economy
exogenous and endogenous variables
general equilibrium – prices and quantities in all market in the model economy are endogenous
competitive equilibrium is Pareto efficient if there are no market failures (monopolies, externalities, public goods, asymmetric information, distortionary taxes)