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Lecture 2: Applied general equilibrium
(Chapter 3)
•
•
•
Introduction of formats
Comparing the formats
Model implementation
–
–
–
–
Classification
The Social Accounting Matrix (SAM)
Missing commodities and markets
Closure rules
Aim of lecture 2
• Highlighting the characteristics of different representations
of general equilibrium models (formats)
• Illustrating the steps from a general equilibrium to an
applied general equilibrium model
Importance of existence proofs
• Non-existence of a solution implies model is inconsistent
and therefore useless
• Existence proofs highlight crucial necessary assumptions
that have to be made
• For applications:
– Necessary assumptions indicate scope for variation of parameters
– Construction of fixed point mapping to prove existence may
suggest algorithm for numerical implementation
Excess demand format

• For every producer j, y *j solves max y p* y j y j Yj
j



• For every consumer i, xi* solves max x 0 ui ( xi ) p* xi  hi* , where
i
hi*  p*i   j ij p* y*j
• All markets are in equilibrium:
*
*
x

y
i i  j j  i i  0
• Define excess demand function z( p)  i xi ( p)   j y j ( p)  i i
• The price p*  0, p*  0 , and the excess demand z( p* ) define an
excess demand equilibrium if z ( p* )  0 .
Excess demand format (continued)
• Debreu (1959) assumes convex production sets and
quasiconcave utility functions. Therefore, excess demand
correspondence is not single-valued.
• This is problematic since compatibility of allocations is not
guaranteed
• Therefore, Debreu assumes consumer demand functions to
be single-valued, and aggregates production by different
producers into one single set
Negishi format
(a) Welfare program
maxxi 0,all i,y j ,all j i i ui ( xi )
subject to
i xi - j y j  i i
(p)
y j Y j
(b) Adjustment of welfare weights  i  S m such that budget
constraints pxi  pi   j  ij  j ( p ) hold for every i,
where  j ( p )  max y py j y j  Y j is the profit function
j
of producer j


Full format
Constrained welfare optimum:
maxxi 0,all i,y j ,all j i i ui ( xi )
subject to
i xi   j y j   i i
(p)
y j Yj
pxi   j  ij py j  pi
( i )
for given   S m and p  S r such that i  0 is a shadow price
and p= p for some scalar  >0 .
Open economy format
(a) Optimization with given consumptions x̂i :
maxm,e0,y j ,all j p ee  p m m
subject to
 j y j  m  e  i ˆxi  i i
( p)
y j Yj
(b) A feedback relation that sets x̂i by solving the consumer
problem:
x̂i  arg max ui ( xi ) pxi  pi   j ij  j ( p ),xi  0 ,all i
where  j ( p )  max py j , y j  Y j




CGE format
• CGE format is a system of simultaneous equations:
– balance equation for factors (f) and goods (g)
g
g
f
i xi ( p , p
f
g
f
i xi ( p , p
g
,hi )  A ( p , p )q  q
,hi )  A f ( p g , p f )q g i if
g
g
f
g
– complemented with individual budgets:
hi  p f if
– and price relations:
p g  p g Ag ( p g , p f )  p f A f ( p g , p f )
CGE format (continued): assumptions
• Constant returns to scale in production
– no profits in equilibrium
• Factors are not produced and used in production
– boundedness of production
• Goods are not available as endowments
• Utility functions are continuous, strictly quasi-concave and
non-satiated. For at least one consumer, utility increases in
all factors
CGE format (continued): extensions
• Decreasing returns
– Firm-specific inputs in CRTS technology imply DRTS in
remaining inputs
– Profits have to be included in budgets
• Markups
– Compensation for inputs not included in the model
– Caused by imperfect competition
• Closure rules
Comparing formats: merits
• Excess demand
– Microfoundations
• Negishi and Full format
– Direct link to welfare analysis
– Weaker assumptions with respect to production technology
– Solvable even if decentralization is problematic
• Open economy format
– Econometric estimation of model parameters
– Possibility to include non-optimizing behavioral rules
• CGE format
– Easiest for application
– Possibility to include non-optimizing behavioral rules
Comparing formats: mathematical requirements
Production sets
Utility functions
Endowments
Excess
demand
format
Possibility of inaction, Continuous, strictly
Strictly positive for
compact, strictly
concave, non-satiated, consumers
convex
increasing in all
commodities*
i  0, i  0 for all i
Negishi
Possibility of inaction, Continuous, strictly
format
compact, convex
concave, nonsatiated,   0 for i=1
i
increasing for I=1
i  0, i  0 for all i
Full format Possibility of inaction, Continuous, strictly
compact, convex
concave, nonsatiated,  i  0 for i=1
increasing for I=1
i  0, i  0 for all i
Open
Possibility of inaction, Continuous, strictly
economy
compact, convex
concave, nonsatiated,  i  0 for i=1
format
increasing for I=1
i  0, i  0 for all i
CGE format CRTS or DRTS
Continuous, strictly
production technology quasiconcave,
 i  0 for i=1
for goods. Factors are nonsatiated,
not produced
increasing for I=1
for factors only
*These are the assumptions made in G/K needed for single-valued excess demand.
all
Mapping from theorems to “work horses”
Concavity of
objective
1.4 Excess demand
3.2 Negishi
3.3 Full format
Assumptions on
u ( x)
Assumptions on u ( x)
Assumptions on u ( x)
Constraint set of welfare
program
Constraint set of welfare program
including budgets
Slater
Homogeneity
Prices can be
constrained to lie on
a simplex.
Welfare weights can be
constrained to lie on a simplex
Welfare weights and prices can be
constrained to lie on a simplex
Maximum
theorem
Continuity of
consumer demand
(strict concavity of
utility function) ;
Continuity of supply
functions (strict
convexity of
production sets)
Continuity of consumer demand
in welfare weights and
endowments, upper
semicontinuity of price
correspondence in welfare
weights, compactness and
convexity of set of prices
Continuity of consumer demand in
welfare weights and endowments,
upper semicontinuity of Lagrangian
multiplier correspondence in
welfare weights and budget prices,
compactness and convexity of set
of Lagrangian multipliers
Kakutani
Fixed point in p
Fixed point in  and p
Fixed point in  ,p, p, 
Driven by excess
demand (Brouwer)
Driven by budget surplus
Driven by p, 
3.4 Open economy
Constraint set of
optimization
program
Upper
semicontinuity ,
compactness and
convex-valued ness
of price
correspondence in
consumption of
individuals.
Fixed point in p
Driven by p
Classifications
• Commodities
– Focus on relevant characteristic that allows aggregation of
different products in some common unit
– Maintain link to data availability
• Agents
– Consumers
• Dissimilarities between groups with respect to reactions on policy
changes (employers vs employees)
• Similarities between agents within the same group (income sources,
consumption patterns)
– Producers
• Different production technologies for single good?
Organizing data: the SAM
Goods
Factors
Firms
Consumers
Goods
Input
Consumption
Factors
Input
Consumption
Firms
Production
(Production)
Consumers
(Endowments) Endowments
(Profits)
*Terms within parentheses are zero in the basic CGE model
*This SAM represents a closed economy without government
The SAM (Continued)
• SAM structure and classification follow model architecture
• SAM entries are in value terms
• Rows and associated columns have equal value since they
represent balances
–
–
–
–
–
Commodity balances in value terms
Budgets
Firm profits
(Government budget)
(Balance of payments)
• Prices follow from confronting SAM with balances in
quantity terms
Missing commodities and markets
• Rough markets
– Only one market for different qualities
• Markets for characteristics
– Consumer preferences in terms of characteristics
– Markets usually in terms of products
– Introduce mapping from characteristics to products
• Commodities traded but not fully represented
– Missing markets for goods implies these go unpriced
• Markups
– Reflect reward for input not accounted for in the model
Closure rules
• Including macro-economic mechanisms in general
equilibrium model
– define a new variable
– define balancing constraint for this variable
– Example: Taxes (closure variable) and government
budget (associated constraint)
– Beware: Closure rule can become dominant mechanism
in model!
Closure rules (continued)
• CGE model with parameters:
• Closure rule:
• Relation:
z ( p; q)  0
g ( p, q )  0
z ( p; q)  0
g ( p, q )  0