University Of Verone

Department Of Economic Sciences

Intertemporal General Equilibrium Model With Imperfect Competition For The Evaluation Of

European Integration

PhD Student Gabriele Standardi

Advisor: Prof. Federico Perali

XXI Cycle

Doctorate in Economics and Finance

Applied Interteporal General Equilibrium Model

Imperfect competition

Product differentation

Scale Economies

This model is used for the evaluation of European trade integration

Six countries: GB, Gr, Fr, It, the rest of the Europe (RE) and the rest of the world (ROW)

Nine sectors of production for each county. Four of these are assumed perfectly competitive: food, beverage and tobacco; agriculture and primary products; other manufacturing industries (textile, wood,

paper, metallurgy, minerals); transports and service.

The other five industries are non-competitive: pharmaceutical products; chemistry other than pharmaceutical products; motor vehicles; office

machinery; other machinery and transport materials

Countries i, j = 1,…W Sectors s, t = 1,…,S

Static model Household

Single representative household, competitive, infinitely lived and utility maximizing. The domestic household owns all the countries’ primary

factors, labor and physical capital, which it rents to domestic firms only, at the same competitive prices regardless of the sector.

The domestic consumer values products of competitive industries from different countries as imperfect substitutes (the Armington assumption,

1969), while he treats as specific each good produced by individual firms operating in the non-competitive industries (the Dixit-Stiglitz

specification, 1977).

Household’s Consumption

.

1 1

.

1 1

.

log log , 1,

, ,

, ,

.

jsi

s

s s

s

fs

f fs s

fs

i si si sic

s S s S

csi jsi jsi

j W

csi js jsi jsi

j W

Max C c

c c s C

c n c s C

s t

, ci i jsi jsi js jsi jsij W s S s S

p C p c n p c

Household’s Investment

.

1 1

.

1 1

.

log log , 1,

, ,

, ,

.

jsi

s

s s

s

fs

f fs s

fs

i si si siI

s S s S

Isi jsi jsi

j W

Isi js jsi jsi

j W

Max I I

I I s C

I n I s C

s t

,

Ii i jsi jsi js jsi jsij W s S s S

p I p I n p I

Firms in competitive industries

In competitive industries, the representative firms of country i, sector s, operate with a Cobb-Douglas constant returns to scale technologies, combining variable capital and labor as well intermediate inputs. Material inputs are introduced in the production function in a way similar to the way consumption goods are treated in the preference of household: with an Armington specification for goods produced by competitive industries and with an Ethier (1982) specification (i.e., with product differentiation at the firm level) in the imperfectly competitive sectors

Competitive firm’s cost minimization

, ,

.

.

subject to

log log log log

minv vis is jtis

v vjti jtis jt jti jtis i is i is

j W t C s CL K x

v vis Lis is Kis is tis tis

t S

tis jtis jti

p x n p x w L r K

Q L K x

x x

1 1

1 1

.

, ,

, ,

where and are share parameters

t

s t

s

xt

x xt t

xt

sj W

tis jt jtis jtisj W

t C

x n x t C

with 1

0 if is non-traded, and the and have the same interpretation as the and

in the consumer's problem.

Cost minimization implies marginal cost pricing (

Lis Kis tist S

x fjtis s s s sj i t

p

) and zero profits ( 0) isj is isv

Non competitive industriesNon competitive industries have a Cobb-Douglas increasing returns to scale production function. In addition to variable costs associated with technological constraints similar to the competitive firms, the individual firm in county i , sector s, has fixed primary costs. Thus, the relationship between variable unit cost and total unit cost or average cost (Vis) becomes:

where , , are, respectively the individual firm's output, fixed labor and fixed capital.

F Fi is i is

is isis

F Fis is is

w L r KV v s C

Q

Q L K

Non competitive firm’s strategy

( ) ( ( ) )

where ( ) is the amount of good produced by the firm secotr and sold to coutry ,

are fixed costs

isj isj isj is isj

isj

p z p v z p fx

z p i s j

fx

log ,

logisj is isj

isj isj

p v zs C j W

p p

The oligopolistic firm maximizes its profits in country j, :isj

In the Bertrand case of non cooperative game, the price-strategy which maximizes the profits yields:

Alternatively, in the Cournot case of non cooperative game, the oligopolistic firm’s profits in country j, and the quantity-strategy which maximizes the profits yields the well-known Lerner’s Equation :

The firm is endowed with the knowledge of preference and technologies of its clients. It then performs a partial equilibrium profit maximization calculation assuming that in each country, each individual client’s current expenditure on the whole industry is unaffected by its own strategic action so that:

( ) ( )

log ,

log

isj isj isj is isj

isj is isj

isj isj

p z z v z fx

p v ps C j W

p z

0 ,

0 ,

0 ,

sj cj j

isj

sj Ij j

isj

sjt jt jt

isj

p Cj W

z

p Ij W

z

v Qj W t S

z

Oligopolistic mark-up

1

'

' 1 1 11 1

Define as the vector on market :

,..., ,..., ,..., ,..., ,..., ,..., ,

where , is the price charged by firm of country on market .

The sector index is not consi

i W

j

n nn fj j j ij ij ij Wj Wj

fij

j

p p p p p p p

p f i j

P

P

' ' ' '

dered in order to not complicate the notation needlessly.

Define in a similar way , , , , as the vectors, respectively, of sales of firm

of country in market , consumption of representat

j j j jt f

i j

Z C I X

ive household of country of good

produced by firm of country , investment of representative household of country of

good produced by firm of country , and inputs demand by firm of country

j

f i j

f i j, sectors .

In market , firms face a demand system that is:

( ( )) ( ( )) ( ( ))

j j j j j j j jt j jt

t

j

Z C P Z I P Z X P Z

Total differentation and some manipulations yelds:

1 1 1ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) j j j j j j j j j jt j jt j j j j

t

dZ E C P C Z E I P I Z E X P X Z E P Z dZ

Non cooperative behavior implies that firm solve the system

with 1 and all other elements of vector set to zero.

Conceptually, the computation of an equilibrium requires solving

one such s

fij j

f

dz dZ

ystem for each firm in all markets . The cost of such

a calculation would be prohibitive without the assumption of symmetry

between domestic firms.

f i j

Static equilibrium conditions

vis

vis

In each competitive market:

, ,

, ,

,

is isj isj isjtj W t S

v Fi is is is

s C s C

v Fi is is is

s C

Q c I x s C i W

K K n K K i j W

L L n L L

, s C

i j W

In each non competitive market:

, , , isj isj isj isjtt S

z c I x s C i j W

Chamberlinian exit/entry condition is

(0) given, ( ) such that ( ) 0

( ) ( ) (0) , 0 1 is is is

is is is

n n

n t n n

The first period ROW wage rate is chosen as the numeraire

Dynamic structure

1

0

( ),

1

subject to

( ) ( ) ( ),

and subject to

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0),

(0), (0) given

t

t tC I s

so o

C te dt

K t I t K t

e p t C t p t I t e w t L t r t K t t dt F

K F

The trade experiment

log log (1 ) ,

log logisj is isj isEU

isj isj isEU

p v p ps C j EU

p z z

with λ = 1 in the calibration. The simulation sets λ = 0. The elasticity on the right side is evaluated using EU-aggregated demand.

The experiment consists to simulate Europe’s move to a single market by forcing firms to switch from their initial segmented market pricing

strategy to an integrated market strategy determined from their average EU-wide monopoly power.

Thus, the fundamental equation in non competitive markets becomes:

Welfare criterion

The welfare gain is determined from the following utility indifference condition:

1

1

0 0

ˆ ( )(1 ) ( ) ,

1 1

where is the equivalent variation welfare measure, most frequently used in the

ˆapplied general equilibrium analysis, ( ) is the reference str

t tC t C t

e dt e dt

C t

eam of consumption

(benchmark equilibrium) and ( ) is the corresponding time profile computed after

the implementation as from date 0 of trade policy simulation (counter factual equilibrium)

C t

t .

Calibration and computational considerations

The number of symmetric firms in non-competitive sectors is inferred from Herfindahl industry concentration indices. Note that, because of the

hypothesis of symmetry among firms within the sector, the firm is an abstraction and, so , nis must be interpreted as an index of product variety

rather than the number of real firms in the market.

The literature includes several sources of econometric estimates of Armington elasticities. Conversely, no reliable estimates on product differentiation, returns to scale, price-costs margins in oligopolistic

industries are available, the strategy is to put exogenously reasonable values for , and then determine jointly the base-year price system and scale elasticities consistent with the data base and the

optimal price-discriminating Cournot-Nash behavior of non competitive firms.

, .x fs s

*

own elasticities cross elasticities

log log ij i j

ij ij

p p

z z

1 1 1

ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) j j j j j j j j j jt j jt j j j jt

Price system

dZ E C P C Z E I P I Z E X P X Z E P Z dZ

( , , , )x fisj isj is s sE e n

denotes the current price trade flows as supplied by the database

(.)denotes a function of which we know the form and the parameter values

isj

isj

e

E

1,

1 ( , , , ) isj

x fis isj isj is s s

ps C

v E e n

is normalized at 1.

where /

( / )

is

is isj isj isj isj isjj W j W

isjisisj

j W j Wis isj is

p

p e e e e p

epe s C

v p v

isj is

isj

p v

p

Finally, the assumption of zero profits determines average costs: is isV p

Dynamic calibration

I make use of results by Mercenier and Michel (1994) on dynamic aggregation.

First Proposition: there exists a sequence of discount factors αn for

which stationary solution of the continuous time optimisation problem is a stationary solution of the discrete time problem. This sequence is unique within the choice of αn >0, and it is defined by the following

recurrence relation:

1 1

1

/(1 ),

with 0 - 2,

where are intervals of time and ( 0,..., ) are possibly unequally spaced dates

n n n

n n n n

n N

t t t n N

Second Proposition: the stationary solution of the infinite horizon continuous time problem is also the constant solution of the finite horizon discrete time problem with βN= αN-1, provided that Proposition 1

holds for n=0,1,…,N-2.

A more empirical result concerns the formula used for temporal aggregation, that is to generate the sample date. The rule of thumb is the

following:

log 11

1 1log

1

n

nN

t

N N

Now, we can write the finite horizon discrete time approximation to the individual household’s intertemporal choice problem:

1 11

0

1

1 0 0

( ) ( )1

1 1

subject to

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,

( ) ( ) ( ) ( ) , ( ), ( ) given

Nn N

n n Nn

n n n n n n n n c n n I n

n n n n n

C t C tMax

F t F t F t w t L t r t K t p t C t p I t

K t K t I t K t F t K t

If the world economy is assumed initially in steady state, these results make the calibration straightforward using the following first order conditions:

1 1

1

( ) ( ), 0

( ) ( )

1( ) ( ) (1 ) ( ) , 0

1

1( ) ( ) ( ) .

n c n

n c n

I n n n n I nn

I N N I n

C t p tn N

C t p t

p t r t p I t n N

p t r t p I t

1 11 11 1

0 0

In the time aggregated framework, the welfare criterion becomes:

ˆ ˆ( )(1 ) ( )(1 ) ( ) ( )1 1,

1 1 1 1

ˆwhere ( ) and ( ) 0,...,

N Nn N n n

n n N n n Nn n

n n

C t C t C t C t

C t C t n N

denote respectively, the benchmark and counter factual

equilibrium profiles of aggregate consumption. The procedure adopted with dynamic discrete

aggregation is to exogenize oligopolistic markups and solve for the intertemporal equilibrium

allocations, prices, and industry structures. Using these newly computed prices and market shares,

the optimal markups are upgraded. Then, Gauss-Seidel algorithm is iterated until convergence

to a fixed point.

ResultsConcerning the results of trade policy experiment, I refer to the works of

Mercenier (1995, 2002).

First work (1995)

Static model

labor and capital move internationally

Second work (2002)

Intertemporal model

labor and capital don’t move internationally

Two equilibria Unique equilibria

Welfare gains from European trade integration (Mercenier’s model, 2002)

  Static model% Equivalent variation

Dynamic model% Equivalent

variation at the steady state

Dynamic model% Equivalent

variation after five years

Wages flexibleGbGrFrIt

RE

 0.850.190.690.500.19

 1.080.640.840.860.02

 0.740.390.550.590.20

Wages indexed in the first five years

GbGrFrIt

RE

   

  

3.412.272.142.640.86

  

2.43 1.60 1.44 1.83 0.70

Remarks

All European countries unambiguously benefit from the trade integration.

As the comparisons between the first two columns indicate, accounting for growth increases the disparity of welfare gains across counties and

confirm that the use of a static model for policy analysis could be misleading.

Remark

Graph 1 shows the solutions of the intertemporal pattern of aggregate consumption computed for different number of dates and the same solution for the static version of the model, the importance of the

intertemporal substitution is very clear. It is optimal for consumers to trade present for future consumption. Clearly, a static model will misse important aspects of the structural adjustments effects of trade policy.

Possible further experiment

Even if this kind of model is mainly used for the analysis of trade policies, I think that its theoretical structure remain valid to evaluate the effects on welfare of European fiscal integration. In particular, it could

be possible to analyse the impact of value-added tax (VAT) harmonization. VAT is the European tax par excellence; despite of many efforts to create a common regime within Europe about its application

and its rates, substantial differences continue to this day across countries.

  Minimum Aliquot Reduced Aliquot Normal Aliquot Special Aliquot

Belgium 1 6 21 12.5

Denmark - - 25 -

Germany - 7 16 -

Greece 4 8 18 -

Spain 4 7 16 -

France 2.1 5.5 20.6 -

Ireland 4 12.5 21 12

Italy 4 10 20 -

Luxembourg 3 6 15 12

Netherlands - 6 17.5 -

Austria - 10/12 20 -

Portugal - 5/12 17 -

Finland - 8/17 22 -

Sweden - 6/12 25 -

U.K. - 5 17.5 -

Following Cavalletti (Fossati, 1991), value-added tax should be treated as ad valorem tax on final consumption. Thus, it need to calculate the aliquot weighing down on the single good. We can compute the VAT rates as ratio between the total amount of VAT

paid by sector s of county i and the value added (production net of intermediate inputs) of sector s of county i. The formula used is the following:

, sisi

si

VATs S i EU

VA

Then, it suffices to insert in the static budget constraint of representetive housold :

(1 ) (1 ) , ,

si

ci i si jsi jsi js si jsi jsij s S s S

i

p C p c n p c j i EU

W

The last expression states that the representative consumer of country i pays the aliquot of her/his country even if the product comes from country j. This is consistent with the

fiscal principle, still in force in European Union, that a domestic firm, which buys a product from a foreign firm, pays according to the domestic aliquot and, so, the final consumer will pay according with the domestic aliquot too. Unfortunately, it doesn’t

account for the goods bought by a person of country i in the country j, for which she/he pays according with the aliquot of country j. This can represent an important distortion.

The role of government is to fix the rates and consume the proceed, according to this equation:

Gi i si jsi jsi js si jsi jsij W s S s S

p G p c n p c

The calibration is simple enough. It suffices to set the values of τis , computed from

the national accounts at the benchmark equilibrium and, then, impose the following condition at the counter factual equilibrium:

where could be an average value of the rate in the EU.si sEU

sEU

i EU

The welfare equivalent variation could be calculated starting the segmented Europe market equilibrium or starting the integrated Europe market equilibrium, that is with λ=0 or λ=1.