Evaluación a través del Sistema de Monitoreo PERLAS

Notes on the structure of the model:

The model is based on an imperfectly competitive structure, using a Dixit-Stiglitz framework. In this framework, as outlined …rst in Dixit and Stiglitz (1977), an industry, i; contains a large number of goods produced by closely competing …rms in the various regions. Each good, g; is produced in one country, c; only. Section 3.1.1 of the main chapter summarises the theory of this type of model.

In this chapter I have used the simpler version of the model, where the number of goods produced within each country and industry, nc;i, is …xed. However, unlike many Armington models, it does allow for monopolistic markups. The full Dixit-Stiglitz variant allows the number of goods/…rms to vary endogenously.

Another variant is to allow capital to ‡ow between countries rather than be …xed within each country. The coding for this variant is also explained in this appendix.

Production of goods

The production function of each …rm combines labour, land and capital using a Cobb-Douglas function to form a value added input: i.e.

V Ag = c;iLg(1 kc;i dc;i)_K kc;i

g Dg dc;i; (3.A2.1)

whereV Ais value added (quantity),Lis labour,Kis capital,Dis land which is sectorally …xed, g denotes the good, i denotes industry and c denotes the country of production.

labour, L, is a Cobb-Douglas aggregate of unskilled labour,LU, and skilled labour, LS. This allows us to write:

V Ag = c;i(LUguc;iLSgsc;i)(1 kc;i dc;i)Kgkc;iDg dc;i; (3.A2.2)

where uand sare share parameters for unskilled and skilled labour within total labour costs, and sum to1 within any industry in any country.

To obtain an equation for the whole industry in country i, we assume all …rms g within iin a given country, c, are identical in terms of cost, inputs, output and market share. We also choose units so that c;i= 1.

Di¤erentiating (3.A2.2) with respect to K; LU; LS and D and setting value of mar- ginal products equal to the wage rate and price of capital gives

LUc;i = V Ac;iP Vc;i uc;i(1 kc;i dci)=W Uc (3.A2.3a) LSc;i = V Ac;iP Vc;i sc;i(1 kc;i dc;i)=W Sc (3.A2.3b) Kc;i = V Ac;iP Vc;i kc;i=Rc;i (3.A2.3c) Dc;i = V Ac;iP Vc;i dc;i=LDPc;i; (3.A2.3d)

where W U and W S denote the wage rates for unskilled and skilled labour, R denotes return on capital andLDPthe sectoral return on land. Both types of labour are assumed to be mobile between sectors, but not between countries, so that wage rates are equal across sectors. Land is sectorally immobile, while we explore two variants in the case of capital: in the …rst, it is mobile between sectors but not between countries, while in the second it is internationally mobile.

Hence, in the …rst variant, we …x R within a country:

Rc;i=RBc: (3.A2.4)

The price of value added is given by

P Vc;i = (W UcLUc;i+W ScLSc;i+RcKc;i+LDPc;iDc;i)=V Ac;i: (3.A2.5)

Higher level of production function

The output of good i is produced by a combination of other goods, ii; and value added, V A. This is done again using a Cobb-Douglas production function

Yc;i= c;iV A vc;i c;i Y ii II Ic;ii;i c;ii;i ; (3.A2.6)

whereY is output,II is the input of goodii into goodiand the coe¢ cients are input shares which sum to 1.

Assuming cost-minimisation, this gives inputs:

IIc;ii;i= Ic;iYc;iP Yc;i=P Uc;ii;i; (3.A2.7)

whereP Y is the output unit variable price andP U is the unit price of inputs, and

V Ac;i = vc;iYc;iP Yc;i=P Vc;i: (3.A2.8)

P P Yc;ii= (V Ac;iiP Vc;ii+X

i

IIc;i;iiP Uc;i)=Yc;ii: (3.A2.9)

This is then adjusted for output tax and subsidies, to give a marginal per unit price including tax and subsidies:

P Yc;ii=P P Yc;ii(1 +OU T T AXc;ii) SU BSIDYc;ii=Yc;ii: (3.A2.10)

Trade and the aggregation of goods

I assume the representative consumer in each country obtains utility by aggregating goods using a two-stage nested utility function: …rst, the various varieties of goods within industry i are aggregated using a Dixit-Stiglitz utility function (see section 3.3.1 in the main part of the chapter for an outline discussion). This is the lower level of aggregation. Then the aggregate goods bundles for each industry, i, are combined to provide aggregate utility using a Cobb-Douglas utility function. This is the top level of the utility function. The total demand in countryCfor produce of industryiis taken to beT Uc;i. This is an aggregate bundle of all the goods,g;which belong to industryi, using a Dixit-Stiglitz demand function: T Uc;i = 0 @X g2i g;cUg;c 1 A 1= ; (3.A2.11)

whereUg;cis use of goodgin countrycand is a parameter re‡ecting qualitative factors (e.g. compatibility of standards) and home bias in consumption. is a substitution parameter, where = ( 1)= , where is the elasticity of substitution between goods g in industryi(assumed to be the same in all countries and industries).

If we assume there arencc;i …rms in countryccmaking goodi, and that the prefer- ence parameter depends only on country of origin, cc, country of use, c, and industry,i, then we can rewrite (A2.11) in order to sum, …rst, the various goods varieties of industry i,g₂(cc_\i);which are produced in countrycc (which all carry the same preference parameter), and then to sum across countries:

T Uc;i = 0 @X cc X g2(cc\i) cc;c;iUg;c;i 1 A 1= : (3.A2.12)

The assumption that all …rms within an industry/country are identical in size allows us to rewrite (3.A2.12) in terms of the total purchases of goods classiby countryccfrom countryc; QUi;c;cc and the total number of …rms in that industry in producing country i:

T Ucc;i= X

c

n_{ci i;c;cc}(QUi;c;cc=nc;i) !1=

; (3.A2.13)

where g is a CES share parameter, and is an elasticity-related parameter, related to the elasticity of substitution by the formula:

= ( 1)= : (3.A2.14)

Total expenditure in country cc on goods in industry i (by …nal consumers and intermediate users) is calculated by summing …nal user price times volume for all goods, g;in industry i.

V Ucc;i= X

This is then used to calculate the price of P U of the aggregate bundleT U:

P Ucc;i=V Ucc;i=T Ucc;i: (3.A2.16)

Consumption (top level of the nested utility function)

Consumers’income is divided between the various industries, i, in order to maximise a Cobb-Douglas utility function

U Tc = Y

i

CN ci;c

i;c ; (3.A2.17)

where U T is utility and CN is consumption of produce of industry i in country c by …nal consumers (in other words, after deducting intermediates use). The c coe¢ cients are expenditure shares, and sum to 1.

Consumers’expenditure on each industry, i, CNic can be calculated relatively sim- ply from the Cobb-Douglas property that _ic is the share of expenditure on i in total consumers’expenditure in countryc,CEc. Hence:

CNi;c= ci;cCEc=P Uc;i: (3.A2.18)

The derivation of total consumers’expenditure is explained below.

Competition and pricing

In a Dixit-Stiglitz model, …rms are imperfect competitors. In basic versions of the model, each …rm produces one good, and the goods are symmetrically competitive, with a constant elasticity of substitution between all goods in an industry consumed in one

country.

The own-price elasticity of demand facing a …rm is derived as follows:

1) If the own price elasticity for the aggregate produce of an industry i is _i, and if competitors do not change their prices in response to …rm, g, changing its price (Bertrand-Nash equilibrium), then the own-price elasticity facing company g would be

+Sg( i );where i is the top-level elasticity of substitution between goodsg, and Sg is the value share of …rm g in demand for industryi. IfSg is small (ienis large) the own price elasticity would be approximately equal to .

2) Within export markets, it is assumed that a …rm has a very small market share and so its own-price elasticity is .

3) By contrast, in the home market countryc, the …rm’s market shareSgc is assumed to be signi…cant. It is calculated as Sgc = (1=nc;i)(1 SMc;i), where SMc;i is the share of imports in consumption ofiin countryc. Since the top level of the consumption function (where di¤erent industries’products are aggregated) is a Cobb-Douglas function in our model, the own price elasticity for the aggregate product of industry i, _i = 1. Consequently, the …rm’s own price elasticity in the home market:

hc;i = + (1=nc;i)(1 SMc;i)(1 ); (3.A2.19)

where, if HU denotes consumption from domestic suppliers and P Ti;c;cc is the price at which it sells (including taxes), then

sales) of its own-price elasticity in the home and export markets.

oc;i= hc;i(HUc;i=Yc;i) + (Yc;i HUc;i=Yc;i): (3.A2.21)

5) In the model variant where the number of …rms is …xed, we …x the value of oc;i.

Monopolistic competition markups: it is assumed that the …rm marks up its production costs by a proportionM Mc;i, where

M Mc;i = 1=(1 (1= oc;i)) 1: (3.A2.22)

The price of good g including monopoly markups is therefore:

P Mc;i=P Yc;i(1 +M Mc;i): (3.A2.23)

It is assumed that no monopoly margin is charged on import tari¤s (the justi…cation being that importers can buy the good in another country if the manufacturer starts price discrimination between markets).

Transport costs

Transport costs are assumed to be proportional to value. Consequently, the price including transport of goods class ifrom country csold in country cc is

P T Ri;c;cc=P Mc;i(1 +T margini;c;cc) (3.A2.24)

whereT marginis the proportional transport cost.

of the goods transported (otherwise known as ’iceberg costs’). Hence, if Xi;c;cc is the quantity ofileaving countrycfor countrycc, the amount which arrives in countryccis:

Mi;c;cc =Xi;c;cc=(1 +T margini;c;cc): (3.A2.25)

This form of treatment means that there is no need for an explicit transport in- dustry, nor for dealing with transport speci…cally in the trade accounts. The costs of transport ‘margin exports’in the GTAP database have been reallocated correspondingly, to maintain balance in the accounts.

Tari¤s

The model allows for tari¤s applied to prices including transport. Tari¤s on imports ofifrom cintocc are expressed as a percentage rate. Consequently, the price including transport costs and tari¤s is

P Ti;c;cc=P T Ri;c;cc(1 +tarif fi;c;cc=100): (3.A2.26)

Finally, the price of produce of industry i from country c consumed in country cc, P U Ui;c;cc, also includes a proportional tax on use of i in cc, which applies to both domestically-produced and imported varieties. Hence, the price facing consumers is

P U Ui;c;cc=P Ti;c;cc(1 +U SET AXcc;i): (3.A2.27)

Exports

countryc as

QUi;c;cc=fXi;c;cc=(1 +T margini;c;cc) ifcc6=c orHUi;c ifcc=cg; (3.A2.28)

where Xi;c;cc is the corresponding volume of exports, and Tmargin is the proportion which ‘melts’(to use the iceberg analogy) en route between the countries.

The equation for aggregating QU within each industry, equation (3.A2.13), has al- ready been explained.

Sales shares

We then di¤erentiate (3.A2.13) setting price equal to marginal utility, to calculate QUi;c;cc as a function of total use of products of industryiin countrycc,T Ucc;i and the relative price of input ofifrom countrycc,P U Ui;c;cc compared to that of aggregate use of iin countrycc,P Ucc;i. Hence, taking

T Ucc;i= X

c

nci i;c;cc(QUi;c;cc=nc;i) !1=

; (3.A2.13)

we di¤erentiate with respect to QU, and set the resulting marginal product equal to P U=P U U, giving

dT Ucc;i=dQUi;c;cc = n1_{ci i;c;cc}QU_i;c;cc1 X c

nci i;c;cc(QUi;c;cc=nc;i)

!(1 )= ;

= n1_{ci i;c;cc}(T Ucc;i=QUi;c;cc)1 =P Ucc;i=P U Ui;c;cc:

QUi;c;cc=T Ucc;inc;i( _i;c;ccP U Ui;c;cc=P Ucc;i)1=(1 ): (3.A2.29)

Aggregate consumer price

The total value of expenditure on good iin countrycis given by

V Ucc;i=X

c

QUi;c;ccP U Ui;c;cc: (3.A2.30)

The aggregate consumer price of iincc,

P Ucc;i=V Ucc;i=T Ucc;i: (3.A2.31)

Factor markets

Both types of labour are immobile between countries, but mobile between industries. The wage is assumed to clear each labour market, so that total skilled and unskilled labour use by all industries equals the skilled and unskilled labour endowment of country c LUc = X i LUc;i; LSc = X i LSc;i: (3.A2.32)

We assume capital is fully mobile between industries. There are two variants - one where total capital within a country is …xed (net capital imported from abroad,KMc =

0) and one where it is allowed to vary. Hence

Kc+KMc = X

i

Kic;i: (3.A2.33)

Where KMc is allowed to be non-zero (so that there are international transfers of capital) the global total ofKM is set to zero.

X c

KMc = 0: (3.A2.34)

The rate of return on capital in each industry is equated to the national rate of return, RBc:

Rc;i=RBc: (3.A2.35)

Where capital is allowed to move internationally, we also set national rates of return equal to the return in the ‘Other OECD’region:

RBc =RBODX: (3.A2.36)

Land is only used in two sectors: agriculture and other primary. Its rent varies according to sector. It is sectorally immobile, so

LDc;i=LDc;i: (3.A2.37)

Variety of goods

Each …rm within a country is of identical size, though the average company size may vary between countries.

For sensitivity analysis, the …xed …rm numbers version of the model assumes the total number of …rms in each country is …xed

nc;i =nc;i: (3.A2.38)

National accounts

Home use of goods from industryiin countryc,HUc;i, is de…ned as total production in countryc less exports.

HUc;i =Yc;i X

cc

Xi;c;cc: (3.A2.39)

Imports of i from country ccto country care equal to exports fromcc toc de‡ated to take account of transport costs.

Where c = cc (ie the variable IDENc;cc equals 1), total use of good i in country c produced in countryccequals home use. Otherwise (whenIDENc;ccequals0), total use equals imports from cctoc.

As well as tari¤s, there are two types of taxes:

Use taxis assumed to be an ad valorem tax on all use of goods class iin country c. Use tax revenue is given by

T U Yc= X i HUc;iP Ti;cc+ X cc P Ti;cc;cEXi;cc;c=(1 +tmargini;cc;c) !! (1 +usetaxc;i): (3.A2.40)

Output tax, OUTTAX is a tax per unit value of output of an industry, as explained in (3.A2.10).

Total consumer expenditurein countryc; CEc, is taken as equalling value added from all industries inC +monopoly pro…ts from all industries in countryC +total tari¤ revenue in countryC +output tax revenue +use tax revenue -total subsidies - the trade balance of country C (assumed to be constant and exogenous) - interest on net capital imports paid at the world rate. Hence,

CEc = X

i

V Ac;iP Vc;i+X

i

Yc;iP Yc;iM Mc;i

+XccX

i

(EXi;cc;cP Mcc;iT Ri;cc;c)=100)

+X

i

OTc;i V Ac;iP Wc;i+X

ii

IIc;ii;iP Uc;ii !

+T U Yc T SU BYc BOTc KMcRBc: (3.A2.41)

The Balance of Trade, BOTc, (including long-term net capital payments) is as- sumed to be …xed. BOTc =X i X cc EXi;c;ccP Mc;i X cc

EXi;cc;cP Mcc;i KMcRBc: (3.A2.42)

The reason this includes long-term net capital payments is to cover the version of the model where capital is internationally mobile: in this case, we would expect interest to be paid at rate RBc on net capital imported from abroad, KMc;and one would expect this to involve country ceither exporting more or importing less.

Key assumed parameter values

Demand side:

The top level utility function is Cobb-Douglas in functional form (so the elasticity of substitution between consumption of the produce of each industry,i, is unity). Share parameters for each product class are calibrated from value shares in total expenditure.

The lower level utility function has an elasticity of substitution between goods g in industry i of :This is assumed to equal 4 in all industries.

Supply side: production technology is assumed to be Cobb-Douglas, so elasticities of substitution between inputs are unity, and share parameters can be directly calibrated from shares in total costs (once monopoly pro…t has been subtracted from costs).

In the absence of other data, …rm sizes have been set equal within each industry across countries, such that Iron and Steel and Heavy Manufacture are seen as the least competitive industries (1 and 3 …rms respectively in our smallest region, Hungary), followed by Other Primary, Light Manufacturing, Textiles and Food Processing (4-6 …rms per industry in Hungary). Services and (especially) agriculture have much smaller …rm sizes, and so are far more competitive. Larger markets (like the EU) have more …rms, and so are more competitive. The main reason for these assumptions is to simulate the pro-competitive e¤ects of trade liberalisation in reducing monopolistic mark-ups in

In document Análisis y evaluación de la cartera de crédito de la Cooperativa de Ahorro y Crédito Fernando Daquilema de la ciudad de Riobamba, provincia de Chimborazo, periodo 2015 (página 118-131)