The Dixit-Stiglitz demand system and monpolistic competition.

Transcription

1 The Dxt-Stgltz demand system and monolstc cometton. Economcs students are generally well traned n erfectly comettve markets. Such markets are often thought to be characterzed by well defned utlty functons and homogenous roducton functons wth constant and decreasng margnal roductvtes and constant returns to scale. Such models are very useful for understandng a wde set of economc mechansms. Increasng returns are known to be feature n many real world roducton rocesses. Often such roducton technologes are sad to generate natural monooles snce an mlcaton of ncreasng returns s that large roducton enttes are more roductve than smaller ones. Stll, we know that many markets are characterzed by many roducers roducng dfferent varetes of the same roducts under ncreasng returns to scale. A stylzed descrton of such markets s the market tye monoolstc cometton. Such markets are characterzed by many roducers who each enoy some market ower but by free entry so that roft oortuntes are lmted. Markets wth monoolstc cometton catures can easly be modeled wth ncreasng returns to scale roducton technologes. Ths has shown very useful for many alcatons. Below we wll consder nternatonal trade. We wll demonstrate that such markets generate trade between smlar countres and trade wthn the same ndustres, so called ntra ndustry trade. These are known to be of great mortance for real world trade flows. Snce such trade can hardly be exlaned by comaratve advantages, trade theory s frutfully sulemented wth monoolstc market aroaches. However, monoolstc markets models have also had wde alcatons n many other economcs models, as e.g. growth theory, envronmental economcs, macroeconomcs and mcroeconomcs. The modelng framework resented below therefore have many alcatons. The demand sde Monoolstc comettve markets must be characterzed by a demand sde that catures many roduct varetes. One secfc aroach for modelng ths s to ntroduce a reresentatve consumer who always demands the exstng varetes. Ths can be nterreted lterally so that one assumes that every consumer refers varety n the consumton basket. Conseuently the demand system we wll descrbe s sometmes referred to as the love of varety aroach. Ths may be msleadng however, snce the underlyng utlty functon mles less love of varety than standard utlty functons of the Cobb-Douglas tye (see below). Another nterretaton s that the demand system s for a reresentatve consumer who s an aggregate of many consumers wth dstnct ndvdual references for each varety. nder some assumtons (whch we wll not dscuss), t can be showed that such an aggregaton may gve rse to the reference structure for a reresentatve consumer that we wll ntroduce. The reresentatve consumers references are descrbed wth the utlty functon: (, ) 2,..

2 Above,, denotes uantty of consumton good and s the elastcty of substtuton among varetes. Generally we wll assume that >. However, we wll show that f, the above utlty functon s a Cobb-Douglas utlty functon for goods and wth eual exendture shares (eual to /). Frst however, note that f, the utlty functon s ust the sum of consumed uanttes of each varety:, when The reason for ths s that f, both the exonents (wthn and outsde the bracket) goes to one. Therefore the case when descrbes the case when the goods are erfect substtutes. In ths case, consumers do not care f one good s substtuted for an eual uantty of another. If however, f, the utlty functon s not easy to nterret. We wll rewrte the utlty functon wth the use of the arameter (-)/: Take natural logarthm of the above exresson to obtan: ln ln ln ote that f, 0. Lettng aroachng zero gves rse to an exresson of the tye (0/0). Ths can be handled wth l Hotal s rule whch allows dervaton of the nomnator and the denomnator wth resect to. Ths roduces the exresson: ln l ' Hotal ln ln ln ln lm 0 e ln The above exresson s a Cobb-Douglas functon for consumton of goods where each has an exendture share eual to (/). We have droed the subscrts and suerscrts from the summaton sgn above and wll only rentroduce t when necessary henceforth.

3 The reference for varety can easly be seen from the utlty functon f we assume that all goods have the same rce so that, and are consumed n the same amount so that. In ths case, the utlty functon can be wrtten: Wrtng consumers ncome as Y, we know that Y and therefore that Y/. Substtutng ths nto the above gves: Y Y Y Snce >, ths mles that ncreases wth the avalable number of varetes. The hgher s, the less does utlty deends on the number of varetes. Ths s n lne wth ntuton. ow assume that ncomes are generated from a mass of labour L that earns w er unt. Ths wll generalze to total ncome n an economy oulated wth L workers. Therefore the budget constrant s. Consumers want to maxmze ther utlty gven ths budget constrant. The corresondng Lagrange functon s: ( ) L Frst order condtons for utlty maxmzaton are: L In the second lne above the frst order condton was somewhat smlfed. In the thrd lne the frst order condton s reeated for another good,. Maxmzaton reures that smlar condtons are fulflled for all goods. From the frst order condtons we obtan:

4 The second lne s smly a reorganzaton of the frst. In the thrd lne we have solved for. In the fourth lne we multly wth. In the ffth we sum over all. The second eualty remnds us that ths summaton s over. Therefore elements n the sum wth subscrts can be multled outsde the summaton sgn. From ths we can solve for : snce The second s the demand functon for good. It deends on the rce,, on total ncome, and on an exresson that s a functon of all rces n the economy. Ths s a functon of a rce ndex for all goods gven by: ( ) Therefore the demand functon can be wrtten as: ote that elastcty of demand s ust :

5 d d Why s a rce ndex for all rces? It can be shown that ths s the cost for obtanng one unt of utlty. Form the frst order condtons we have: Insert ths exresson nto the utlty functon: Agan we have set the elements contanng subscrt outsde the summaton snce summaton runs over. From the above, solve for : E In the second euaton we multled the soluton for wth. In the thrd lne we summed over all s. Snce the sum over all s and over all s nvolves summng over all rces the sums can be exressed as n the last eualty n thrd lne. In the second lne we

6 set and wrte the corresondng sum as E. These are the exendtures for one unt of utlty and therefore the rce ndex for all goods. The roducton sde (to be added) Eulbrum (to be added) Trade (to be added).