Taste for variety: Model of Grossman and Helpman.

Variety expansionIntroduction◮ In the end of 1980-s economists mainly agreed upon theunsatisfactory results of neoclassical growth theory;◮ According to empirical findings,less than 50 percent of growthmight be attributed to the growing capital - labour ratio;◮ Hence, the half of growth rate is due to Solow residual;◮ Thus the models endogenizing technology as a source ofgrowth emerged;◮ The first generation of such models were of taste-for-varietytype;◮ The main idea of these was to find some source of growth,other that capital productivity (K/L ratio);◮ The model of Grossman and Helpman achieved nice formaldescription of these ideas in 1990.

Variety expansionThe ModelOverall structure◮ There are two countries in the world economy;◮ Each country is engaged in three types of productiveactivities:◮ Final output, being consumption good, Y ;◮ The continuum of varieties of differentiated middle products(intermediaries);◮ Research and development (R&D).◮ There is a single primary factor of production;◮ This factor is fixed, constant, specific to the country i:L i = const.◮ Capital consists of all the intermediaries, as in Romer’s model.

Variety expansionThe ModelFinal goods producers◮ The final good is produced in a perfectly competitive way;◮ Hence the prices are set through the marginal pricing rule,P = MC;◮ Each such producer solves the problem of profit maximization:( ( ∫ nΠ i (Y i ) = p Yi BA i L 1−β∫ n− w i L Yi −0Y i0p xi (ω)x i (ω)dω →) β/α )x i (ω) α dω −max . (2)L Yi ,x i (ω)

Variety expansionThe ModelMiddle product demand IIFrom Eq. 4 we derive demand for the middle products ω = k,ω = l:( ∫x i (k) = p xi (k) − 11−α ×(βp nYi BA i L 1−βx i (l) = p xi (l) − 11−α ×(βp Yi BA i L 1−βY iY i0( ∫ n0) β/α−1 ) 1x i (ω) α 1−αdω; (5)) β/α−1 ) 1x i (ω) α 1−αdωDenoting the term in brackets as X one may rewrite demand asx i (k) =(6)( pxi (l)) 11−αx i (l). (7)p xi (k)

Variety expansionThe ModelMiddle product demand IIINow insert this last equation into the expression for the output:X =(( n∫ ) β/α )βp Yi BA i L 1−β1Y ix i (ω) α 1−αdω0( n∫ ) 1x i (ω) α 1−αdω0(8)This using Eq. 7 will give the demand for each intermediateproduct as function of total output, X and prices:x i (k) = X ×n∫0p xi (k) − 11−αp xi (ω) 1− 11−α dω(9)

Variety expansionThe ModelFinal output price◮ Using Eqs. 3, 4 we obtain the price of the output as afunction of factor prices and productivity:p Yi =( wi) 1−β ( ∫ nA i0p X (ω) 1−ɛ dω) β/(1−ɛ);(10)ɛ = 11 − α > 1.◮ Middle products are freely traded between both countries andhave the same price in both countries,p xi = p xj = p X . (11)

Variety expansionThe ModelMiddle products producers◮ Middle products producers engage in oligopolistic competitionin every country;◮ Every producer sets the price of his/her product as tomaximize profits:π i (ω) = (p X (ω) − w i a LXi ) ×p X (ω) −ɛβ ∑ n∫p X (ω) 1−ɛ dω i0p Yi Y i . (12)where a LXi is the unit labour requirement in country i forproduction of intermediaries.◮ The profit is per unit profit (in brackets) times the demand forthe middle product.

Variety expansionThe ModelPrices for middle products◮ The price is set according to the fixed mark-up rule:αp X (ω) = w i a LXi ; (13)◮ From this it is straightforward that all the varieties fromcountry i has the same price, p Xi ;◮ With equal prices for all the nation-specific varieties, theequation for final good’s price, Eq. 10, is condensed intop Yi =( wiA i) 1−β (∑◮ And variety price in country i isjn j p 1−ɛX j) β/(1−ɛ)(14)p Xi = 1 α w ia LXi . (15)

Variety expansionThe ModelOutput of intermediaries◮ With prices given by two last equations, the aggregate outputof middle products is:X i =∑jn ip −ɛX in j p 1−ɛ β ∑X j jp Yj Y j (16)◮ Being the function of:◮ Number of producers in the country, n i ;◮ Price of the variety (identical across varieties), p Xi ;◮ Total output of final product in both countries.

Variety expansionThe ModelComments◮ So far the model follows that of Romer (1990) very closely;◮ The role of fixed human capital is delegated to labour with nosignificant differences;◮ There is no difference in levels of production of differentvarieties;◮ This is the direct consequence of homogeneity of knowledge;◮ In economic terms, the demand system for intermediariesyields similar demands and prices;◮ There is no difference in productivities gain of varieties beingused in final good production;◮ This implies similar prices, which imply similar levels ofproducing these varieties.

Variety expansionThe ModelResearch sector◮ Resources devoted to research generate blueprints;◮ These blueprints expand the measure of differentiatedproducts;◮ Research is made by private, profit maximizing entrepreneurs;◮ Their profit is th fraction of profits of oligopolies producingthe newly invented middle product:◮ They receive indefinite patent protection;◮ Blueprints are not tradable, thus all varieties have to beproduced in the country of origin;◮ There is the free entry of new entrepreneurs into the R&Dsector in any country.

Variety expansionThe ModelExpected profits from R&D◮ The free entry requires, that the number of oligopolisticproducers is defined from the expected zero profit condition;◮ Profits of inventors are equal to costs of research then;◮ Zero profit condition implies then:∫ ∞te −(R(τ)−R(t)) π i (τ)dτ = c ni (t), ∀t. (17)◮ Expected discounted stream of profits from producing themiddle product is equal to costs of inventing this product atany given time t.

Variety expansionThe ModelNo-arbitrage conditionDifferentiating Eq. 17 w. r. t. to time we obtain the so-calledno-arbitrage condition:π i (t) + c ˙ ni (t)c ni (t)= ˙ R(t). (18)instantaneous return on shares of variety producing firms must bethe same as the instantaneous interest rate.

Variety expansionThe ModelKnowledge growth◮ The R&D activities in each country has two effects:◮ The increase i the measure of varieties, ni which may bepatented;◮ And the growth of productivity Ai , which decreases theamount of labour needed for the invention of new products.◮ This second effect is immediately widespread across the worldand is non-excludable;◮ It is the by-product of new products inventions;◮ This last depends on the achieved productivity of labour (dueto the second effect):˙ n i (t) = L ni K(t)/a Lni ; (19)where K is the current stock of knowledge and a Lni is theproductivity parameter.

Variety expansionThe ModelKnowledge growth II◮ Knowledge is growing proportional to the achieved knowledge;◮ There are no diminishing returns;◮ By choosing the units for K such as the proportionality factoris 1, we have:K(t) ˙ = ∑ L ni K(t)/a Lni . (20)isince knowledge is not country-specific.◮ The acquisition of knowledge is costless and free;◮ Hence the costs of product development are:c ni = w i a Lni /n. (21)

Variety expansionThe ModelHouseholds◮ Households consume only the final product, Y ;◮ They have identical, homothetic preferences worldwide:◮ Final products from two countries are imperfect substitutes forthem;◮ They maximize life-time time-separable utility subject tointertemporal budget constraint:U t =∫ ∞te −ρ(τ−t) log u(y 1 (τ), y 2 (τ))dτ, (22)where y i (τ) is the consumption of final good from country iat the time τ.

Variety expansionThe ModelIntertemporal optimization of consumption◮ Assuming linearly homogeneous instantaneous utility function,the solution has two stages;◮ First, one maximizes instantaneous utility at time τ for somefixed E(τ);◮ This generates the indirect utility function, v(p Y1 (τ), p Y2 (τ));◮ At the second stage one has to choose optimal time patternof expenditures to maximize this indirect utility;◮ This generates the condition for expenditures plan.

Variety expansionThe ModelDerivation of optimal conditionThe intertemporal indirect utility has the form∫ ∞V t = e −ρ(τ−t) {log v(p Y1 (τ), p Y2 (τ)) + log E(τ)}dτ. (23)tand have to be maximized subject to the constraint on resources:∫ ∞∫ ∞e −(R(τ)−R(t)) E(τ)dτ ≤ e −(R(τ)−R(t)) w(τ)Ldτ + Z(t). (24)ttwhere Z(t) is the value of consumer’s asset holdings at time t.

Variety expansionThe ModelOptimal expenditures◮ The maximization of intertemporal consumption path leads tothe condition on expenditures:E(t) ˙E(t) = R(t) ˙ − ρ; (25)◮ Which means, that expenditures should grow at the same rateas the interest rate on assets holdings minus discount rate;◮ Consumers may invest their holdings into firms’ shares with Rrate of return or into riskless bonds with ρ rate of return;◮ The condition on optimal expenditures is the no arbitragecondition, meaning, that composition of portfolios should notmatter.

Variety expansionEquilibriumEquilibrium definitionIn this model equilibrium is defined by the set of equations:◮ Prices for goods (both final and middle ones) are defined;◮ Asset returns are related by no arbitrage condition;◮ Market clearing for final goods;◮ Market clearing for factors markets.Two first points are derived within the model already:( wi) 1−β (∑ ) β/(1−ɛ);p Yi =n j p 1−ɛAX jijp Xi = 1 α w ia LXi ;π i (t) + c ˙ ni (t)c ni (t)E(t) ˙E(t) == ˙ R(t);˙ R(t) − ρ. (26)

Variety expansionEquilibriumProducts markets clearing◮ At every point in time markets for final goods in bothcountries should be in equilibrium;◮ Static equilibrium implies, that:in both countriesp Yi Y i = s i (p Y1 , p Y2 )E; (27)◮ The function s i (•) is the function (fraction of expenditures)derived from intertemporal optimization of consumption;◮ It is homogeneous of degree zero;◮ Clearing condition means that consumption of final goods isequal to the share of world spending allocated to final productby consumers.

Variety expansionEquilibriumDemand for labour◮ Labour market clearing condition equates labour supply anddemand in each country;◮ Demand of labour by final goods producers is:L D Y ii= (1 − β)p Yi Y i /w i ; (28)◮ The demand for labour by middle goods producers is:L D X ii= a LXi X i ; (29)◮ Demand for labour by product developers is:L Dn ii= (a Lni /n) n˙i . (30)

Variety expansionEquilibriumLabour market clearingThe labour market clearing requires that total demand given by thesum of Eqs. 28,29,30 above equal to total labour supply:(1 − β)p Yi Y i /w i + a LXi X i + (a Lni /n) n˙i = L i (31)where L i is constant and fixed.

Variety expansionEquilibriumSteady state conditions◮ The necessary conditions for convergence of the model to thesteady state growth path (or BGP) are:p X1 = n(a LX1 /a Ln1 );p X2 = n(a LX2 /a Ln2 ). (32)◮ These are conditions on prices of middle products (all of them)as function of relative labour productivities in both countries;◮ These conditions imply that relative prices of middle productsare constant along the convergent path;◮ This implies that relative wages and relative final goods pricesare also constant;◮ Then expenditure shares s i are also fixed.

Variety expansionEquilibriumBalanced growth rates◮ Denote by g the growth rate of the number of productsworldwide:g(t) =n(t)/n(t) ˙ = K(t)/K(t); ˙(33)since knowledge grows in the 1-to-1 proportion to the numberof products;◮ Then prices of middle products (and all other prices) alsogrow at this rate:p X1 ˙ /p X1 = p X2 ˙ /p X2 = w˙1 /w 1 = g. (34)

Variety expansionEquilibriumComparative advantageDenotee = E/n; σ i = n i /n, b i = (a Lni /a LXi ) α , (35)◮ The first is expenditure-to-variety;◮ The second denotes the fraction of worldwide varieties beingproduced in country i;◮ The last denotes the comparative advantage of thecountry in conducting R&D;◮ With b 1 > b 2 country 1 has a comparative advantage ininventions.

Variety expansionEquilibriumGrowth rateWith these notions the growth rate of the world economy may bedescribed asg = ∑ in˙i /n = H − β e σ − 1 − β se, (36)αwhere:H = ∑ L i /a Lni - total effective labour force;is = ∑ s i /b i - product shares; σ = ∑ σ i b i - weighted average ofiicomparative advantages.

Variety expansionEquilibriumCommentsThe equation for overall growth rate 36 implies that:◮ The intersectoral structure of allocation is summarized in σ;◮ It is in inverse proportion to the growth rates;◮ It grows only if the number of differentiated products in acountry with comparative disadvantage exceeds that in theother country;◮ Spending per middle product worldwide decrease growth rates;◮ The two variables which govern the world economy, are e andσ as some overall charactersitics of allocation of expendituresand production.

Variety expansionEquilibriumExpenditures growth rateThe evolution of the world economy may be described by the pairof equations:◮ The first relates rate of growth of expenditures per middleproduct to the growth rate:ėeβe= Ė/E − g =ασ + 1 − β se − H − ρ; (37)α◮ This rate is greater, the greater is spending per product;◮ It is smaller, the greater is the share of products beingproduced in the county with comparative disadvantage;

Variety expansionEquilibriumDistribution of varieties dynamics◮ The second describes the evolution of production of middleproducts across countries;◮ Rates of change of product shares are:σ˙i /σ = n˙i /n i − ṅ/n; (38)◮ At the same time, the sum of changes is zero;◮ Hence, one may characterize the rate of product shareschange by a single equation:˙σ = ∑ iσ˙i b i = h − 1 − β (α e − σ H − 1 − β )α se , (39)h = ∑ iL i /a Lni b i .

Variety expansionEquilibriumDynamics◮ The whole system is governed by equations on e and σ;◮ This is a 2-dimensional autonomous system;◮ The curve ė = 0 has a positive slope in e − σ space;◮ However, ˙σ = 0 curve has a different form depending onparameters;◮ There are two distinct cases of dynamics;◮ They are defined by the conditions:hH < 1 s ; h H > 1 s . (40)

Variety expansionEquilibriumGeneral remarks◮ In these different cases authors define long term determinantsof the growth rates;◮ New intermediate products permit greater furtherspecialization in production in two countries;◮ The one with more efficient (comparatively) R&D producesmore intermediaries while the other produces final products;◮ The specialization leads to welfare gains in both countries;◮ The improvements in R&D being made in the backwardcountry will not speed the overall growth;◮ Thus complete concentration of R&D activities in the leadingcountry is efficient.

Variety expansionEquilibriumLiterature◮ Grossman G. and Helpman E. (1990) Comparative Advantageand Long-run Growth. The American Economic Review, Vol.80, No. 4, pp. 796-815;◮ Romer P. (1990) Endogenous Technological Change. Journalof Political Economy, Vol. 98, No. 5, Part 2, pp. S71-S102;◮ Ethier W. (1982) National and International Returns to Scalein the Modern Theory of Trade. The American EconomicReview, Vol. 72, No. 3, pp. 389-405.