Homework 5 Solution (pdf)

* −σyy(p / p)τAnd so, c h = and c f =* * 1 σL + L (( p / p)τ )−1−σ* * 1−σLτ+ L ( p / p)Substituting these into the market-clearing condition yields* 1−σy* y(p / p)τy = L+ L.* * 1−σ1−σ* * 1−σL + L (( p / p)τ ) Lτ+ L ( p / p)And simplifying1 x(1) 1 = +,1−σ 1−σ1+lrx 1/ l + x / r*where x = p / p , = 1−σ*r τ < 1, and l = L / L > 1 .Clearly, (D+D * )/y (the RHS of (1)) is increasing in*x = p / p . And evaluating the total*demand for each home variety at x = p / p = 1 shows that it falls short of supply.y 1 1 1/ l + 1/ r + 1+lr( + ) = yy1+ lr 1/ l + 1/ r 1/ l + 1/r + r + l< .*Therefore, in trade equilibrium we must have x = p / p > 1. Of course, the same is true*for w / w .c) Derive a version of the gravity equation in this monopolistic competition model withtransportation cost. Make sure that one of the terms in the equation is the product ofGDPs of each country. Discuss. How does τ affect trade?* *** −σ* L w f L GDP [ α(σ −1)/β ]( p / p)τ p LwVT = 2 D pNτ= 2 c p τ =( ) τ** 1−σ* * 1−σw ασ w Lτ+ L ( p / p)w ασ=p σ −11w σβ Lτ1(1/ p)τ*+ L ( p*−σ*2GDP×GDP*−σ* 1−σ/ p)p( )w1−σ2 * τ1= GDP×GDP( )1−σ* *L L ( p / p)1 −στ +wNormalize w ≡1 (we can do that but just for one country):*GDP × GDPVT = 2.* 1L + L ( x / τ )−σ*d) Determine d ( p / p)/ dτ when τ = 1. Discuss.1 x1 = +1−σ 1−σ1+lrx 1/ l + x / rImplicitly differentiating (1) yields2 2 2dxl /(1 + l)− l /(1 + l)= −dr12 1/ l + 1−(1 −σ)r=(1 −σ) l /(1 + l)−2(1/ l + 1)l(1− l)1−l= −= < 0 ,2(1 −σ) l − l − l σ σ (1 + l)and

dxdτ τ = 1=dxdrr = 11−σ1−l(1 −σ) =σ 1+lσ −1l −1= ∈(0,1).σ 1+le) Compare utilities for each consumer in autarky and under trade when τ = 1. What isthe effect of higher transportation cost, τ , on welfare in each country? Use Excel (or*other software) to plot p / p and consumer utility under trade as functions of τ (e.g., set*σ = 2 and L / L = 2 ).Demand by each home consumer for the foreign varieties is* −σ* * 1−σ−f , h w(piτ)w ( p / p )( p ) τci=*=N1−σ* * 11−σN * 1−σp + ( p τ ) p Np + N ( p τ )∑i=i∑i=i[ ( 1) / ]( / )L(p / p ) + Lτand demand by each foreign consumers for the foreign varieties is* * −σf , f w ( pi)ci=*N 1−σN * 1−σ( p τ ) + ( p )∑ = i ∑ =iiiσ−σ* −σα σ − β p p τ f , h= = c* 1−σ* 1−σ( )[ (p N(pτ) + N ( p ) L(p / p)And so, the welfare of the home country under trade isuhome== N ( c1)/τ]+ L** 1−σw pα σ − βf , f= == c*1−σ* * 1−σ* σ −11−σ*N h,h h,h( σ −1) / σ N∑ ( ) += 1 ∑ =)ic i( c+ N( c)*i 1h,h ( σ −1) / σ * f , h ( σ −1)/ σy= N (L + L (( p / p)τ )y=ασ( σ −1) /y=ασ( σ −1) /y=ασ( σ −1) /y=ασ( σ −1) / σσσf , hi)( σ −1) / σy(p / pL(p / p )) τ+ Lτ* −σ( σ −1) / σ *( σ −1) / σ) + N ()* * 1−σ* 1−σ* 1−σ1[ L(*L + L ( xτ)1−σ1 ([ L()* 1−σL + L x r1 ([ L()* 1−σL + L x r* 1−σL + L x r* 1−σ( σ −( L + L x r))( σ −1) / σσ −1) /σ −1) /σ[ 1) / σ( σ −1) / σ 1/ σy L= (1 + l(xτ)ασ( σ −1) / σ 1/σy Lwhere Q = .ασ)1−σ1/ σ−σ* τ / x+ L (σ −1* 1Lx + Lτ)( σ −1) / σ−σ−σσ * τ / x ( σ −1) / σ+ L ( ) ]σ −1*Lx + L r−σσ * ( xτ) ( σ −1) / σ+ L ( ) ]* 1−σL + L rx( σ −1) / σy* 1−σ)] = ( L + L ( xτ) )ασ= x1−σ1/ σQ (1 + l(τ ) ) ,1/ σ],.

The welfare of the foreign country under trade isuforeign=N h,f h,h( σ −1) / σ N∑ ( ) +i=1 ∑i== N(cc i( ch,f)+ N( c*1f , fi( σ −1) / σ * f , f ( σ −1)/ σ))( σ −1) / σ( σ −1) / σ 1/ σy L1−σ1/ σ1−σ1/ σ=(( τ / x)+ l)= Q (( τ / x)+ l).ασDifferentiation yieldsd ( uhome )=1 1−σ1/ σ −1−σdx 1−σ(1 ( ) ) (1 )(1 −σ−σQ + l xτl − σ x τ + x τdτ σdτ)1 1−σ1/ σ −11−σ−σ− dx= Q (1 + l(xτ) ) l(1− σ ) x τ ( x1 τ + 1) < 0 ,σdτd ( uforeign )=1 1−σ1/ σ −1σ −2dx 1−σσ −(( / ) + ) (1 − )( −+1 −σQ τ x l σ x τ x τ ) 1−dxdτdxl −1> ( l −1),dτdxdττ = 1< 1.τ = 1τ = 1τ = 1,*The steps follow by σ > 1, τ ≥ 1, and l = L / L > 1. And so, we showed that the smallercountry suffers more due to the increase in the transportation cost.τ = 1τ = 14. Now let us go back to the HO model with 2 x 2 x 2 (“constant returns to scale anddifferent factor endowments” world). Assuming balanced trade, derive the volume oftrade (nominal for concreteness):VT = 2 p1(y1− d1),where it is held that the home country is labor-abundant and good 1 is labor-intensive.Assume that FPE obtains. You should obtain a nice expression in terms of the shares of

factor endowments of the home country in the world (home + foreign) endowment, andsome endogenous variables that do not vary with the distribution of factor endowmentsacross countries (the world endowment of each factor is fixed) as long as FPE obtains,and the home country remains labor-abundant. Discuss the determinants of trade in thiscase. Compare this with the gravity equation derived in question 3c.Solution:**VT = 2 p1(y1− d1)= 2 p1(y1− s1(y1+ y1)) = 2 p1(y1(1− s1)+ s1y1) ,wL + rKwhere s1 =is the share of the home country in the world income,w wwL + rK*L w *= L + L , and K w = L + K . To determine output of good use the full employment ofresources in the home country in HO model:y1a1L( w,r)+ y2a2L( w,r)= Ly1a1K ( w,r)+ y2a2K( w,r)= KInverting yields:⎛ y1 ⎞ 1 ⎛a2KL − a2LK⎞⎜ ⎟ =⎜⎟ .⎝ y2⎠ a1La2K− a2La1K⎝ a1LK − a1KL ⎠To determine the demand for good 1 in the home country, use the homotheticity of thepreferences:www wL + rK* wL + rK a2KL − a2LKd1 = s1y = ( y1+ y1) = .w ww wwL + rKwL + rK a a − a aVT = 2 p1(y1− d1)2( waL+ ra1K)=( a2a1La2K− a2La1K2( waL+ ra1K)=w( a a − a a )( wL + rK=( a=( a1L1L1Laa2K2K2K2( wa− aL − a1LwL + rKK −wL + rK2K( a2L1K− a1 wwK 2Lw w 2K2L(( a)L − aK)(wL+ rK1 w ww 2K2L2L1Kww− ( wL + rK)(a2KL − a2LK))+ ra)1 L 1Kwww( aw w 2KLwL + a2KLrK − a2LKwL2La1K)( wL + rK )www− wLa2KL + wLa2LK− rKa2KL +2( wa− a+ ra)( a)r[LK− KL ] − aL)w[KLK− a))2LrKa− LK1L1Kw ww ww w 2K2L2La1K)( wL + rKww1K)( a2Kr + a2Lw)K L LK[− ]w ww2La1K)( wL + rK ) K K2( wa1L+ ra= .( a a − a1L2KAnd so, given that the FPE holds and the world endowments do not change, thevolume of trade is greater for “more” dissimilar countries. It depends on the product ofthe home capital endowment and the difference between the home country and the worldwL Llabor/capital ratios, K( − ) . In contrast, with the gravity equation in the case ofwK K2LKrK])Kww)

intra-industry trade and increasing returns to scale (see Problem 3c), the country size doesw wnot matter as long the relative endowments do not change: LK − KL = z , orw wwL = K( L / K ) + z / K . Then the amount of trade does not depend on the size of thehome country, as measured by K given that( L,K ) ∈ FPE .w wL = K( L / K ) + z /Kw, and