Problem Set 3. 1 Offshoring as a Rybzcynski Effect. Economics 245 Fall 2011 International Trade

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1 Due: Thu, December 1, 2011 Instructor: Marc-Andreas Muendler E-mal: Economcs 245 Fall 2011 Internatonal Trade Problem Set 3 November 15, Offshorng as a Rybzcynsk Effect There are two ndustres 1 and 2 and two factors of producton: non-offshorable labor N and offshorable labor L. Non-offshorable labor earns a wage s and offshorable labor earns a wage w. Each ndustry s producton functon Q = AF (N, L ) s homogeneous of degree one. The foregn country s producton functons are dentcal up to a Hcks-neutral productvty parameter: Q = A F (N, L ). Suppose A < A. Throughout ths queston, assume that factor-prce equalzaton occurs: w/w = s/s = A/A and that A and A are constant. Introduce one addtonal type of trade: frms can have L-tasks performed offshore. Onshore and offshore tasks are not perfect substtutes. They are combned through a Cobb-Douglas producton functon L = (L on ) 1 γ (L off ) γ, where L off denotes offshored labor and γ (0, 1) captures the ntensty of onshore labor. When offshored, a foregn unt of labor costs βw for the home economy to contract foregn labor at a dstance. To standardze the analyss, consder ndustry 1 to be relatvely ntensve n offshorable L-labor and ndustry 2 ntensve n non-offshorable N-labor. Only the offshorng cost parameter β n the model s free to change. 1. Derve the elastcty of substtuton between L on [Hnt: Derve optmal nputs L on express L on as a functon of L off.] and L off and L off : d ln L on /d ln L off for gven L. as functons of p Q / L and each other, and 2. Under these functonal forms, s there offshorng for βw > w? In other words, s A < A a necessary condton for offshorng? Is there two-way offshorng (home contractng from abroad and abroad contractng from home)? Show that your answers hold for any functon L = G(L off lm L off ; L on ) that satsfes the Inada condtons lm L off ; L on )/ L off = 0. G(L off 1 0 G(L off ; L on )/ L off = and

2 3. Use the frst-order condtons for optmal nputs to show that ˆL off = ˆL on ˆβ = 1 1+γ ˆL 1 γ 1+γ ˆβ, where hats over a varable x denote relatve changes ˆx d ln x. 4. Show that, n equlbrum, total offshorable labor supply to the home economy s L = L 1 + L 2 = (L on ) 1 γ (L off ) γ. [Hnt: Use the fact that L on /L off s a constant across both ndustres for gven β.] 5. Factor market clearng s equvalent to a L1 Q 1 + a L2 Q 2 = (L on ) 1 γ (L off ) γ a N1 Q 1 + a N2 Q 2 = N for the unt labor requrements a Lj = L j /Q j = [(L on ) 1 γ (L off ) γ ]/Q j and a Nj = N j /Q j. Show that α L1 ˆQ1 + (1 α L1 ) ˆQ 2 = (1 γ)ˆl on off + γ ˆL α N1 ˆQ1 + (1 α N1 ) ˆQ 2 = ˆN for adequately defned α L1 and α N1. State α L1 and α N1. 6. What does nelastc labor supply and the absence of cross-border mgraton mply for and ˆN? ˆL on 7. Use nelastc labor supply and the result from 3 to show that ( ) ( ) ˆQ1 γ 1 αnx = ˆQ 2 α LX α NX α ˆβ. NX Under the assumptons made n the begnnng, what are the sgns of ˆQ 1 and ˆQ 2? 8. Much of the emprcal lterature on wage nequalty and trade uses wage-bll shares n estmaton. Defne the onshore wage-bll share of non-offshorable labor n ndustry as θ on N = sn wl on + sn. Show that the relatve change n the wage-bll share of non-offshorable labor s ˆθ N on = (1 θn)(ŝ on ŵ + ˆN on ˆL ). 2

3 on 9. Suppose factor-prce equalzaton holds. Use the above results to derve and ˆθ N2 as functons of θn on, parameters and ˆβ. on How do the ˆθ N responses to ˆβ dffer n sgn? How do ther responses to ˆβ dffer n magntude? 2 Helpman, Meltz & Yeaple (AER 2004) and Horzontal FDI There are two countres, and there s a contnuum of frms n each country. In each country lves a measure of L d consumers, who nelastcally supply one unt of labor and own the shares of domestc frms. The L d representatve consumers have dentcal CES preferences over a contnuum of varetes [ 2 U d = s=1 q sd (ω) σ 1 σ ω Ω sd dω ] σ σ 1 wth σ > 1, where s denotes the source country and d the destnaton country of a varety shpment. Each frm produces one varety ω. A frm s producton technology s constant returns to scale gven the frm s productvty φ. Frms draw φ from a Pareto dstrbuton F (φ) = 1 (b s /φ) θ. It wll be convenent to call all frms ω wth a gven productvty level the frms φ. Frms choose to enter ther respectve home market and any foregn destnaton. There are two modes of entry nto the foregn destnaton: exports from the respectve home market, or horzontal foregn drect nvestment. There are ceberg transportaton costs τ sd between countres for exportng. There s a fxed cost F D to enter the domestc market, a fxed cost F X for exportng to the foregn market, and a fxed cost F I to enter the foregn market through horzontal FDI. 1. Show that demand for a varety q sd (ω) s q sd (ω) = (p ( ) sd) σ 1 (P d ) 1 σ y dl d wth P d p 1 σ 1 σ sd dω. ω Ω sd 2. Show that proft maxmzaton of frm wth productvty φ mples: p sd (φ) = σ τ sd w s φ wth σ = σ σ Show that a frm s gross operatonal profts from producng n source country s and shppng to destnaton market d are ( ) σ 1 Pd φ y d L d Π(τ sd w s ) σ τ sd w s σ. 3 ˆθ on N1

4 4. Show that net profts are Π(τ ss w s, F D ) for natonal non-exporters, Π(τ sd w s, F X ) for exporters, and Π(τ dd w d, F I ) for horzontal multnatonals, where Π(τ ss w s, F D ) = Π(τ sd w s, F X ) = Π(τ dd w d, F I ) = ( ) σ 1 Ps φ y s L s σ w s σ F D ( ) σ 1 Pd φ y d L d σ τ sd w s σ F X ( ) σ 1 Pd φ y d L d σ w d σ F I. 5. Derve the followng break-even ponts for a frm as productvty thresholds: φ D (breakeven between shutdown and natonal non-exportng), φ X (break-even between natonal nonexportng status and exportng), and φ I (break-even between exportng status and horzontal multnatonal status). What chan of nequaltes do (w d /w s ) and the fxed costs need to satsfy so that φ D < φ X < φ I? 6. Is t possble to fnd condtons so that φ D < φ I < φ X? Is t possble to fnd condtons so that φ X < φ D < φ I? 3 Translog Cost Functons Burgess (REStat 1974) has extended Chrstensen, Jorgenson & Lau s (REStat 1973) sngle-product translog (transcendental logarthmc) cost functon to the case of multple products (such as products shpped to N dfferent destnaton markets or made n N dfferent source countres): ln C j = α α k ln Q k j + τ l ln w l + l=1 λ kl ln Q k j ln Q l j χ kl ln Q k j ln w l δ kl ln w k ln w l, (1) where the subscrpt j denotes a frm or an ndustry, dependng on applcaton, Q l j s output at or for locaton l, and w l s a factor prce at or for locaton l. There are N locatons that dfferentate the product. 1. Is the cost functon (1) separable n ndvdual products for product-level cost functons c l j( ) so that C j (Q j ; w) = l cl j(q l j; w)? 4

5 N 2. For (1) to be homogeneous of degree one n factor prces for any gven output vector Q j, parameters must satsfy certan condtons. What condton does N l=1 τ l have to satsfy? What does N l=1 χ kl have to satsfy for all k? What condton do the sums N δ kl, N l=1 δ kl and N l=1 δ kl have to satsfy? By symmetry, we must have δ kl = δ lk. How many symmetry restrctons are there for N locatons? Now consder captal K l a quas-fxed factor n the short run. Followng Brown & Chrstensen (equaton of chapter 10 n Berndt & Feld 1981: Modelng and measurng natural resource substtuton), one can augment (1) to a short-run translog multproduct cost functon ln C V j = α α k ln Q k j + τ l ln w l + l=1 λ kl ln Q k j ln Q l j κ k ln Kj k + ζ kl ln Kj k ln w l χ kl ln Q k j ln w l δ kl ln w k ln w l µ kl ln Kj k ln Q l j (2) ψ kl ln Kj k ln Kj. l 1. What addtonal condton on N l=1 ζ kl s now needed for lnear homogenety of (2) n factor prces? 2. Use Shepard s Lemma to derve frm or ndustry j s demand for factor l from (2). 3. Show that the cost share of factor l n j s total costs C V j s θ l j = τ l + χ kl ln Q k j + ζ kl ln Kj k + δ kl ln w k. 4. The constant-output cross-prce elastcty of substtuton between factors l and k s defned as ε lk ln / ( ) Xl j 2 C j Cj = w k, ln w k w l w k w l where X l j s factor demand. Show that the second equalty follows from Shepard s Lemma. Derve the cross-prce elastcty of substtuton (l k off dagonal) and the own-prce elastcty (l = k on dagonal) for the translog cost functon C V j. 5

6 5. The partal Allen-Uzawa elastcty of substtuton between two factors of producton l and k s defned as / ( ) σlk AU 2 C j Cj C j C j = ε lk, w l w k w l w k θj k where ε lk s the (constant-output) cross-prce elastcty of factor demand and θ k j s the share of the kth nput n total cost. Show that the second equalty follows from Shepard s Lemma. Derve the Allen-Uzawa elastcty on and off the dagonal for the translog cost functon C V j. 6. Morshma elastctes are superor to Allen-Uzawa elastctes. Blackorby & Russel (AER 1989) show that, among other benefts, Morshma elastctes preserve Hcks s noton that the elastcty of substtuton between two factors of producton should completely characterze the curvature of an soquant. Allen-Uzawa elastctes fal n ths regard when there are more than two nputs. The Morshma elastcty of substtuton can be derved as a natural generalzaton of Hcks s two-factor elastcty and s defned as / ( ) / ( ) σlk M 2 C j Cj 2 C j Cj w l w l = ε w l w k w k ( w l ) 2 kl ε ll, w l where ε kl s the (constant-output) cross-prce elastcty of factor demand. Show that the second equalty follows from Shepard s Lemma. Derve the Morshma elastcty on and off the dagonal for the translog cost functon C V j. [Note: Morshma elastctes are nherently asymmetrc because Hcks s defnton requres that only the prce w l n the rato w l /w k vary.] 6