Agriculture and aggregation

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1 Agrculture and aggregaton Juan Carlos Córdoba a,marlarpoll b, a Department of Economcs, Rce Unversty, Houston, TX, USA b Department of Economcs, Unversty of Pttsburgh, Pttsburgh, PA, USA Aprl, 2008 Abstract We derve a multplcatve representaton of the aggregate producton functon n a two-sector economy. Ths representaton ncludes a dversfcaton component that s a functon of factor allocaton across captal-ntensve nonagrculture, and land-ntensve agrculture. Snce ths component s large n very poor countres, ther productvty levels are lower than the mpled by Solow resduals. Keywords: Aggregate producton functon; Dversfcaton; Solow resduals; Development accountng JEL classfcaton: O11; O13 1 Introducton It s a well-known fact that n less developed countres agrculture accounts for a larger share of output and employment than n rcher economes. In addton, agrculture s more ntensve n the use of land than nonagrculture. These facts suggest the exstence of dfferences n sectoral composton across countres, and n producton technologes across sectors. How do these sectoral dfferences affect the aggregate representaton of the producton functon? And, how do they affect our understandng of the sources of cross-country ncome dfferences? Ths paper studes the sources of cross-country ncome dfferences n a world composed by a large number of small open economes, each of whch produces an agrcultural and a nonagrcultural good. We characterze the aggregate producton functon for the general case n whch sectors dffer n ther land, captal, and labor ntenstes. We assess the extent to whch the Solow resduals obtaned from the standard one-sector model properly characterze the underlyng technologcal dfferences Correspondng author. Tel.: ; fax: E-mal address: rpoll@ptt.edu (M. Rpoll) Ths paper corresponds to the frst part of a manuscrpt ttled "Agrculture, aggregaton, wage gaps, and crosscountry ncome dfferences" 1

2 across countres. Fnally, we use the derved aggregate producton functon to perform a varance decomposton of output per worker. The analyss yelds three man results. Frst, when sectoral technologes are Cobb-Douglas, we can derve a multplcatve representaton of the aggregate producton functon that s sutable for varance decomposton. Ths representaton ncludes a dversfcaton component that captures the effcency gans of operatng both sectors, and s a functon of factor allocaton across captalntensve nonagrculture, and land-ntensve agrculture. Second, the dversfcaton component s negatvely correlated wth per capta ncome. Ths negatve correlaton s manly drven by the very poor countres n the sample. We fnd that the technologcal gap between, for nstance, Uganda and US n nonagrculture s 15% larger than what s mpled by Solow resduals. Fnally, we fnd that although productvty n very poor countres s lower than the one mpled by Solow resduals, development accountng exercses for the cross-secton of countres usng our methodology are only margnally dfferent from those usng one-sector models. Our paper s related to recent work on agrculture and development (Chanda and Dalgaard, 2008; Golln et al., 2004; Restucca et al., 2008; Temple and Woessman, 2006; and Vollrath, 2008), but t s closest to Casell (2005). A dstngushng feature of our work s that we explctly derve a multplcatve representaton of the aggregate producton functon for a two-sector model, and characterze the Solow resdual as a functon of factor allocaton across sectors. We develop multplcatve results that allow the applcaton of varance decomposton technques from one-sector models nto two-sector models. The remander of the paper s organzed as follows. Secton 2 derves the aggregate producton and provdes a quanttatve assessment of the dversfcaton component. Secton 3 presents a development accountng exercse usng a cross-secton of countres n Concludng remarks are n Secton 4. 2 Aggregate producton functon In ths secton we derve the aggregate producton functon for a two-sector, three-factor economy. Our results can be easly extended to the more general case n whch the number of sectors s smaller than the number of factors. 1 Consder a world composed of a large number of small open economes. 1 When the number of sectors s larger than the number of factors, some of the sectors wll not operate. 2

3 There are two goods, agrculture a and nonagrculture n, and three factors of producton, physcal captal K, effectve labor H, and arable land T. Factors are moble across sectors wthn a country, but nternatonally mmoble. Goods can be traded nternatonally, and output prces p a and p n are determned n nternatonal markets. A country can produce good usng a constant returns to scale producton functon A F (K,T,H ), where A s the country-specfc TFP of sector. Functon F s dentcal across countres. Assume that F (K,T,H )=K αk T αt H αh wth α K + α T + α H =1and 0 <α f < 1 for f = {K, T, H}. We assume agrculture to be land ntensve n the sense that α T a α T n. 2.1 Effcent level of producton Denote by A p A the prce-adjusted TFP level n sector. The central planner n each country chooses the allocaton of factors across sectors that maxmzes the aggregate level of output Y, Y = G(K, T, H; A) max {K,H,T } =a,n [A a F a (K a,t a,h a )+A n F n (K n,t n,h n )], (1) subject to, K a + K n = K; H a + H n = H; T a + T n = T ; (2) and gven A [A a,a n ]. We assume that A a and A n are such that an effcent allocaton s nteror. If functons F are dentcal across sectors (F = F ), an nteror soluton requres the restrcton A a = A n. As a result, the aggregate producton functon G takes the form G(K, T, H; A) = AF (K, T, H). To analyze the more general case n whch F a 6= F n,denotek K/H and t T/H, and defne B A n /A a.letk K /H and t T /H be optmal allocatons for = {a, n}. A property of these allocaton rules s that they depend only on the ratos (k, t; B) but not on the exact levels (K, T, H; A),.e.,k = k (k, t; B) and t = t (k, t; B). Proposton 1 For any = {a, n}, effcent producton satsfes Y = G(K, T, H; A)= A {z} TFP F (K, T, H) {z } Factors Z (k, t; B) {z } Adjustment (3) where k α K µ Z (k, t; B) =µ t α T α K k t k k + α T t t + α H. (4) 3

4 Proof: Optmal allocatons k,t for = {a, n} and h a can be found usng three frst-order condtons of the form: F a j (k a,t a, 1) = BF n j (k n,t n, 1) for j = {K, T, H}, and resource constrants: k ah a + k n (1 h a)=k, andt ah a + t n (1 h a)=t. In the Cobb-Douglas case, ths fve-equaton system can be reduced to one nonlnear equaton n h a. Gven h a, the system of equatons can be used to obtan k,t. Fnally, for any sector the aggregate producton functon can be wrtten as Y = G(K, T, H; A)=A F (K, T, H) α K F K (K,T,H ) F K (K, T, H) + αt F T (K,T,H ) F T (K, T, H) + αh FH (K,T,H ) FH, (K, T, H) whch n the Cobb-Douglas case has the form specfed n (3) and (4). The man vrtue of equaton (3) s that t provdes a multplcatve formulaton of aggregate producton n three fundamental components. The frst two are the tradtonal TFP and factors components. They descrbe the total producton of the economy f all of factors are placed n only sector, a reference sector. Such decson s suboptmal snce, by assumpton, the soluton s nteror. Term Z, whch s necessarly larger than 1, adjusts the output mpled by the frst two terms to obtan the effcent level of output. Thus, Z captures gans from dversfcaton. 2.2 Solow resduals The typcal decomposton of the sources of cross-country ncome dfferences (Klenow and Rodrguez- Clare 1997; Hall and Jones 1999) assumes an aggregate producton of the form Y = AK α T β H 1 α β, (5) where α ' 1/3, β s typcally set to zero, and A s the Solow resdual. A lmtaton of usng (5) for cross-country comparsons s that t may not descrbe well the producton possbltes of agrcultural countres, n whch land plays a more central role than captal. The aggregate producton functon of these countres s better descrbed by (3). In order to make (3) consstent wth extensve growth accountng lterature that regards (5) as a satsfactory descrpton of ndustralzed countres, we make assumptons to guarantee that (3) s approxmately equal to (5) for countres n whch agrculture s a smaller share of output. Ths s guaranteed by choosng nonagrculture as the reference sector, and by assumng that the 4

5 nonagrcultural producton functon s gven by (5). Under these condtons, (3) can be wrtten as Y = A n Z n K α T β H 1 α β, (6) whchcombnedwth(5)mples A = A n Z n (k, t; B). (7) Ths equaton provdes two man nsghts. Frst, Solow resduals are a re-scaled verson of nonagrcultural TFPs. Second, the sze of the re-scalng, Z n, s not dentcal for all countres and t depends on factor endowments. If, as we confrm below, Z n s larger for poorer than rcher countres, then Solow resduals underestmate underlyng productvty dfferences between rch and poor countres n the nonagrcultural sector. 2.3 Quanttatve exercse In ths secton we compute Z n for a sample of countres. We construct p a Y a and p n Y n followng Casell (2005), who uses FAO 1985 data to measure PPP agrcultural output. 2 Aggregate K s computed usng the perpetual nventory method on nvestment data from Penn Word Tables. Average human captal h = H/L s measured usng schoolng years from Barro and Lee (2000), and followng Hall and Jones (1999) to adjust for Mncer returns. We use data on arable land T and labor force n agrculture L a from the World Bank. Gven a specfc set of nputs shares α f,weusetheeffcency condtons and resource constrants of the model to compute K,T, and H. In addton, Z n s obtaned usng equaton (4). Regardng nput shares, we consder two scenaros. Frst, we assume as most accountng exercses that labor share s 2/3 for all countres (α H a = α H n =0.66). We also assume that agrculture s land-labor ntensve and nonagrculture s captal-labor ntensve and set α K a = α T n ' 0. Although these assumptons better descrbe agrcultural technologes n poorer economes, agrculture s a very small fracton of output n rcher countres. Fgure 1 portrays the correspondng Z n relatve to the US n 1985 at dfferent levels of ncome. Two observatons emerge. Frst, Z n s larger n poor than n rch countres. For example, Z n s around 1.15 for Uganda. Ths means that the technologcal gap between Uganda and US n 2 See p

6 nonagrculture s 15% larger than what s mpled by Solow resduals. For the other very poor countres, ths number ranges between 3 and 7%. Ths mples that the dsperson of nonagrcultural TFPs s larger than the dsperson of Solow resduals. The second concluson from Fgure 1 s that Z n quckly decreases wth the level of ncome, and for most countres outsde the poorest range, t s n the order of 1%. Consder now a second scenaro that relaxes the assumpton of an equal labor share n both sectors. Golln (2002) reports that n most countres, regardless of ther level of development, α H a <α H n. For nstance, for the US n 1992, α H a ' 0.25 whle α H n ' 0.7. Wefnd that f α H a =0.33 and α T a =0.69, thenz n s sgnfcantly larger than n the frst scenaro. Specfcally, t s around 1.34 n Uganda, and between 1.08 and 1.17 for most other poorer countres. In sum, our exercses suggest that very poor countres are not properly represented n standard models that abstract from land and agrculture. 3 Development accountng Tradtonal development accountng exercses wrte (5) as y Y L = e A ex (8) where ex (K/Y ) α 1 α β (T/Y) β 1 α β h, and ea A 1 1 α β. The analogous formulaton for (6) s y Y L = A X Z. (9) where X (K/Y ) αk n /αh n (T/Y) αt n /αh n h s the factor ntensty, A A 1/αH n n s TFP, and Z Z 1/αH n n s the dversfcaton component. Followng Klenow and Rodríguez-Clare (1997), the contrbutons of factors, TFP, and the dversfcaton components to output dsperson can be defned as: Φ X = cov ln y, ln X var (ln y) ; Φ A = cov ln y, ln A var (ln y) ; Φ Z = cov ln y, ln Z. var (ln y) where Φ X + Φ A + Φ Z =1. Snce Z s larger for poorer countres (Fgure 1), then Φ Z s negatve. Accordng to equaton (7), standard accountng measures the contrbuton of TFP as Φ A + Φ Z, and as a result, t downplays 6

7 the role of TFP. Equaton (4) suggests that snce Z n depends on factor endowments, then part of Φ Z should be added to Φ X. Snce t s unclear how to exactly assgn Φ Z,wedvdeΦ Z equally between TFP and factors components. 3 Specfcally, we defne the contrbutons of factors, Φ X, and TFP, Φ A,as Φ X = Φ X + Φ Z /2; Φ A = Φ A + Φ Z /2. Table 1 reports development accountng for the two-sector economy. For nonagrculture we use the standard values α H n =0.66 and α K n =0.33. The frst row reproduces results for a one-sector model, accordng to whch TFP accounts for 64% of the varance of output per worker. Ths percentage ncreases slghtly to 67% n our two-sector model. The relatve small ncrease n the contrbuton of TFP s due to the fact that, except for the very poor countres, the adjustment factor Z n s quanttatvely small. 4 Concludng remarks Ths paper derves a representaton of the aggregate producton functon n a two-sector model. The representaton has two specal attrbutes: frst, t s sutable for varance decomposton; and second, t explctly derves the dversfcaton component Z n for the case of Cobb-Douglas technologes. Although theoretcally, the presence of ths dversfcaton effect should ncrease the role of TFP n explanng cross-country ncome dfferences, we fnd the quanttatve effects to be small. However, the dversfcaton component s large for the group of very poor countres. Research n understandng the specal characterstcs of these very poor economes s warranted. References [1] Barro, R., Lee, J-W., Internatonal data on educatonal attanment: Updates and mplcatons. CID Workng Paper # 42. [2] Casell, F., Accountng for cross-country ncome dfferences. In: Aghon, P., Durlauf, S. (Eds.), Handbook of economc growth. Elsever, Klenow and Rodrguez-Clare splt the covarance term equally between factors and TFP. 7

8 [3] Chanda, A., and Dalgaard, C. J., Dual economes and nternatonal total factor productvty dfferences: Challengng the mpact from nsttutons, trade and geography, forthcomng, Economca. [4] Golln, D., Parente, S., and Rogerson, R., Farm work, home work and nternatonal productvty dfferences. Revew of Economc Dynamcs 7, [5] Golln, D., Gettng ncome shares rght. Journal of Poltcal Economy 110, [6] Hall, R., Jones, C. I., Why do some countres produce so much more output per worker than others?. Quarterly Journal of Economcs 114, [7] Klenow, P. J., Rodríguez-Clare, A., The neoclasscal revval n growth economcs: Has t gone too far?. In: Bernanke, B., Rotemberg, J. J. (Eds.), NBER Macroeconomcs annual MIT Press, Cambrdge, [8] Restucca, D., Yang, D.T., and Zhu, X., Agrculture and aggregate productvty: A quanttatve cross-country analyss. Journal of Monetary Economcs 55, [9] Temple, J., and Woessman, L., Dualsm and cross-country growth regressons. Journal of Economc Growth 11, [10] Vollrath, D., How mportant are dual economy effects for aggregate productvty?, forthcomng, Journal of Development Economcs. 8

9 Fgure 1: Dversfcaton term Z GDP per worker relatve to the US Table 1. Development accountng α K a α T a α H a Factors TFP % 64% % 65% % 67% 9