A Two-Sector Model of Home Production

Transcription

1 A Two-Sector Model of Home Production Cak Hong Leo Lam, Cong K. Yip Department of Economics, Cinese University of Hong Kong January 3, 207 Abstract: Tis paper develops a two-sector model of ome and market production to examine te role played by ome production on te development of te aggregate economy. Based on te time-use data across countries, we allow for a fraction of te ome-produced good to be sold to te market sector so as to understand te relation between te size of te ome production sector and economic growt. Wen te size of te ome sector canges, its macroeconomic e ects depend on te capital intensities of te two sectors. We nd tat te e ect of ouseold production on te SNA (System of National Accounts) -output can be decomposed into a direct e ect from its scale and an indirect e ect troug te sectoral reallocation of labor, working at opposite directions. (Very preliminary; please do not quote) JEL Classi cation: E23, E32, O, O7. Keywords: ome production, time allocation, factor intensity, macroeconomic stability. Correspondence: Cong K. Yip, Department of Economics, Te Cinese University of Hong Kong, Satin, Hong Kong. Tel: (852) , Fax: (852) , congkeeyip@cuk.edu.k.

2 Introduction Time allocation on ouseold activities is always te focus of aggregate economic analysis. An interesting aspect about te time allocation pattern of te macroeconomy is tat it as been fairly stable over time. For illustration purpose, we use te American Time Use Survey wic is te most compreensive and consistent annual time-use survey since We plot te time-use pattern in Figure and nd tat te time use allocation from 2003 to 204 as very little variation. Figure about ere In addition, it is also observed tat more time is allocated to leisure in developed countries, wereas more time is allocated to marketized ouseold production [te category of SNA (System of National Accounts) productive activities in time-use data sets] in developing countries. To illustrate tese patterns, we ave picked six countries as examples, including United States, Spain, Japan, Cina, Pakistan and Gana. We believe tis set of countries is representative because it includes OECD countries in di erent geograpical locations (US, Japan and Spain), fast-growing countries (Cina) and developing countries (Gana and Pakistan). From Figure 2, it is clear tat developed countries ave iger leisure tan developing countries. Figure 2 about ere Altoug te SNA productive activities are iger in developed countries tan developing countries (except for Cina), te locations (establisment versus ome) were tey are carried out are very di erent between te two groups. We nd tat te SNA productive activities of establisment in Cina is similar to te oter developed countries, wereas te ome-based SNA productive activities in developing countries are generally iger tan tose in developed countries. Since bot te ouseold SNA productive activities and te non- SNA productive activities are carried out witin te ouseold, let us combine tem into a single category. Tis grouping can indirectly provide a measure for te size of te ouseold sector. From Figure 3, we can ten see tat countries from earlier stages of development would ave a iger portion of teir ouseold production marketized. Figure 3 about ere In general, ouseolds allocate teir time into tree major uses: work, leisure and rest. From te macroeconomic perspective, work can be subdivided into ome production and market production. Tus, te focus of our paper is to Countries oter tan US usually will not perform te time use survey every year. Tey usually carry out te survey on a ve-year basis. However, we could presume tat te stability in time-use allocation is universal, so tat we can comparison time-use pattern across countries of data from sligtly di erent years. We ave compared te 2008-US, 2008-Spain, 2006-Japan, 2008-Cina, 2007-Pakistan and 2009-Gana. 2

3 develop a dynamic model to study te e ect of te scale of ouseold production on te economic performance of a country. Our main nding is tat tere are two con icting components in te e ect of te scale of ouseold production on te SNA-output: a direct e ect from te scale and an indirect e ect troug te sectoral reallocation of labor. Te direct e ect can be easily understood. Wen tere is less ouseold production being marketized, national income would decrease by te value of per-capita ouseold production since it is part of te te SNA-output. For te indirect positive e ect from te reallocation of labor, it is sown tat it depends on te size of te SNA-output. Wen te proportion of ome production increases, more labor is reallocated to te market sector so tat te SNA-output would increase by te product of te marketized per-capita output and te reallocated labor. It is reasonable to believe tat te direct negative e ect would be dominated by te indirect e ect positive e ect troug te reallocation of labor for advanced economies. Te intuition is tat te more developed a country is, te larger te relative size of its market sector. Tis provides us wit te result wic is consistent wit te observation from te time-use survey: te more ouseold production is marketized, te poorer a country in terms of te SNA production is. Related Literature Evidence on te importance of ome production in te macroeconomy is well documented. For instance, Hill (985), wo analyzes te Micigan Time Use Survey, and observes tat a married couple spends on average 28.2% or 4.6% of total time working at ome, depending on weter or not sleep is included in total time, wic is only sligtly less tan wat it works in te market. Eisner (988) reports a wide range of estimates of te size of te ome sector, from 20% to 50% of GNP. Greenwood and Hercowitz (99), wo sow tat te capital used in te ome sector, de ned as consumer durables plus residential capital, is bigger tan te capital used in te market sector, de ned as business non-residential capital; teir average ratio for te period in te US is about.3. McGrattan et al. (997), using maximum likeliood tecniques on postwar US time series, nd tat a model witout ome production is rejected in favor of a model wit ome production. Using te Panel Study of Income Dynamics (PSID) for te years , Rupert et al. (995) also nd tat, at least for single females and married couples, tere is evidence of a fairly ig willingness to substitute between ome and market consumption. Recently, Ngai and Pissarides (20) explores te relation between market ours and ome ours and nds tat tere are large di erences in te allocation of work time across countries due to di erences in taxes and subsidies. Tere as been work on studying te role of ome production in an aggregate model. By introducing a ome good sector in an oterwise standard real business cycle model, Benabib, Rogerson and Wrigt (99) found tat its quantitative performance was signi cantly improved in terms of te uctuations of output, labor ours, investment and consumption. Greenwood and Hercowitz (99) developed a Beckerian macroeconomic model of ouseold production to 3

4 study te allocation of capital and time across te market and ome sectors. Perli (998a) developed a propagation mecanism based on non-market production to ave increasing returns of labor productivity. As a result, equilibrium indeterminacy is possible so tat consumption and labor can move togeter over some periods of time to generate output persistence tat observed in te US data. Perli (998b) proposed tat cycles driven by self-ful lling propecies can exist wit external e ects in labor and capital tat are sensibly small wit ome production. Following Benabib et al. (99), Perli introduced a ome sector in wic te agents use labor and capital to produce a good wic is used only for teir own personal satisfaction. In equilibrium, a propecy can be self-ful lled wen te combination of more labor and a iger return to labor exists. Te mecanism is tat wen agents worore by drawing labor out of leisure and non-market work, tis causes a cange in te desired allocation of capital. If te agents are going to invest more in te market tan at ome, te aggregate labor demand may sift up enoug to sustain te propecy as an equilibrium. Tis is te same logic as proposed by Benabib and Farmer (996) in a two-sector model wit sector speci c externalities. Recently, Ngai and Pissarides (2008), by di erentiating ome production from leisure due to its marketization, combined structural transformation and marketization forces to account for te trends in time use observed in long-run data and teir relation to economic growt. Speci cally, te U-sape of te market ours over time comes from te fact tat te ome production of agriculture and manufacturing is marketized in early industrialization; ten te ome production of services rst rises but nally is marketized. Te organization of te paper is as follows. Te next section gives te basic description of te two-sector model and conducts te steady-state equilibrium analysis. Section 3 provides te comparative statics: te e ect of te scale of ome production on various macroeconomic variables. Section 4 studies te dynamical system of te model, and Section 5 concludes. 2 A Two-Sector Model of Home Production In tis section, we present te set-up of te basic two-sector model: a sector wit ome production and anoter wit market production. Te two sectors are closely linked on four aspects, including allocation of capital, allocation of labor inputs, allocation of consumption bundle and te proportion of ome-produced goods wic is sold in te market. 2. Te Basic Setup Houseold Suppose tat te representative agent cooses te aggregate consumption pats C t, working ours L t and capital K t to maximize te discounted lifetime 4

5 utility: Z [ln C t v (L t )] exp ( t) dt (P) subject to 0 _K t r t K mt + w t L mt C mt K t + p t ( ") Y t ; (C) C t "Y t ; (C2) were v (L t ) is a convex function (v 0 > 0; v 00 0) and > 0 is te constant time preference rate. Te rst constraint is a standard budget constraint wit total income coming from production of market goods (r t K mt + w t L mt ) and ome goods (p t ( ") Y t ). Te ome production sare parameter " indicates te sare of ome production tat are for self uses wereas te remainder ( ") will be sold in te market wit an implicit relative price p t. For ome production, we adopt a standard neoclassical production function: Y t H (K t ; L t ) : Aggregate consumption is a composite bundle of market goods C mt and ome goods C t : C t C(C mt ; C t ) Te factor market clearing conditions are L t L mt + L t ; K t K mt + K t : Firms We assume standard neoclassical tecnology for te production of te market goods: Y mt F (K mt ; L mt ), Since factor markets are perfectly competitive and factor are fully mobile across sectors, we ave r mt and w mt, were r t and w t are te rental rate of capital and wage rate of labor, respectively. 2.2 Functional Forms Di erent speci c functional forms are applied to te model for computation purpose. We adopt te following: Houseold: Constant Elasticity of Substitution Consumption: C t C mt + ( )C t i 5

6 were > > 0; < Indivisible Labor: v (L t ) BL t Firm: Cobb-Douglas Production Function: so we get: Y mt F (K mt ; L mt ) A m K m Y t H (K t ; L t ) A K mt Lmt m t L t r mt A m ( K mt L mt ) m and w mt A m ( K mt L mt ) m 2.3 Hamiltonian Set-up For convenience, time notation (:) t is omitted. To solve te maximization problem (P), we can de ne as te costate variable associated wit K and set up te current-value Hamiltonian: ^H [ln C B(L m +L )]+(rk m +wl m C m K+pA (K K m ) L pc By solving for te necessary rst-order conditions, we ^H : Cm + m : ( C m + ( )C )C i C m. i ( )C (.2) m )A m m B ^H : p( ) ( ") A )"A k B ^H : m A m km m p ( ") A k "A m (.5) K : _ + ma m m (.6) Te necessary conditions (.) to (.6), togeter wit oter equilibrium and market clearing conditions, can be used to solve te problem (P). Importantly, we ave to ensure te su cient condition tat te optimal Hamiltonian is concave in K. Ten we specify oter conditions: 6

7 Equilibrium conditions: : _K rk m + wl m + p ( ") A k L C m K (.7) : C "A k L (.8) De ne : k K L ; Km L m ; k K L ; l Lm L : Factor market clearing implies: K K m + K (.9a) L L m + L (.9b) k l + k ( l) (.9c) Perfect factor mobility implies: MP K m pmp K or MP L m pmp L (.0) (.0) and (.5) imply: p Terefore from equation (.) and (.2): C m C p (p) (.) C ( + p) Tis implies tat te relative consumption ratio is determined by te relative price between market and ome produced goods. From equation (.3) and (.4), we get: p m A m km m A k Tis equation sows tat te relative price is determined by te sectoral production tecnology, output elasticities of capital and labor and capital intensity. 2.4 Steady-state Analysis We will sow an equilibrium analysis in te steady state, in wic te growt rates of various variables is zero. In te steady-state, 0, so we ave: _ K _ K 7

8 k ( + p ) A km m k m p m A m () m A (k ) Solving te above equations simultaneously, we get: km ( A m m + ) m k ( A m m + ) m m m km; were is de ned as m m p (A m m ) m A ( + ) m m From equation (.), we are also able to solve for: ( C m ) [p C ] (A m m ) m [ A ( + ) m m ] And from equation (.8) we get: ( C L ) " y Terefore, we ave: " A ( A m m + ) m m m ( C m L ) ( C m C ) ( C m C ) [p ] " y Tis result is important tat in te following sections we would apply it to understand te e ect of " on various variables. 8

9 _ K Given K 0, we can also solve te labor allocation l and capital intensity k from (7) and (9): A m (k m) m l k ( C m ) l L k l k k km k Solving simultaneously, we get: + p ( ")A (k ) l k and ( l) k km k 0 ymk k + ( Cm L ) km p ( ")y ym + Cm L p ( ")y (km k ) X(") Z(") l k k km k ( Cm L ) p ( ")y + (ym km) + ( Cm L ) p ( ")y + 2 (0; ) 3 Comparative Statics Tis section would present te e ect of " on oter variables. Terefore, we would calculate various @" Cm L ) C L ) C @" Te detailed steps are straigtforward but tedious (we placed to te appendix part wic is available upon request). We only summarize te results and give te intuition ere. Cm L [p > 0 ] y () Summary sows te e ect of " on te market consumption per ouseold labor. It seems tat tis comparative static does not ave muc economic meaning but it is crucial to be sown ere because it is a stepping stone for calculating te oter results. [(p > 0 ) + p ] y (y m km) [Z(")] 2 (2) 9

10 Positive e ect on l of ": Tis means wen tere is lower portion of ouseold production is marketized, more labor is allocated to te market production and ence less to te ouseold production. Tis result is meaningful in te sense tat te labor allocation, in tis model, varies according to marketized products. It is because of te competitive wage across sector, return to labor in bot sector depends on te value of te products in te market. Wen te ouseold production is not marketized, labor would rater engage in market production. [(p ) + p ] y (y m km) (km k ) [Z(")] 2 E ect on k of " depending on capital intensity: Witout te capital intensity di erence (km k ), k is positively related to " by te l e ect. Wen tere is a lower proportion of ouseold production is marketized, labour would rater work in te market tan ouseold. And if te capital intensity of market sector is iger tan tat of te ome sector, te overall capital intensity would increase due to te expansion of market sector troug l e ect. Anoter important igligt is tat bot km and k are unrelated to ", tis means tat te size of marketized ome sector only in uence te overall capital intensity but not sectoral capital intensity. Here we de ne te per capita SNA-output(y s ): Summary 4 Te main result: were y s [Y m + p ( ") Y ]L l y m + ( l)p ( ") y l y m + ( l)p y ( l)(p y [y m + p y ( ")] l " ( l)p y Direct and indirect e ect on y s : We de ne y s to be te per-capita SNA-income. Te comparative statics include two e ects: a direct e ect of " and an indirect e ect via l. Te direct e ect of " is ( l)py, wic can also be understood as te value of ouseold production per capita. Wen tere is more ouseold production being marketized (" decreases), te SNA-output would increase by te value of per-capita ouseold production from tis direct e ect. For te indirect e ect given by [y m + p y ( ")] l ", te term [y m + p y ( ")] is te marketized output per-capita and l " is te e ect of " on labor allocation. Wen te proportion of marketized ouseold production increases (" decreases), te SNA-output would 0

11 decrease by te product of te marketized per-capita output and te reallocated labor. Tis is because te reallocation of labor toward te market good as a positive net e ect on overall market production. It is reasonable to believe tat te direct e ect from " would be dominated by te indirect e ect positive e ect troug te reallocation of l for advanced economies. Te intuition is tat te more developed a country is, te larger te relative size of its capital-intensive sector. In terms of our notation, tis translates into a very smaller number of ( ")py relative to y m. Tis provides us wit te result wic is consistent wit te observation from te time-use survey: te more ouseold production is marketized, te poorer a country is. We next study te e ects of " on di erent per capita consumption variables: Summary Cm L Cm L " [(y m km) (p y k )] Z Summary C L C L " [(y m km) (p y k )] Z Summary C L C L " [(y m km) (p y k )] Z E ect on per-capita consumption depending on sectoral di erence in net income: From te above comparative statics, te e ect of " on per capita consumption depends on tree elements: te equilibrium level of te per capita consumption, te size of te non-marketed ouseold production(") and te sectoral di erence in net output value [(ym km) (p y k )]. To ave a better illustration, we can express te e ect in form of elasticity, de ned as: "Cx L " ( Cx L ) Summary 8 we would obtain te following results: "Cm "C "C [(y m k m) (p y k )] Z Te above elasticities sow a result tat te percentage cange in per-capita consumption given a cange in " would be te same across all sector. Te key element a ecting te elasticity is te te sectoral di erence in net output value, wic may also be understood as te relative strengten of te market and ome sector. If [(ym km) (p y k )] is positive wic implies tat te <

12 market sector is stronger tan te ome sector, increased " would give rise to te per-capita consumption across all sector, vice versa. To understanding te term [(ym km) (p y k )], we start from te steady state equilibrium and obtain p y ( m) ( ) y m and k ( m) ( ) m (ym km) (p y k) ( m ) ( m ) y m ( ) ( ) m m m y m m ( ) m [ m y m ] m ( ) m m y m m ( ) Recall te dynamic equation: _ Am m km m ( + ) m m y m ( + ) m Tis sows tat in steady state equilibrium, m y m ( + ) i So we get tat my m is positive, and terefore te wole term (ym km) (p y k ) only depends on ( m ). Tis term is directly related to te sectoral capital intensity ( k ), wic means tat saying tat ( m ) > 0 if and only if ( k ) > 0. y s [Y m + p ( ") Y ]L l y m + ( l)( ") py Here we would like to study two extreme cases, were te cannel between market and ome production is fully open or sut down (" 0 and " ) : y s j "0 l y m + ( l) py y s j " l y m 2

13 Recall tat te equilibrium labor allocation l : [p l (ym km) + [p ] "y p ( ")y + k ] "y p ( ")y + k We can study te extreme cases of l by taking limit: lim "! l [p (y m k m) + [p ] y + k ] y + k (p y lim "!0 l k ) (ym km) (p y k ) Recall tat l is de ned between 0 and, tis makes te above result di cult to andle. If we adopt te non-negativity constraint of l to te limit lim l, we "!0 must ensure te denominator is negative under wic m <. However, tis will render a result tat lim l > wic is also contrary to te assumption of "!0 0 l. A better way to andle te limit is to allow it to be negative. We sould be alerted tat we allow negative lim l does not implies tere exist l to be "!0 negative in our model. Recall tat: [(p ) + p ] y (y m km) [Z(")] 2 > 0 l is strictly monotone to ". Te negative lim "!0 l implies tat tere exists a critical value (^") were l crossed from positive to negative and we can obtain te value by setting l 0: [p [p l 0 (ym km) + [p ] "y + k p ( ")y ^" p p y k + p ] "y p ( ")y + k y ] "y p ( ")y + k Tis result gives us an implication tat te market sector would sut down (l becomes 0) before te cannel is fully open (" becomes 0). Anoter important result is tat lim "!0 l is restricted to be negative so (y m k m) (p y k ) must be positive. Hence, m >,wic means te market sector is more capital intensive tan ome sector ( > k ), is obtained. Te e ect of " on overall capital intensity and per-capita consumption becomes certain. < 3

14 Remark 9 Since we conclude tat te market sector is more capital intensive tan ome sector ( > k ), > Cm L > C L ) > C L ) Te intuition is tat wen more labor-intensive good is consumed directly and sifted away from market production (" increases), market production becomes more capital intensive. As more labor is allocated toward te capital-intensive > 0), more capital is needed so tat k increases as well. As a result, all types of consumption per capita increase. Te net e ect on per-capita SNA-income ten depends on te two con icting forces: te direct negative e ect of te ome-production good vs te indirect positive e ect of te market-production good. We summarize all of our comparative statics in Table. 4 Local dynamics In tis section, we would sow tat te steady state is a saddle by caracterizing te model as a two-by-two system of and K, and ten linearizing it around its steady-state ( ; K ). Rearranging Equation (.3), we can express in terms of : From Equation (7) and (5): B ( m )A m () k k () p p() Summing up equation (.) and (.2), we get: m From Equation (6): C m + pc Cm C p"y L "y L C m pc p"y L p"y L L L () 4

15 Looking into te _ K equation: _K m A m ( K m ) m K m + ( m )A m ( K m ) m L m ( + c ) C m K + p( ")A K L m L m m A m km m L m + pa k L [( + c ) C m + pc ] K y m L m + py L py L + ( ) py L K ( m ) L m K + ( ) K ( m ) m Terefore we can caracterize te 2x2 system: K L _ + A m m [ ()] m i + ( ) ( m ) _K py L m K K Jacobian Matrix: _ a 0 _K a 2 a 22 K K a A m m ( m ) 2 m " m B ( m ) A m a 22 y m > 0,(at steady-state, y m + ) m m + Tus te determinant of te Jacobian matrix is negative so tat te steady state is a saddle. 5 Concluding Remarks In tis paper, we ave developed a two-sector model of ome and market production wit endogenous leisure. Te novel feature of te model is tat a fraction of te ome production can be marketized. It is sown tat te market sector is capital intensive relative to te ome production sector. As a result, our comparative statics ndings sow tat as te less ome production is marketized, per capita capital and output rise. Tis also leads to an increase of all types of per capita consumption. However, te net e ect on marketized output is ambiguous. Tis is consistent wit te cross-country time use data tat we present in te introduction. Te intuition is tat wen more labor-intensive # < 0 5

16 good is consumed directly and sifted away from market production, market production becomes more capital intensive. As more labor is allocated toward te capital-intensive > 0), more capital is needed. As a result, all types of consumption per capita increase. Te net e ect on percapita SNA-income ten depends on te two con icting forces: te direct negative e ect of te ome-production good vs te indirect positive e ect of te market-production good. Our model can be readily applied to policy analysis. We plan to study bot income and consumption taxation in te above framework so as to sed ligt on te stability nature of balanced budget policies in developed and developing economies. Tere as been a debate in te literature weter consumption taxation is preferred to income taxation in terms of macroeconomic stability (e.g., Giannisarou, 2007). Wit our two-sector framework, we can explore te relation between te stability nature of di erent balanced budget policy rules and te development stage of te macroeconomy. 6

17 Time Allocation in USA % 0% 20% 30% 40% 50% 60% 70% 80% 90% 00% SNA procuctive activities Non SNA production activities Leisure Meal Study and training Sleep Oter Personal care Figure Time Allocation Gana-2009 Pakistan-2007 Cina-2008 Spain-2008 Japan-2006 US % 0% 20% 30% 40% 50% 60% 70% 80% 90% 00% SNA procuctive activities(establisment) SNA procuctive activities(ouseold) Non SNA production activities Leisure Personal care Meal Study and training Sleep Oter Figure 2

18 portion of marketed and non marketed ouseold productive activities Gana Pakistan Cina Spain Japan US 0% 0% 20% 30% 40% 50% 60% 70% 80% 90% 00% SNA procuctive activities(ouseold) Non SNA production activities Figure 3 Table. (before proving α m > α ) Variables(Steady-state) Comparative Statics of ε Effect k m 0 0 k 0 0 ( p 0 0 C m C ) 0 0 l k ys [(p β β ) φ +p ] y (y m δk m ) [Z(ε)] 2 + [(p β β ) φ +p ] y (y m δk m ) (k m k ) [Z(ε)]? 2 [y m + p y ( )] ( C m L ) [(y εδ l ε ( l) p y k ) ( p y δk )] σ εcm m m ( C Z? δk )] Z? L ) σ εc [(y m δk m ) (p y ( C L ) σ εc [(y m δk m ) (p y δk )] Z? Table.2 (after proving α m > α )? k [(p β β ) φ +p ] y (y m δk m ) (k m k ) [Z(ε)] 2 + ( C m L ) σ εcm [(y m δk m ) (p y δk )] ( C Z + δk )] Z + L ) σ εc [(y m δk m ) (p y ( C L ) σ εc [(y m δk m ) (p y δk )] Z +

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