On Indeterminacy in One-Sector Models of the Business Cycle with Factor-Generated Externalities

Transcription

1 On Indeterminacy in One-Sector Models of the Business Cycle with Factor-Generated Externalities Qinglai Meng Chong K. Yip The Chinese University of Hong Kong June 2005 Abstract By relaxing the restrictions commonly imposed on the magnitude of capital externalities with Cobb-Douglas technology in one-sector models, we find that indeterminacy can arise in the following two cases: i) the felicity function is separable in consumption and leisure and there are negative capital externalities; ii) the felicity function is non-separable and the social returns to scale in capital is greater than one. In both cases indeterminacy can happen when the labor-demand curve slopes down. We also give a necessary condition for indeterminacy with general utility function and Cobb-Douglas technology, which nests the results of the one-sector models studied in the literature. JEL classification: E00; E32, D11 Keywords: Externalities; Indeterminacy; Preferences Department of Economics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Meng: meng2000@cuhk.edu.hk. Yip: chongkeeyip@cuhk.edu.hk. 1

2 1 Introduction It has been well known that under certain market imperfection conditions such as external effects models of business cycle can be subject to indeterminacy, in that from the same initial condition there exist a continuum of equilibria all converging to a common steady state. The existence of indeterminate equilibria in turn is associated with the possibility of self-fulfilling prophecies. 1 In their seminal work, Benhabib and Farmer (1994) show, in a one-sector growth model with endogenous labor supply, that indeterminacy can occur with large positive labor-input externalities and with positive capital-input externalities in technology. In particular, for indeterminacy to arise the labor externalities have to be so large that the social returns to scale in labor should be greater than 1 and the labor-demand curve should be upward-sloping. Furthermore, the labor-demand curve should slope up more steeply than the Frisch labor supply curve. In this paper we generalize Benhabib and Farmer (1994) by relaxing some of the restrictions in their model. In particular, we ease the restrictions commonly imposed on the magnitude of capital externalities with Cobb-Douglas technology,andatthesametimeweusemoreflexible preference specifications. We find that indeterminacy can arise in the following two cases: i) the felicity function is separable in consumption and leisure and there are negative capital externalities; ii) the felicity function is non-separable and the social returns to scale in capital is greater than one. In both cases indeterminacy can happen when the labor-demand curve slopes down. These indeterminacy solutions have largely been overlooked in the literature. In addition, we give a necessary condition for indeterminacy with general utility function and Cobb-Douglas technology, which nests the results of the one-sector models studied in the literature. This paper is also related to recent studies on one-sector models, such as Bennett and Farmer (2000) and Hintermaier (2003), that explore the possibility of indeterminacy under preference specifications beyond the separable utility with unit consumption elasticity adopted in Benhabib and Farmer (1994). Hintermaier (2003) establishes, in a general setup which does not impose specific functional forms on utility, that there are no concave utility functions compatible with indeterminacy if the social returns to scale in labor is less than 1. In this paper, in contrast to these studies we do not confine apriorithe signs and magnitudes of factor externalities. Our investigation is thus complementary to Bennett and Farmer (2000) and Hintermaier (2003). 1 See Benhabib and Farmer (1999) for an extensive survey of the literature on indeterminacy and sunspots. 2

3 The next section studies the case with separable utility, and Section 3 investigates the case with non-separable utility. Section 4 presents a necessary condition for indeterminacy in the one-sector models. Section 5 concludes. 2 Separable Utility and Indeterminacy under Arbitrarily Close to Constant Returns We follow Benhabib and Farmer (1994) by assuming a large number of competitive firms, each of which produces a homogenous commodity using capital (k t )andlabor(l t ) with a Cobb-Douglas technology, y t = E t kt a lt b (1) where a + b =1,and k t and l t denote the average economy-wide levels of capital and labor. The term E t α a k t lβ b t (2) represents the factor-generated externalities, taken as given by each firm. In addition, assume that a, b, α, β>0, i.e., both the private and social production functions are increasing in the factor inputs. These assumptions are standard. However, in this paper we depart from many studies of one-sector models by relaxing the restrictions imposed on the relative magnitudes of externalities. Specifically, we do not limit to the assumption that a<α<1, asisusuallyimposedinthe literature, nor do we place any restrictions on the relative magnitudes of β and b. 2 Letting w t and r t denote the wage rate and the rental rate for capital, from the firm s maximization problem, we have (in equilibrium l t = l t, k t = k t ) w t = bkt α l β 1 t, r t = akt α 1 l β t (3) 2 In fact, many studies on one-sector models, such as Benhabib and Farmer (1994), Bennett and Farmer (2001), Wen (2001) and Hintermaier (2003), assume that social and private partial elasticities of output are related by the same positive factor of proportionality for capital and labor, i.e., α/a = β/b > 0. Such a specification is related to the notion that the production is subject to an externality from average output, that is, production function = k a l b ( k a lb ) φ,whereφ>0. 3

4 The representative consumer maximizes the present value of utility subject to the budget constraint Z 0 U(c t,l t )e ρt dt, k t =(r t δ)k t + w t l t c t, (4) and an initial condition for capital and the usual no-ponzi-scheme constraint. Here ρ is the discount rate, δ is the depreciation rate. In this section, we assume that the instantaneous utility is separable in consumption and leisure, i.e., U(c t,l t )=u(c t ) v(l t ), (5) where u 0 > 0, u 00 0, v 0 > 0, v The separable utility assumption here nests Benhabib and Farmer (1994) as a special case. 3 The first-order conditions for the consumer s problem are u 0 (c t )=λ t, (6) v 0 (l t )=λ t w t, (7) λ t = λ t (ρ + δ r t ). (8) where λ t is the co-state variable. From eqs. (3), (4), and (6)-(8), the system can be reduced to two differential equations, namely, λ t = λ t [ρ + δ ak α 1 t l(λ t,k t ) β ]=λ t [ρ + δ r(λ t,k t )], (9) k t = k α t l(λ t,k t ) β δk t c(λ t )=y(λ t,k t ) δk t c(λ t ), (10) where the implicit functions c t = c(λ t ) and l t = l(λ t,k t ) can be obtained by solving eqs. (6) and (7) jointly (using the expression for w t in eq. (3)). We assume the existence of a unique steady state, and analyze the local dynamics of the system around the steady state. To proceed, we denote by x the steady-state value of a variable x t. By linearization we have 4 3 They use U(c, l) =lnc (1 + χ) 1 l 1+χ (χ 0). 4 In this paper, we shall use the original variables for linearization without resorting to log-linearization. 4

5 λ t = J λ t λ k t k t k. (11) Here the elements of the Jacobian matrix J are j r 11 j 12 λ = λ λ r k = (ρ + δ)γ λ k (ρ + δ)(α + αγ 1) y j 21 j 22 λ dc y dλ k δ c λ [ (ρ+δ)γ ρ+δ aδ + 1 α(ρ+δ) σ ] a (1 + γ ) δ where γ = β/[1 β+v 00 (l )l /v 0 (l )], σ = u 00 (c )c /u 0 (c ) 0. Bydenotingχ = v 00 (l )l /v 0 (l ) 0, the determinant and trace of J aregivenbytheexpressions. (12) Det(J) = (ρ + δ)[ρ + δ(1 a)] aσ (1 β + χ [α + β 1 (1 α)χ βσ ], (13) ) ρ + δ Tr(J) = 1 β + χ [(1 + χ ) α β] δ. (14) a Since one variable of the two-dimensional system (11) is predetermined, and the other is free, local equilibrium indeterminacy occurs when both roots of J have negative real parts, which requires that Det(J)> 0 and Tr(J)< 0. We therefore have the following results: PROPOSITION 1: With the Cobb-Douglas technology and separable utility, local indeterminacy occurs if (i) 1 β + χ < 0, andσ > α+β 1 (1 α)χ β, β +(1 β + χ ) ρ+δ δ < (1 + χ ) α a ;or (ii) 1 β + χ > 0, andσ < α+β 1 (1 α)χ β, β +(1 β + χ ) ρ+δ δ > (1 + χ ) α a. Note first that the result in case (i) of Proposition 1 encompasses as a special case the indeterminacy result in Benhabib and Farmer (1994). They use u =logc and v =(1+χ) 1 l 1+χ, where χ 0 is a constant parameter. In our model, σ is the steady-state value of the inverse of the intertemporal elasticity of consumption, and it can be different from 1. χ is the steady-state value of the inverse of the labor supply elasticity. The condition 1 β + χ < 0 corresponds to the labor market condition in Benhabib and Farmer (1994), and it says that the labor-demand curve is upward-sloping, and slopes up more steeply than the Frisch labor-supply curve. Notice that when σ =1, in order for Det(J)> 0 and Tr(J)< 0, wehave1 >α>a. 5 This is precisely the 5 α>aholds since 1+χ = β +(1 β + χ ) <β+(1 β + χ ) δ ρ+δ < (1 + χ ) α a. 5

6 assumption made in Benhabib and Farmer (1994), so are many other studies on one-sector models in the literature. A novel result in this paper is case (ii) of Proposition 1, which is the focus of our discussion below in this section. In stark contrast to case (i) and the result in Benhabib and Farmer (1994), the condition 1 β +χ > 0 implies that the labor-demand curve is less steep than the Frisch laborsupply curve. If χ =0, then this condition is equivalent to β<1, i.e., the labor-demand curve is downward sloping. For χ 6=0, indeterminacy can also easily happen when the labor-demand curve is downward sloping. Moreover the following corollary can be readily established: COROLLARY 1: The necessary conditions for case (ii) of Proposition 1 to occur are a>α, β>b,andσ < 1. Thus, in case (ii), by allowing for larger-than-unit intertemporal elasticity of consumption (at the steady state) and negative capital externalities indeterminacy can occur. Note also that positive labor externalities are required for indeterminacy, and although the capital externalities are negative, in order for Det(J)> 0 we have α + β>1, i.e., there are increasing returns at the social level due to positive labor externalities. In fact, the degree of the social returns to scale needed for indeterminacy can be arbitrarily close to constant returns to scale, depending on relevant parameters of the model. To see this, and for simplicity, we let δ =0, then the firstandthethird inequalities in case (ii) are equivalent to β 1 <χ < βa α α, and the second inequality is equivalent to βσ +(1 α)χ <α+ β 1. Aslongasα and β satisfy 1 >β>α, a> α β,andα + β 1 > 0, then all the inequalities in case (ii) hold for sufficiently small values of σ and χ (that is, for sufficiently large steady-state values of the intertemporal elasticity in consumption and the labor supply elasticity). These results are similar to those found in two-sector models such as Benhabib and Nishimura (1998) and Harrison (2001), where they use the utility function with constant consumption and labor supply elascities. In contrast, such results do not hold in case (i) of Proposition 1. 6 We list below two numerical examples with indeterminacy for case (ii) in Proposition 1: Example 1: a =0.3,b=0.7,α=0.25,β =0.85,σ =0.11,χ =0,δ =0.025,ρ=0.0045; 6 In fact, from the second inequality in case (i), βσ +(1 α)χ >α+ β 1, and given the values of α and β such that α + β 1 > 0, indeterminacy can not happen if both σ and χ are sufficiently small. For example, if χ =0, then there is lower bound for σ required for indeterminacy and it can not happen for sufficiently small σ. 6

7 Example 2: a =0.3,b=0.7,α=0.25,β =0.76,σ =0.01,χ =0,δ =0.025,ρ= where in Example 1 the degree of social returns to scale is 1.1, and in Example 2 the degree of social returns to scale is 1.01 (with smaller σ relative to Example 1). To see more clearly the result in case (ii) of Proposition 1, we can compare with the case without externalities. Without externalities α = a and β = b, then from (12), the Jacobian matrix is given by, +, (15) + + where Tr(J)= j 11 + j 22 = ρ>0, anddet(j)= j 11 j 22 j 12 j 21 < 0, so that there is equilibrium uniqueness. With externalities and under the conditions in case (ii), the Jacobian matrix is given by. (16) + + In this case Tr(J)< 0 as j 11 dominates j 22, which in turn is due to the presence of negative capital externalities in the social marginal product of capital ( y k δ). However,Det(J)> 0 holdsasthe private marginal product of capital is increasing in k, i.e., k r > 0 (due to the presence of positive labor externalities or increasing returns to scale) so that the sign for j 12 changes and the term j 12 j 21 dominates j 11 j 22. The role which the negative capital externalities play here in generating indeterminacy in the one-sector framework is similar to that of the positive capital and labor externalities in the twosector model in Benhabib and Nishimura (1998). In their model, there are decreasing returns at the private level and constant returns at the social level and, with factor externalities, the production function of the investment good sector can be decreasing in aggregate capital because of the Stopler-Samuelson effect, which is necessary for their indeterminacy result. Here although the social production function is still increasing in the capital input, it is dampened because of the negative capital externalities to such a degree that j 22 is dominated by j 11. It is important to compare the result in case (ii) in Proposition 1 with the that obtained in Kehoe (1991). Kehoe (1991, ) presents an example of one-sector growth model with 7

8 exogenous labor supply where indeterminacy happens when production is subject to a negative externality from capital. There a quadratic production function (in k and K) isusedsothatitis possible that the private marginal product is increasing in k but the social production is decreasing in k. Here we use Cobb-Douglas technology. In contrast, it can shown that indeterminacy can not arise in Kehoe (1991) with Cobb-Douglas technology. Finally, we note that Hammour (1989) points out that increasing returns in dynamic one-sector models is often destabilizing and must be coupled with congestion effects of one form or another to keep the equilibrium path of the economy within reasonable bounds. The negative capital externalites can be viewed as a type of congestion effects. It is an unresolved empirical matter whether external effects from inputs are typically positive or negative. Harrison (2003) finds that it is not unreasonable to pose negative values for externalities in a two-sector model. 3 Non-separable Utility In the previous section we show that, with separable utility, indeterminacy can happen when negative capital externalities are present. In this section, we will demonstrate that with nonseparable utility, indeterminacy can happen when capital externalities are positive and large, in that the social returns in capital are larger than 1 (i.e., α>1) Thus, with non-separable utility, large capital externalities do not necessarily lead to unbounded growth as is generally believed. Furthermore, a stationary, indeterminate steady state can arise. 7 We assume that the felicity function is of a particular type, as specified in King, Plosser and Rebelo (1988), i.e., U(c t,l t ) = c1 σ t 1 σ v(l t), for σ>0, σ 6= 1, (17) = logc t v(l t ), for σ =1, where for v>0, andforσ 6= 1, (1 σ)v 0 < 0, andσvv 00 /(σ 1) (v 0 ) 2 > 0. Although this class of utility function is non-separable for σ 6= 1, it is separable for σ =1andthusalsoincludesasa special case the utility function in Benhabib and Farmer (1994). 7 Pelloni and Waldmann (1998) consider a one-sector endogenous growth model with α =1using the non-separable utility and the CES production function. They show that an indeterminate balanced growth path can arise in such a setup. 8

9 When σ 6= 1,the first-order conditions are c σ t v(l t )=λ t, (18) c1 σ t 1 σ v0 (l t )=λ t w t, (19) λ t = λ t (ρ + δ r t ), (20) where w t and r t are given in eq. (3). The system can again be reduced to two differential equations for λ t and k t, in the same forms as eqs. (9) and (10), while the implicit functions c(λ t,k t ) and l(λ t,k t ) and their partials are now determined from eqs. (18) and (19). We compute the trace and the determinant of the Jacobian matrix M of the linearized system to obtain the following expressions: Tr(M) = a(ρ + δ) σ µ ω µ β a 1 β + ξ δα ασ ω β 1 β + ξ + δ (α 1), (21) Det(M) = a(ρ + δ)c ω α 1 ασ 2 k ( ), (22) 1 β + ξ in which, evaluated at the steady state, ξ = l σ 1 σ vv ( σ 0 σ 1 vv00 v 02 ) > 0, and ω = ασ a [1 + (1 σ)(ρ+δ) vv 00 ρ+δ(1 a) b(v02 v 02 )]. Notice that in this case ξ is the steady-state value of the labor elasticity. It turns out that when the depreciation rate for capital is zero we can obtain useful analytical results on indeterminacy. When δ =0we have ω > 0. 8 Then the expression for Det(M) is relatively simple, and it has the same sign as ( 1 β+ξ α 1 ). On the other hand, the indeterminacy result under the King-Plosser-Rebelo utility function when σ =1is already included in case (i) of Proposition 1 (with σ =1,andforδ 0). We therefore summarize these results in the following proposition: PROPOSITION 2: With the Cobb-Douglas technology and the utility function of the King- Plosser-Rebelo type, local indeterminacy occurs if 8 We show here that ω > 0: Forσ<1, itisobviousthatω > 0. Forσ>1, wehave1+(1 σ) b( v02 vv 00 v 02 )= 1+(1 σ) b +(σ 1) b( vv00 v 02 ) > (1 b)+ 1 σ b>0. However, for δ>0, we are unable to show analytically that ω > 0. 9

10 (i-a) For σ =1and δ 0, the conditions 1 β+ξ < 0, andα<1, β+(1 β+ξ ) δ ρ+δ < (1+ξ ) α a hold; or (i-b) For σ 6= 1and δ =0, the conditions 1 β + ξ < 0, andα<1, β<ω hold; or (ii) For σ 6= 1and δ =0, the conditions 1 β + ξ > 0, andα>1, β>ω hold. By including the separable utility function, case (i-a) of Proposition 2 contains the result in Benhabib and Farmer (1994) as a special case just as in Proposition 1, which is the overlap of the two propositions. 9 In addition, in cases (i-a) and (i-b), the condition 1 β + ξ < 0 implies that the labor-demand curve is upward sloping, and α<1 is required for indeterminacy. From Proposition 2 indeterminacy can not arise when β<1 and α<1. However, indeterminacy can occur when β<1+ξ and α>1 in case (ii). The condition 1 β + ξ > 0 implies that the labor-supply curve is steeper than the labor-demand curve, while α>1 says that social returns to scale in capital is greater than 1. Indeterminacy can in fact easily happen when the labor-demand curve is downward sloping. We can also obtain the following corollary: COROLLARY 2: The necessary conditions for case (ii) of Proposition 2 to occur are σ<1 and β>b. Proof : See the appendix (part A). While the results of case (ii) in Proposition 2 and Corollary 2 hold when δ =0, simulations suggest that indeterminacy can happen under the same conditions therein for a large range of values for δ>0. Letting v(l t )=(1 l t ) σ, a numerical example on indeterminacy is (ρ =0.0045): a =0.8,b =0.2,α =1.0001,β =0.2899,σ =0.1,δ = in which the degree of social returns to scale is As in the previous section, we can also look at how indeterminacy comes about in case (ii) of Proposition 2 by comparing with the case without factor externalities. In the latter case, for the corresponding Jacobian matrix of the {λ t,k t } system, we have j 11 < 0, j 12 > 0, j 21 > 0 and 9 Notice that for the separable utility with σ =1in this section and the one with σ =1in the previous section we have ξ = χ. 10

11 j 22 > 0, and trace is positive and the determinant negative, so that there is equilibrium uniqueness. Under the conditions in case (ii), because of large and positive externalities from capital, while the signs for other elements of the Jacobian matrix remain unchanged, we now have j 12 < 0, thatis, the private marginal product is increasing in k. This makes it possible for the determinant to be positive. On the other hand, because of the presence of labor externalities, labor supply changes so much more than the case without labor externalities that j 11 dominates j 22 and the trace becomes negative (see (21)). 4 A Necessary Condition for Indeterminacy Existing research on indeterminacy of one-sector models, including the analyses in the previous sections of this paper, has so far been concentrated on the relative slopes of the labor-demand curve and the Frisch labor-supply curve in giving intuitive explanations about indeterminacy. In this section, we provide an alternative and complementary necessary condition for indeterminacy, using the framework with the general utility function and the Cobb-Douglas technology with externalities. In particular, whereas the production function is given by (1), we assume that the felicity function takes the general form, U(c t,l t ),whichsatisfies, U c > 0,U l < 0,U cc 0,U ll 0,U cc U ll U 2 cl 0,U lu cl U c U ll 0. (23) These assumptions on the felicity function are standard: it is continuously differentiable, strictly increasing, and concave in consumption and leisure. The last inequality means that consumption is a normal good. We state and prove the following theorem (assuming the existence of a unique steady state): THEOREM: With the Cobb-Douglas technology and the general felicity function, a necessary condition for local indeterminacy is that, evaluated at the steady state, Ul U cc Uc Ucl is, leisure is a strictly normal good in the long run (at the steady state). > 0, that Proof: We show that indeterminacy can not occur if Ul U cc Uc Ucl =0. From the first-order conditions, we have two differential equations for λ t and k t, in the same forms as eqs. (9) and (10). When Ul U cc Uc Ucl =0, the Jacobian matrix N of the linearized system is given by (see part B in the appendix) 11

12 N = 0 λ k (ρ + δ)[ α(1+η ) 1 β+η 1] a+η α(1+η ) (ρ + δ) δ 1 U cc a(1+η ) 1+η β, (24) where η =( U c Ull U l U cl Uc Ul )l 0. ThenTr(N)= a+η α(1+η ) a(1+η ) 1+η β (ρ + δ) δ, anddet(n)= λ U cck (ρ + δ)[ α(1+η ) 1+η β 1]. Suppose that indeterminacy can happen in this case, then Det(N)> 0, orequivalently α(1+η ) 1+η β a+η > 1. This implies that Tr(N)> a(1+η )(ρ + δ) δ>ρ>0, which is a contradiction.k The theorem characterizes the essential role that labor plays in making indeterminacy possible. In one-sector growth models with exogenous labor supply, Boldrin and Rustichini (1994) establish that with a general production function and positive capital externalities, local indeterminacy can not arise regardless of the magnitude of externalities, and Kehoe (1991) shows that indeterminacy can happen with negative capital externalities. However, the production function is quadratic in k and K in the example given in Kehoe (1991), where indeterminacy can not occur with Cobb- Douglas production function. Thus with Cobb-Douglas technology indeterminacy can not arise in one-sector models when labor is inelastic, whether the capital externalities are positive or negative. The above theorem says that with Cobb-Douglas production function a necessary condition for indeterminacy is that leisure must be a strictly normal good in the long run regardless of the relative slopes of labor supply and labor demand curves, and it is a stronger result than those implied in Boldrin and Rustichini (1994) and Kehoe (1991). To see why Ul U cc Uc Ucl > 0 is necessary for nonuniqueness, we look at the result in case (ii) of Proposition 1. Indeterminacy happens when there is an increase in the shadow price agents increase their investment, and marginal product of capital will also rise. Labor supply must rise as well to accommodate the increase in capital, which can happen only if Ul U cc Uc Ucl > 0. If it were true that Ul U cc Uc Ucl =0we would have j 11 =0in the Jacobian matrix in (16), which would make it impossible for the trace to be negative. Similar explanations can be given to the result of case (i) of Proposition 1, as well as those in Proposition 2 as to why U l U cc U c U cl must be strictly positive for indeterminacy to arise. Before concluding this section we note that a class of utility function that satisfies U l U cc U c U cl =0is the quasilinear utility function, i.e., U(c, l) =G(c g(l)), whereg 0 > 0 >G 00, and g 0 > 0 > g For the quasilinear utility functions U l U cc U c U cl =0for all (c, l) combinations while our result only requires that 12

13 5 Conclusion We show in this paper that in the one-sector model with factor-generated externalities, indeterminacy can arise when the felicity function is separable in consumption and leisure and there are negative capital externalities, or when the felicity function is non-separable and the social returns to scale in capital is greater than one. In both cases indeterminacy can happen when the labordemand curve slopes down. These findings have not been observed before. In addition, we present a necessary condition on indeterminacy with general utility function and Cobb-Douglas technology, which nests the results of the one-sector models studied in the literature. All the results in this paper are obtained based on using the Cobb-Douglas production function with factor externalities. It remains to be seen whether any new results can be achieved by utilizing alternative production functions and/or preference specifications. We plan to pursue this line of research in the future. U l U cc U c U cl =0at the steady state. The utility function, U(c, l) =G(c g(l)), was popularized in macroeconomic analysis by Greenwood, et al. (1988). 13

14 6 Appendix Part A We prove here Corollary 2 in Section 3. We first show that σ < 1 must hold. Under the indeterminacy conditions in case (ii) of Proposition 2, we have, 1+ξ >β>ω,thatis, ³ 1 ασ "1+(1 # σ) b (v0 ) 2 vv 00 µ 1 σ a (v 0 ) 2 b>0. (25) σ Suppose that σ>1, thenv 00 > 0 and 1/σ > (v0 ) 2 vv 00 (v 0 ) 2. Then (25) implies that ³ 0 < 1 ασ µ 1 σ [1 + (1 σ) b/σ] b =1 ασ ³ ασ µ1 a σ a σ b a σ < 1 ασ ³ ασ µ1 a σ =1 α a σ a < 0, which yields a contradiction. We next show that β>bmust hold. Suppose that b>β,thenwehave " # b > β > ω = ασ 1+(1 σ) b (v0 ) 2 vv 00 a (v 0 ) 2 > ασ ασ [1 + (1 σ) b/σ] > a a = αb a >b, which is again a contradiction. Part B [b +(1 σ) b/σ] The first-order conditions with respect to c t and l t in the setup of Section 4 are U c = λ t,and U l = λ t w t, which define the implicit functions c(λ t,k t ) and l(λ t,k t ).IfU l U cc U c U cl =0,then the partials evaluated at the steady state are dl dλ = Ul U cc Uc Ucl λ (UccU ll U cl 2)+U ccu l [(1 =0, (26) β)/l ] dl dk = αl k ( 1 1+η ), (27) β dc dλ = 1 Ucc, (28) 14

15 dc dk = U cl dl Ucc dk, (29) which can be used to obtain the Jacobian matrix (24). The intermediate steps are omitted. 15

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