Structural Change, Industrial Upgrading, and Middle-Income Trap

Abstract

Motivated by several stylized facts about middle-income trap, we develop a simple multi-sector general equilibrium model of structural change and industrial upgrading. The model features the distinction between production service and consumption service and the input-output linkages between different sectors. We show that the role of production service is asymmetric at different levels of development. Whereas an underdeveloped sector of production service is not a binding obstacle for development (sometimes even beneficial) at an early stage of development, it becomes a key bottleneck when the economy reaches a middle-income status. To escape the middle-income trap, government intervention is needed to prevent premature de-industrialization and facilitate beneficial industrial upgrading. Moreover, it also requires a timely reduction of entry barrier to the production service and improvement in its productivity. These theoretical findings are shown to be consistent with the stylized facts and also useful to China. The analysis provides a justification for the government’s strategic use of industrial policies to avoid middle-income trap.

Appendices

Appendix 1

It is easy to show that ${\hat{F}}_{\mathrm{max}}$, the cutoff value for entry cost in the social planner problem, is uniquely determined by

$$ {\displaystyle \begin{array}{c}{A}_h^{\theta }{A}_s^{1-\theta }{\hat{F}}_{\mathrm{max}}^{-\frac{1-\sigma }{\sigma}\left[\alpha \theta +\beta \left(1-\theta \right)\right]}{A}_m^{\alpha \theta +\beta \left(1-\theta \right)}{l}_s^{\left(1-\alpha \right)\theta +\left(1-\beta \right)\left(1-\theta \right)+\frac{1}{\sigma}\left[\alpha \theta +\beta \left(1-\theta \right)\right]}{\left(\frac{\theta }{1-\theta}\frac{1-\alpha }{1-\beta}\right)}^{\left(1-\alpha \right)\theta}\\ {}\cdotp {\left(\frac{1-\sigma }{\sigma}\right)}^{\frac{1-\sigma }{\sigma}\left[\alpha \theta +\beta \left(1-\theta \right)\right]}{\left(\frac{\theta }{1-\theta}\frac{\alpha }{\beta}\right)}^{\alpha \theta}{\left\{\frac{\beta }{1-\beta }{\left[\left(\frac{\theta }{1-\theta}\frac{\alpha }{\beta }+1\right)\frac{\beta }{1-\beta}\right]}^{\frac{1-\sigma }{\sigma }}\right\}}^{\alpha \theta +\beta \left(1-\theta \right)}\\ {}=\frac{\epsilon }{\epsilon -1}L\left\{{A_b}^{\frac{\epsilon -1}{\epsilon }}-{\left[\frac{\left(\frac{\theta }{1-\theta}\frac{\alpha }{\beta }+1\right)\beta {A}_b{A}_m{l}_s^{2-\left[\left(1-\alpha \right)\theta +\left(1-\beta \right)\left(1-\theta \right)\right]}}{{\left(\frac{\theta }{1-\theta}\frac{1-\alpha }{1-\beta}\right)}^{\left(1-\alpha \right)\theta}\left(1-\theta \right){\left(1-\beta \right)}^2}\right]}^{\epsilon -1}\right\}\end{array}} $$

(51)

where l_s is uniquely determined by Eq. (43). When F ≤ ${\hat{F}}_{\mathrm{max}}$, it can be shown that the real output per capita and welfare of a representative household is given by

$$ {\displaystyle \begin{array}{c}w={A}_h^{\theta }{A}_s^{1-\theta}\frac{F^{-\frac{1-\sigma }{\sigma}\left[\alpha \theta +\beta \left(1-\theta \right)\right]}}{L}{A}_m^{\alpha \theta +\beta \left(1-\theta \right)}{l}_s^{\left(1-\alpha \right)\theta +\left(1-\beta \right)\left(1-\theta \right)+\frac{1}{\sigma}\left[\alpha \theta +\beta \left(1-\theta \right)\right]}{\left(\frac{\theta }{1-\theta}\frac{\alpha }{\beta}\right)}^{\alpha \theta}\\ {}\cdotp {\left(\frac{\theta }{1-\theta}\frac{1-\alpha }{1-\beta}\right)}^{\left(1-\alpha \right)\theta }{\left(\frac{1-\sigma }{\sigma}\right)}^{\frac{1-\sigma }{\sigma}\left[\alpha \theta +\beta \left(1-\theta \right)\right]}{\left\{\frac{\beta }{1-\beta }{\left[\left(\frac{\theta }{1-\theta}\frac{\alpha }{\beta }+1\right)\frac{\beta }{1-\beta}\right]}^{\frac{1-\sigma }{\sigma }}\right\}}^{\alpha \theta +\beta \left(1-\theta \right)}\\ {}+\frac{\epsilon }{\epsilon -1}{\left[\frac{\left(\frac{\theta }{1-\theta}\frac{\alpha }{\beta }+1\right)\beta {A}_b{A}_m{l}_s^{2-\left[\left(1-\alpha \right)\theta +\left(1-\beta \right)\left(1-\theta \right)\right]}}{{\left(\frac{\theta }{1-\theta}\frac{1-\alpha }{1-\beta}\right)}^{\left(1-\alpha \right)\theta}\left(1-\theta \right){\left(1-\beta \right)}^2}\right]}^{\epsilon -1}\end{array}} $$

(52)

where l_s is uniquely determined by Eq. (43).

Appendix 2

Suppose F = 0 and σ = 1, then we are in the perfectly competitive market environment with constant-returns-to-scale technologies. Both the first and second welfare theorems can be applied. The decentralized market equilibrium allocation is Pareto efficient and is identical to the solution to the following artificial social planner problem:

$$ \underset{lb, lm, lh, ls,n, mh, ms, ch, cs, cb\ }{\max }{C}_h^{\theta }{C}_s^{1-\theta }+\frac{\epsilon }{\epsilon -1}{C}_b^{\frac{\epsilon -1}{\epsilon }} $$

(53)

subject to

$$ {c}_bL={A}_b{l}_b $$

(54)

$$ {c}_hL={A}_h{m_h}^{\alpha }{l_h}^{1-\alpha } $$

(55)

$$ {c}_sL={A}_s{m_s}^{\beta }{l_s}^{1-\beta } $$

(56)

$$ {m}_h+{m}_s={A}_m{l}_m $$

(57)

$$ lb+ lm+ lh+ ls=L $$

(58)

and no-negativity conditions for all relevant variables.

Define $\overset{\sim }{H}\left({A}_m,{A}_h,{A}_s\right)$ as the same function H(A_m, A_h, A_s) given by Eq. (18) except that σ is substituted out with unity and χ = 0 based on Eq. (19). The socially efficient allocation is as follows: When $1-{A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon}\left({A}_m,{A}_h,{A}_s\right)>0$, or equivalently, when Eq. (21) is satisfied with σ = 1, we have

$$ {l}_b={A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon }\ \left({A}_m,{A}_h,{A}_s\right)L $$

(59)

$$ {l}_m=\left[\alpha \theta +\beta \left(1-\theta \right)\right]\left[1-{A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon }\ \left({A}_m,{A}_h,{A}_s\right)\right]L $$

(60)

$$ {l}_s=\left(1-\theta \right)\left(1-\beta \right)\left[1-{A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon }\ \left({A}_m,{A}_h,{A}_s\right)\right]L $$

(61)

$$ {l}_h=\left(1-\alpha \right)\ \theta \left[1-{A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon }\ \left({A}_m,{A}_h,{A}_s\right)\right]L $$

(62)

$$ {m}_h=\alpha \theta {A}_m\left[1-{A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon }\ \left({A}_m,{A}_h,{A}_s\right)\right]L $$

(63)

$$ {m}_s=\beta\ \left(1-\theta \right){A}_m\left[1-{A_b}^{\epsilon -1}{\overset{\sim }{H}}^{-\epsilon }\ \left({A}_m,{A}_h,{A}_s\right)\right]L $$

(64)

There are strictly positive entries to the production service sector, but the firm number (n) is indeterminate because of the constant-returns-to-scale technology plus free entry with F = 0.

Appendix 3

1.1 H and F are completely specialized in b and h

It is easy to show that in equilibrium, we have

$$ {\displaystyle \begin{array}{c}{p}_b=\frac{w}{A_b};{p}_m={n}^{1-\frac{1}{\sigma }}\frac{w}{{\sigma A}_m};{p}_s=\frac{{p_m}^{\beta }{w}^{1-\beta }}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }};\\ {}{p_h}^{\ast }=\frac{w^{\ast }}{{A_h}^{\ast }};{p_s}^{\ast }=\frac{w^{\ast }}{{A_s}^{\ast }}\end{array}} $$

The demand functions for b, h, and s are as follows when both b and h are consumed in both countries:

$$ {c}_b={\left(\frac{{p_h}^{\ast \theta }{p_s}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon } $$

(65)

$$ {c}_h=\frac{\theta \left[w-{p}_b{\left(\frac{{p_h}^{\ast \theta }{p_s}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon}\right]}{{p_h}^{\ast }} $$

(66)

$$ {c}_s=\frac{\left(1-\theta \right)\left[w-{p}_b{\left(\frac{{p_h}^{\ast \theta }{p_s}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon}\right]}{p_s} $$

(67)

$$ {c_b}^{\ast }={\left(\frac{{p_h}^{\ast \theta }{{p_s}^{\ast}}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon } $$

(68)

$$ {c_h}^{\ast }=\frac{\theta \left[{w}^{\ast }-{p}_b{\left(\frac{{p_h}^{\ast \theta }{{p_s}^{\ast}}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon}\right]}{{p_h}^{\ast }} $$

(69)

$$ {c_s}^{\ast }=\frac{\left(1-\theta \right)\left[{w}^{\ast }-{p}_b{\left(\frac{{p_h}^{\ast \theta }{{p_s}^{\ast}}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon}\right]}{{p_s}^{\ast }} $$

(70)

Market clearing conditions are as follows:

$$ {c}_bL+{c_b}^{\ast }{L}^{\ast }={A}_b{l}_b $$

(71)

$$ {c}_hL+{c_h}^{\ast }{L}^{\ast }={A_h}^{\ast }{l_h}^{\ast } $$

(72)

$$ {c}_sL={Y}_S={A}_s{m_s}^{\beta }{l_s}^{1-\beta } $$

(73)

$$ {c_s}^{\ast }{L}^{\ast }={Y_S}^{\ast }={A_s}^{\ast }{l_s}^{\ast } $$

(74)

$$ {D}_m={Y}_m\Longrightarrow {m}_s={A}_m{l}_m $$

(75)

$$ nF+ lb+ lm+ ls=L $$

(76)

$$ {lm}^{\ast }+{ls}^{\ast }={L}^{\ast } $$

(77)

We always have

$$ {Y}_m={n}^{\frac{1}{\sigma }}\frac{{\sigma A}_mF}{1-\sigma } $$

(78)

H does not produce h, so m is used for producing s only. Using Shephard’s lemma, we can obtain the aggregate demand for m

$$ {D}_m={m}_s={D}_s\frac{\partial {p}_s}{\partial {p}_m}=L\frac{{\beta c}_s{p}_s}{p_m} $$

(79)

Since D_m = Y_m, we have ${n}^{\frac{1}{\sigma }}\frac{{\sigma A}_mF}{1-\sigma }=L\frac{{\beta c}_s{p}_s}{p_m}$, which implies

$$ \frac{nF}{\left(1-\sigma \right) L\beta \left(1-\theta \right)}=1-{A_b}^{\epsilon -1}{n}^{\left(1-\frac{1}{\sigma}\right)\beta \left(1-\theta \right)\epsilon }{\left({\left[\frac{w^{\ast }}{w\theta {A_h}^{\ast }}\right]}^{\theta }{\left[\frac{1}{\left(1-\theta \right){\left({\sigma A}_m\right)}^{\beta }{A}_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right]}^{1-\theta}\right)}^{\epsilon } $$

(80)

So, an equilibrium exists only if the right-hand side of the above equation is strictly positive. In particular, when there exist two roots (n₁ and n₂), with n₁ < n₂, the equation above implies that ${n}_2=\Xi \left(\frac{w^{\ast }}{w}\right)$ with ${\Xi}^{\prime}\left(\frac{w^{\ast }}{w}\right)$ < 0 and ${n_1}^{\prime}\left(\frac{w^{\ast }}{w}\right)$ > 0. We focus on the high-equilibrium n₂.

On the other hand, balanced trade implies

$$ L{c}_h{p}_h^{\ast }={L}^{\ast }{c_b}^{\ast }{p}_b $$

(81)

which is equivalent to

$$ {\displaystyle \begin{array}{c}{A_b}^{1-\epsilon }{\left[{A}_h^{\ast \theta }{\uptheta}^{\theta }{\left(1-\theta \right)}^{1-\theta }{A}_s^{\ast 1-\theta}\right]}^{\varepsilon }=\\ {}\frac{L^{\ast }}{L\theta}{\left(\frac{w^{\ast }}{w}\right)}^{\varepsilon }+{\left[\frac{\varLambda^{\left(1-\frac{1}{\sigma}\right)\beta }}{{\left({\sigma A}_m\right)}^{\beta }{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right]}^{\left(1-\theta \right)\varepsilon }{\left(\frac{w^{\ast }}{w}\right)}^{\varepsilon \left[\theta +\left(1-\frac{1}{\sigma}\right)\beta \left(1-\theta \right)\epsilon \right]}\end{array}} $$

(82)

where

$$ \Lambda \equiv \frac{{\mathrm{L}}^{\ast}\frac{1-\sigma }{F}\beta \frac{1-\theta }{\theta }}{{A_b}^{1-\epsilon }{\left[{A}_h^{\ast \theta }{\uptheta}^{\theta }{\left(1-\theta \right)}^{1-\theta }{A}_s^{\ast 1-\theta}\right]}^{\epsilon }} $$

(83)

Therefore, n = $\Lambda {\left(\frac{w^{\ast }}{w}\right)}^{\varepsilon }$. Thus, Eq. (82) implies that there exists a unique equilibrium when $\theta +\left(1-\frac{1}{\sigma}\right)\beta \left(1-\theta \right)\epsilon \ge 0$ because the right-hand side strictly increases with $\frac{w^{\ast }}{w}$. It implies $n=\Psi \left(\frac{w^{\ast }}{w}\right)\equiv \Lambda {\left(\frac{w^{\ast }}{w}\right)}^{\varepsilon }$ with ${\Psi}^{\prime}\left(\frac{w^{\ast }}{w}\right)>0$.

$$ {\displaystyle \begin{array}{c}\frac{\partial }{\partial F}\left(\frac{w^{\ast }}{w}\right)<0;\frac{\partial }{\partial {L}^{\ast }}\left(\frac{w^{\ast }}{w}\right)?0;\frac{\partial }{\partial {A}_b}\left(\frac{w^{\ast }}{w}\right)<0;\frac{\partial }{\partial L}\left(\frac{w^{\ast }}{w}\right)>0;\\ {}\frac{\partial }{\partial {A}_h^{\ast }}\left(\frac{w^{\ast }}{w}\right)>0;\frac{\partial }{\partial {A}_s^{\ast }}\left(\frac{w^{\ast }}{w}\right)>0;\frac{\partial }{\partial {A}_s}\left(\frac{w^{\ast }}{w}\right)=\frac{\partial }{\partial {A}_b^{\ast }}\left(\frac{w^{\ast }}{w}\right)=\frac{\partial }{\partial {A}_h}\left(\frac{w^{\ast }}{w}\right)=0;\\ {}\frac{\partial }{\partial {A}_m}\left(\frac{w^{\ast }}{w}\right)>0\end{array}} $$

Specifically, when θ = 0, no trade occurs. When θ = 1, $\frac{w^{\ast }}{w}={\left[{A_b}^{1-\epsilon }{A}_h^{\ast \varepsilon}\frac{L}{L^{\ast }}\right]}^{\frac{1}{\varepsilon }}.$

So, ${n}_2=\Xi \left(\frac{w^{\ast }}{w}\right)$ and $n=\Psi \left(\frac{w^{\ast }}{w}\right)$ must determine a unique solution, denoted by $\overset{\sim }{n}$(F, L, L^∗, A_s, A_m, A_b, A^∗, A^∗) and $\overset{\sim }{\frac{w^{\ast }}{w}}$ (F, L, L^∗, A_s, A_m, A_b, A^∗, A^∗), if they cross. Moreover, it is easy to show that the following is true.

$$ {\displaystyle \begin{array}{c}\frac{\partial }{\partial F}\left(\frac{w^{\ast }}{w}\right)<0;\frac{\partial }{\partial {L}^{\ast }}\left(\frac{w^{\ast }}{w}\right)<0;\frac{\partial }{\partial {A}_b}\left(\frac{w^{\ast }}{w}\right)<0;\\ {}\frac{\partial }{\partial L}\left(\frac{w^{\ast }}{w}\right)>0;\frac{\partial }{\partial {A}_h^{\ast }}\left(\frac{w^{\ast }}{w}\right)>0;\frac{\partial }{\partial {A}_s^{\ast }}\left(\frac{w^{\ast }}{w}\right)>0;\frac{\partial }{\partial {A}_s}\left(\frac{w^{\ast }}{w}\right)>0;\frac{\partial }{\partial {A}_m}\left(\frac{w^{\ast }}{w}\right)>0\end{array}} $$

1.2 H and F both produce h and b

In this case, we can show that

$$ {\displaystyle \begin{array}{c}{p}_b=\frac{w}{A_b};{p}_m={n}^{1-\frac{1}{\sigma }}\frac{w}{{\sigma A}_m};{p}_h=\frac{{p_m}^{\alpha }{w}^{1-\alpha }}{A_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }};{p}_s=\frac{{p_m}^{\beta }{w}^{1-\beta }}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }};\\ {}{p_b}^{\ast }=\frac{w^{\ast }}{{A_b}^{\ast }};{p_h}^{\ast }=\frac{w^{\ast }}{{A_h}^{\ast }};{p_s}^{\ast }=\frac{w^{\ast }}{{A_s}^{\ast }}\end{array}} $$

In equilibrium

$$ {p}_b={p_b}^{\ast}\Longrightarrow \frac{{A_b}^{\ast }}{A_b}=\frac{w^{\ast }}{w} $$

(84)

$$ {p}_h={p_h}^{\ast}\Longrightarrow \frac{{\left[{n}^{1-\frac{1}{\sigma }}\frac{{\sigma A}_m}{w}\right]}^{\alpha }{w}^{1-\alpha }}{A_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}=\frac{w^{\ast }}{{A_h}^{\ast }}. $$

(85)

therefore,

$$ n={\left\{{\sigma A}_m{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}\right\}}^{\frac{\sigma }{\sigma -1}} $$

(86)

and

$$ {p}_m={\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}w $$

(87)

$$ {p}_h=\frac{{A_b}^{\ast }}{A_b{A_h}^{\ast }}w;{p}_s=\frac{{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}w}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }} $$

(88)

The equilibrium conditions are as follows:

$$ {l}_m=\frac{n\sigma F}{\left(1-\sigma \right)}=\frac{\sigma F}{\left(1-\sigma \right)}{\left\{{\sigma A}_m{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}\right\}}^{\frac{\sigma }{\sigma -1}} $$

(89)

$$ {n}^{\frac{1}{\sigma }}\frac{{\sigma A}_mF}{\left(1-\sigma \right)}={m}_h+{m}_s $$

(90)

$$ {m}_h=L\frac{\alpha {y}_h{p}_h}{p_m}=L{y}_h\frac{\alpha \frac{{A_b}^{\ast }}{A_b{A_h}^{\ast }}}{{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}} $$

(91)

$$ L{y}_h={A}_h{m_h}^{\alpha }{l_h}^{1-\alpha } $$

(92)

$$ {\displaystyle \begin{array}{c}{m}_s=L\frac{{\beta c}_s{p}_s}{p_m}=\frac{L\beta \left(1-\theta \right)}{{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}}\\ {}\cdotp \left(1-{A_b}^{-1}{\left[\frac{A_b{\left(\frac{A_b^{\ast }}{A_b{A_h}^{\ast }}\right)}^{\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }}\right]}^{\varepsilon }{\left(\frac{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right)}^{\left(1-\theta \right)\epsilon}\right)\end{array}} $$

(93)

$$ {\displaystyle \begin{array}{c}{m}_h={\left\{{\sigma A}_m{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}\right\}}^{\frac{1}{\sigma -1}}\frac{{\sigma A}_mF}{\left(1-\sigma \right)}-\frac{L\beta \left(1-\theta \right)}{{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}}\\ {}\cdotp \left(1-{A_b}^{-1}{\left[\frac{A_b{\left(\frac{A_b^{\ast }}{A_b{A_h}^{\ast }}\right)}^{\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }}\right]}^{\varepsilon }{\left(\frac{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right)}^{\left(1-\theta \right)\epsilon}\right)\end{array}} $$

(94)

Thus, lh > 0 if

$$ \frac{{\left\{{\sigma A}_m{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}\right\}}^{\frac{\sigma }{\sigma -1}}F}{L\beta \left(1-\theta \right)\left(1-\sigma \right)}>1-{A_b}^{-1}{\left[\frac{A_b{\left(\frac{A_b^{\ast }}{A_b{A_h}^{\ast }}\right)}^{\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }}\right]}^{\varepsilon }{\left(\frac{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right)}^{\left(1-\theta \right)\epsilon }. $$

(95)

Moreover,

$$ {c}_b={\left(\frac{{p_h}^{\theta }{p_s}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon }={\left[\frac{A_b{\left(\frac{A_b^{\ast }}{A_b{A_h}^{\ast }}\right)}^{\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }}\right]}^{\varepsilon }{\left(\frac{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right)}^{\left(1-\theta \right)\epsilon } $$

(96)

$$ {c}_h=\frac{\theta \left[w-{p}_b{\left(\frac{{p_h}^{\theta }{p_s}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon}\right]}{p_h}=\theta \left[\frac{{A_h}^{\ast }{A}_b}{{A_b}^{\ast }}-\frac{{A_h}^{\ast }{c}_b}{{A_b}^{\ast }}\right] $$

(97)

$$ {c}_s=\frac{\left(1-\theta \right)\left[w-{p}_b{\left(\frac{{p_h}^{\theta }{p_s}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p}_b}\right)}^{\epsilon}\right]}{p_s}=\left(1-\theta \right)\left[\frac{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}}\left(1-\frac{c_b}{A_b}\right)\right] $$

(98)

$$ {c_b}^{\ast }={\left(\frac{{p_h}^{\ast \theta }{{p_s}^{\ast}}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p_b}^{\ast }}\right)}^{\epsilon }={\left[\frac{{A_b}^{\ast }{\left(\frac{1}{{A_h}^{\ast }}\right)}^{\theta }{\left(\frac{1}{{A_s}^{\ast }}\right)}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }}\right]}^{\varepsilon } $$

(99)

$$ {c_h}^{\ast }=\frac{\theta \left[{w}^{\ast }-{p_b}^{\ast }{\left(\frac{{p_h}^{\ast \theta }{{p_s}^{\ast}}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p_b}^{\ast }}\right)}^{\epsilon}\right]}{{p_h}^{\ast }}=\theta \left[{A_h}^{\ast }-\frac{{A_h}^{\ast }{c_b}^{\ast }}{{A_b}^{\ast }}\right] $$

(100)

$$ {\displaystyle \begin{array}{c}{c_s}^{\ast }=\frac{\left(1-\theta \right)\left[{w}^{\ast }-{p_b}^{\ast }{\left(\frac{{p_h}^{\ast \theta }{{p_s}^{\ast}}^{1-\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }{p_b}^{\ast }}\right)}^{\epsilon}\right]}{{p_s}^{\ast }}=\left(1-\theta \right)\left[{A_s}^{\ast }-\frac{{A_s}^{\ast }{c_b}^{\ast }}{{A_b}^{\ast }}\right]\\ {}\frac{\frac{1-\alpha }{\alpha }{\left\{{\sigma A}_m{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}\right\}}^{\frac{\sigma }{\sigma -1}}\frac{F}{1-\sigma }}{\frac{1-\alpha }{\alpha } L\beta \left(1-\theta \right)\left(1-{A_b}^{-1}{\left[\frac{A_b{\left(\frac{A_b^{\ast }}{A_b{A_h}^{\ast }}\right)}^{\theta }}{\uptheta^{\theta }{\left(1-\theta \right)}^{1-\theta }}\right]}^{\varepsilon }{\left(\frac{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta }{\alpha }}}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right)}^{\left(1-\theta \right)\epsilon}\right)}>1\end{array}} $$

(101)

Market clearing conditions are as follows:

$$ {c}_bL+{c_b}^{\ast }{L}^{\ast }={A}_b{l}_b+{A_b}^{\ast }{l_b}^{\ast } $$

(102)

$$ {c}_hL+{c_h}^{\ast }{L}^{\ast }=L{y}_h+{A_h}^{\ast }{l_h}^{\ast } $$

(103)

$$ {c}_sL={Y}_S={A}_s{m_s}^{\beta }{l_s}^{1-\beta } $$

(104)

$$ {c_s}^{\ast }{L}^{\ast }={A_s}^{\ast }{l_s}^{\ast } $$

(105)

$$ nF+ lb+ lh+ lm+ ls=L $$

(106)

$$ {lb}^{\ast }+{lh}^{\ast }+{ls}^{\ast }={L}^{\ast } $$

(107)

So

$$ {\displaystyle \begin{array}{c}{n}^{\frac{1}{\sigma }}\frac{{\sigma A}_mF}{\left(1-\sigma \right)}={m}_h+{m}_s=L\left(\frac{\alpha {y}_h{p}_h}{p_m}+\frac{{\beta c}_s{p}_s}{p_m}\right)\\ {}{\left\{{\sigma A}_m{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}\right\}}^{\frac{\sigma }{\sigma -1}}\frac{{\sigma A}_mF}{\left(1-\sigma \right)}=L\left(\frac{\alpha {y}_h\frac{{A_b}^{\ast }}{A_b{A_h}^{\ast }}}{{\left[\frac{{A_b}^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{1}{\alpha }}}+{\beta c}_s\frac{{\left[\frac{A_b^{\ast }{A}_h{\alpha}^{\alpha }{\left(1-\alpha \right)}^{1-\alpha }}{A_b{A_h}^{\ast }}\right]}^{\frac{\beta -1}{\alpha }}}{A_s{\beta}^{\beta }{\left(1-\beta \right)}^{1-\beta }}\right)\end{array}} $$

(108)