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Structural Change, Industrial Upgrading, and Middle-Income Trap

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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.

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Notes

  1. 1.

    There is a discernible pattern of economic convergence among the high-income countries. However, even within this group, the per capita income gap between the richest ones and the poorest ones is also large. In addition, some high-income countries fail to continue to converge to frontier countries. These important phenomena are beyond the scope of this paper and will be left for future research.

  2. 2.

    New structural economics advocates the use of neoclassical approach to study the determinants and impacts of structure and structural transformation in an economy (Lin JY 2011).

  3. 3.

    These 49 economies are Argentina, Australia, Austria, Belgium, Brazil, Bulgaria, Canada, Chile, China, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, India, Indonesia, Ireland, Israel, Italy, Japan, Korea, Latvia, Lithuania, Luxembourg, Malta, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Romania, Russia, Slovak Republic, Slovenia, South Africa, Spain, Sweden, Switzerland, Taiwan, Thailand, Turkey, UK, USA, and Vietnam.

  4. 4.

    Production service has recently become an official category in various economies. For example, China’s National Bureau of Statistics has provided the list of subsectors that are classified as production service since 2014. The list is almost identical to that with the upstreamness score higher than 3.3 based on our own calculation.

  5. 5.

    Ideally, it would be also useful to compare the value-added share of production service of ME and MT for the same per capita GDP level when both are in the middle-income status. However, such information is not sufficiently available in the current data set. Moreover, for a given GDP per capita level, the value-added share of production service could be different in different years even for the same country, if it experiences big enough economic fluctuations. This is particularly likely during the 2008 global financial crisis. We will leave this for future research.

  6. 6.

    For simplicity, physical capital is not explicitly modeled as one of the production factors. It is reserved for future research. For more discussions on the role of capital intensities, refer to Ju et al. (2011, 2015) and Li et al. (2016).

  7. 7.

    Although pecuniary externality caused by increasing returns to scale exists both in our model and in the study of Murphy et al. (1989), there are several crucial differences: First, in our model, pecuniary externality is amplified through the channel of input-output linkages and further augmented through the channel of the non-homothetic preferences in the context of two processes: structural change and industrial upgrading, whereas neither of these two channels nor these two processes are simultaneously considered in the study of Murphy et al. (1989), as their model assumes that the traditional sector and the modern sector produce identical final goods and there is no sector producing intermediate goods. Second, the demand spillover mechanism highlighted in their model crucially relies on that the net profits of entering firms in the modern sector strictly increase with the number of entrants and must be strictly positive after sufficient entry, which results from the model assumption that the maximum number of entrants is exogenous and fixed. More profits imply higher income and hence higher demand. In contrast, our model allows for free entry with no upper limit of firm entry, so the net profits of entering firms are always zero, independent of the number of entrants. This technical difference in modeling reflects that we highlight more on the supply side rather than the demand side: that is, more firm entry in the upstream sector enhances specialization and competition, which reduces the cost of intermediate inputs for downstream sectors, leading to a lower price of final consumption goods and a higher real income. Third, Murphy et al. (1989) focus on the market inefficiency with the symptom of delays in industrial upgrading, but our model shows that inefficiency may also come from premature upgrading and premature de-industrialization, which will be explicitly explained later.

  8. 8.

    It is both interesting and challenging to empirically validate the positive role of effective government coordinations, which is beyond the scope of this paper and left for future research. Be reminded that multiple market equilibria are not the only main point of this model. In fact, we will show that even the high market equilibrium is not Pareto efficient and international trade also plays key roles.

  9. 9.

    It can be shown that the value-added share of production service in total service is \( \frac{\left(1-\theta \right)\left(1-\beta \right)}{\theta \alpha +\left(1-\theta \right)} \).

  10. 10.

    Appendix 2 characterizes what happens if production service is perfectly competitive without entry cost (σ = 1 and F = 0).

  11. 11.

    \( {\hat{F}}_{\mathrm{max}} \) is endogenously determined. Refer to the appendix to see the equation that uniquely determines \( {\hat{F}}_{\mathrm{max}} \).

  12. 12.

    Other factors such as imperfections in labor market (Hukou system) and associated regulations in the social service sector (including schooling, Medicare, and pension) may also account for underdevelopment of the service sector, which is beyond the scope of this paper and deserves separate explorations in the future.

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Acknowledgments

We are grateful to Dani Rodrik and Karl Aiginger for their many insightful suggestions. For the helpful discussions and comments, we would also like to thank Robert Barro, Joe Kaboski, Jhoon-Wha Lee, Diego Restuccia, Pengfei Wang, Ping Wang, as well as the seminar and conference participants at the University of Hong Kong, HKUST, SAET conference, International Conference of Macroeconomics at Zhejiang University, Asian Development Bank Institute Conference on Middle-Income Trap, NBER-CCER annual conference, and various universities in mainland China. All the remaining errors are our own.

Funding

This work received financial supports from the National Science Foundation Project “Escaping the Middle-income Trap” (project number 71350002) and HKUST-IEMS Research Grant “Structural Change and Middle-Income Trap.”

Author information

Correspondence to Yong Wang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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 ls 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 ls 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(Am, Ah, As) 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

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 Dm = Ym, 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, As, Am, Ab, A, A) and \( \overset{\sim }{\frac{w^{\ast }}{w}} \) (F, L, L, As, Am, Ab, A, A), if they cross. Moreover, it is easy to show that the following is true.

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

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)

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Lin, J.Y., Wang, Y. Structural Change, Industrial Upgrading, and Middle-Income Trap. J Ind Compet Trade (2020). https://doi.org/10.1007/s10842-019-00330-3

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Keywords

  • Structural change
  • Industrial policies
  • Middle-income trap
  • Chinese economy
  • Economic growth

JEL Classification

  • O11
  • O14
  • O33
  • O41