## The Philippine Review of Economics

**Editor-in-Chief**
EMMANUEL F. ESGUERRA

**Editorial Advisory Board**
EMMANUEL S. DE DIOS

RAUL V. FABELLA HAL CHRISTOPHER HILL

CHARLES Y. HORIOKA KIAN GUAN LIM ROBERTO S. MARIANO JOHN VINCENT C. NYE GERARDO P. SICAT JEFFREY G. WILLIAMSON

**Associate Editors**
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HONLANI RUTH R. RUFO

A joint publication of the
**University of the Philippines**

School of Economics
and the **Philippine Economic Society**

**ARTICLES IN THIS ISSUE**

Shared prosperity characterized by four development goals:

pro-poor growth, pro-poor development, inclusive growth, and inclusive development

**Nanak Kakwani**
**Zakaria Siddiqui**

Piketty inequality, meta- market failures and the new role of the state

**Raul V. Fabella**

Diamond and Dybvig in developing economies and in a digital world

**Margarita Debuque-**
**Gonzales**

Toward a general neoclassical theory of economic growth

**Delano S. Villanueva**

Measuring fiscal policy sustainability in developing Asia: what does the Markov Switching Augmented Dickey- Fuller Test tell us?

**Dannah Ysabel M. Premacio**
**Ezra Rebecca G. Vidar**
**Toby C. Monsod**

The 16^{th} century *Carrera del *
*Pacífico*: its sailor-merchants
and their trade goods

**Kristyl Obispado**

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**Aims and Scope:** The *Philippine Review of Economics* (^{PRE})
invites theoretical and empirical articles on economics and
economic development. Papers on the Philippines, Asian and other
developing economies are especially welcome. Book reviews will
also be considered.

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Philippine Economic Society. Its contents are indexed in the *Journal *
*of Economic Literature*, EconLit, and RePec. PRE’s readership
includes economists and other social scientists in academe, business,
government, and development research institutions.

**Publication Information**: The PRE (p-ISSN 1655-1516; e-ISSN
2984-8156) is a peer-reviewed journal published every June and
December of each year. A searchable database of published articles
and their abstracts is available at the PRE website (http://pre.econ.

upd.edu.ph).

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**Copyright:** The *Philippine Review of Economics* is protected
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## The Philippine Review of Economics

Vol. LX No. 2 December 2023

p-ISSN 1655-1516 e-ISSN 2984-8156 DOI: 10.37907/ERP3202D

1 Shared prosperity characterized by four development goals: pro-poor growth, pro-poor development, inclusive growth, and inclusive development

Nanak Kakwani Zakaria Siddiqui

25 Piketty inequality, meta-market failures and the new role of the state

Raul V. Fabella

## 39 Diamond and Dybvig in developing economies and in a digital world

Margarita Debuque-Gonzales

64 Toward a general neoclassical theory of economic growth

Delano S. Villanueva

81 Measuring fiscal policy sustainability in developing Asia: what does the Markov Switching Augmented Dickey-Fuller Test tell us?

Dannah Ysabel M. Premacio Ezra Rebecca G. Vidar Toby C. Monsod

104 The 16

^{th}

## century Carrera del Pacífico: its sailor- merchants and their trade goods

Kristyl Obispado

Toward a general neoclassical theory of economic growth

Delano S. Villanueva*

International Monetary Fund**

The Harrod-Domar (H-D) growth model assumes a fixed capital-output ratio, signifying absence of substitutability between capital and labor, leading to a “knife-edge” problem wherein balanced growth of capital (fixed warranted rate) and labor (fixed natural rate) occurs only by accident, preventing the attainment of macroeconomic stability with full employment.

The neoclassical Solow-Swan (S-S) growth model provides an elegant solution to the H-D problem by endogenizing the warranted rate via the saving-investment relation, wherein capital growth is a function of a fully adjusting income-capital ratio (inverse of the H-D capital-output ratio)—

allowing for smooth substitutability between capital and labor while keeping
the natural rate exogenously fixed. The S-S model implies a positive, albeit
*temporary* output growth effect of a higher saving rate. The present paper
extends the capital-labor ratio’s influence onto the natural rate via effects on
labor productivity through a modified Arrow learning by doing framework,
and via labor participation through real wage adjustments. Thus, the positive
output growth of a higher saving rate, although *temporary* in the short run as
in the S-S model, is *permanent* in the long run through adjustments in both
the warranted and natural rates—a generalization of the Solow-Swan model.

**JEL classification**: E130, O410

**Keywords**: Harrod-Domar, neoclassical growth model, Solow-Swan, warranted rate, natural
rate, balanced growth, learning by doing, labor participation

## 1. Introduction

The basic Solow [1956]-Swan [1956] or S-S growth model provides an elegant
solution to the “knife-edge” Harrod [1939]-Domar [1946] or H-D problem^{1} by
endogenizing the warranted rate via the saving-investment relation, wherein
capital growth is a function of a fully adjusting income-capital ratio (inverse of the
H-D capital-output ratio)—allowing smooth substitutability between capital and
labor while keeping the natural rate exogenously fixed. This makes the warranted

* Address all correspondence to dansvillanueva@gmail.com.

** Former Advisor, IMF

1 Explained below.

rate a negative^{2} function of the capital-labor ratio, allowing short-run adjustments
in output or income during the transition to the exogenously fixed natural rate—

sum of labor augmenting productivity/technical change and working population growth. The present paper extends the effects of the capital-labor ratio to the natural rate via capital intensity effects on labor augmenting productivity and labor participation. Thus, equilibrium growth is obtained through adjustments in both warranted and natural rates—a generalization of the S-S growth model.

The model is not meant to take account of aggregate demand—neither is the
S-S model. Like the S-S model, it is a supply-side model, albeit a generalized one
(where the natural rate is made endogenous via endogenous labor productivity
and labor participation). Questions on real GDP, inflation, and unemployment,
as opposed to *capacity or potential* GDP, are not addressed. For a merger of the
textbook short-run macroeconomics of aggregate demand and the S-S textbook
long-run macroeconomics of aggregate supply, as called for by Solow [2022] in
the 1992 addendum to his Nobel Prize speech , and for a numerical application to
the Philippines, see Villanueva et al. [2023], Chapters 7 and 10, respectively.

To put this paper in the simplest perspective, assume a constant-returns (unit- homogeneous) aggregate production function,

*Y* = *K*^{α}*L*^{(1−α)}, (1)

where *Y* is output, *K* is capital stock, *L* = *APN* is effective labor (in efficiency
units), *A* = a labor productivity or technology multiplier, *P* = labor participation
rate, *N* = total population, *α* = elasticity of output with respect to capital,
1−*α* = output elasticity with respect to labor, and 0 < *α* < 1 is a constant.

Conlisk [1968] had shown that in a general production function *Y* = *F* (*K*, *L*)
with constant rates of factor-augmenting technical change attached to *K* and *L* and
subject to degree β returns to scale,^{3} the existence of a well-behaved and balanced
growth equilibrium^{4} requires a unitary elasticity of substitution *ε*(*k*) = (*k*̇/*k*)/(*u**̇**/u*)
where *u *= *F*2/*F*1 and *k *= *K*/*L*.^{5} If *F* is Cobb-Douglas (*β* = 1) as in (1), then *ε*(*k*) = 1
and *α* is the constant income share of capital.^{6}

Income growth at any moment of time is equal to the weighted sum of capital
growth and labor growth, the weights being *α* and 1−*α*, respectively (a dot over a
variable denotes time derivative, i.*e*., *K*

̇

=*dK*/

*dt*):

2 Owing to diminishing marginal product of capital. The H-D growth model assumes a fixed capital-output ratio, signifying absence of substitutability between capital and labor.

3 *β* < 0 signifies decreasing returns to scale,* β* = 1 constant returns to scale, and *β* > 1 increasing returns to scale.

4 Defined as *g*^{*}= where (*K**̇** /K*)^{*}= (*L**̇ **/L*)^{*}= (*Y**̇ **/Y*)^{*} = (*A**̇ **/A*)^{*} + (*P **̇ **/P*)^{*} + (*N**̇ **/N*)^{*}, “ ^{* }” indicates equilibrium and
(*K**̇** /K*)^{*}, (*A**̇ **/A*)^{*}, (*P **̇ **/P*)^{*}, and (*Y**̇ **/Y*)^{*} are functions of capital intensity *k*^{*}* *= *K/L* , as postulated in Section 2.

5 This requirement applies to growth models with *increasing returns to capital* (*β* >1), e.g., Romer [1986],
Lucas [1988], Grossman and Helpman [1990, 1991], Rivera-Batiz and Romer [1991], Barro and Sala-i-
Martin [1995], and Aghion and Howitt [1998]. See discussion in Conlisk [1968].

6 See Chapter 6 of Villanueva et al. [2023] and Conlisk [1968].

*Y*

̇

*Y* *L*

̇

*K*

̇

*L*

= *α* *K* + (1−*α*) , (2)

Capital growth is the *warranted rate* and labor growth is the *natural rate*. The
warranted rate (saving-investment) is a monotonically declining function of the
capital-labor ratio,

where *s* is the fixed gross saving/income ratio and *δ* is a constant depreciation
rate.^{7} From the definition *L* = *APN*, labor growth or the natural rate, is given by

Assume, as in the S-S model, that *Ȧ /A* = *λ *, *Ṗ */*P* = 0 , and *Ṅ */*N* = *n *, where *λ* and *n*
are constants. In the steady state, *k* is constant at *k*^{*},^{8} implying that

and by the constant-returns assumption, using (1) and definition, *g*^{*} = *λ* + *n*,

where *g*^{*} defines the equilibrium growth rate of income *Y*, or the balanced growth
path.

The knife-edge H-D problem (*P*

̇

/*P*= 0 by assumption) is expressed by the condition

where *v* = *K*/*Y* is the fixed H-D capital-output ratio in (5). Since both sides are
constants, equilibrium growth is not assured.^{9}

The S-S model solves the knife-edge H-D problem by employing a neoclassical
production function with smooth substitutability between capital and labor, i.e.,
1/*v* is a monotonically decreasing function of *k*, such that the warranted rate fully
adjusts from any initial level of *k*, making balanced growth possible.^{10} However,
equilibrium income growth *g*^{*} remains exogenous because the natural rate, being
fixed at *λ* + *n*, serves as a bottleneck to the growth process, making the positive
growth effect of a higher saving rate *s* temporary.^{11}

The present paper extends the capital-labor ratio influence on the natural
rate via capital intensity effects on labor productivity (*A*

̇

*/A*) through a modified

7 The income-capital ratio *Y/K* declines with *k* owing to diminishing marginal product of capital—(3).

8 The Inada [1963] conditions assure the existence of a unique value of *k*^{*}.

9 Equality signifies simultaneous achievement of balanced growth, macroeconomic stability, and full employment. Inequality signifies either inflationary spiral or continuous deflation with unemployment.

10 Refer to footnote 7.

11 As Solow [1991:4] calls it.

*K*

̇

*K* *Y*

= *s* *K*− *δ*, (3)

*L*

̇

*L* *A*

̇

*A* *P*

̇

*P* *N*

̇

= + +*N*. (4)

*K*

̇

*K* *L*

̇

*Y* *L*

* = *s* *K*^{*} − *δ* = ^{*} = *λ *+* n* (5)

*K*

̇

*K* *L*

̇

*L* *Y*

̇

* = *sk*^{*(α-1) }− *δ* = ^{*} = *Y*^{*} = *g*^{*}* *= *λ* +* n*, (6)

*sv* − *δ* ⋛ *λ*+ *n*, (7)

Arrow [1962] learning by doing framework [Villanueva 1994], and on labor
participation (*P*

̇

/*P*) through real wage [Villanueva 2020]. Thus, equilibrium income growth is obtained through adjustments in both the warranted rate and in the natural rate.

The 1960s through 1990s saw attempts to solve the S-S model’s exogeneity of
the natural rate *A*

̇

_{/A + P}

̇

_{ /P + N}

̇

/*N*, by making labor-augmenting technical change

*A*

̇

/*A*endogenous.

^{12}An early learning by doing model by Arrow [1962] assumes that learning has a positive effect on the equilibrium growth of output. If labor productivity

*A*changes according to

*Ȧ*/

*A =*

∅

^{(K}

̇

/*K*), where

∅

is a learning coefficient, then equilibrium output growth*g*

^{*}is a multiple of the S-S steady-state growth rate

*n*+

*λ*, since 0 <

∅

< 1 by assumption:Note the absence of the saving rate *s* in (8). However, if the Arrow [1962] model
is interpreted as a change in learning *Ȧ /A* being proportional to the capital-labor
ratio *k* = *K/L*,^{13} and not to the growth rate of the capital stock *K*

̇

*/K*, then the present paper is the Arrow model extended to the case of endogenous labor participation

*P*

̇

*/P*(elaborated below).

Nelson and Phelps [1966], Conlisk [1967], and Villanueva [1994] advanced
early growth models with endogenous labor-augmenting technical change,
deriving the key result that an increase in the saving rate raises equilibrium output
growth. Agénor [2004:466-471] refers to Villanueva’s [1994] model as a variant
of the Conlisk [1967] model and “an extension of Arrow’s [1962] learning by
doing model,…[wherein] the productivity of workers increases when the relative
availability of capital goods (for instance, the stock of high-performance computers)
rises”, leading to enhanced equilibrium growth effects of saving and investment
rates. More precisely, Villanueva [1994] interprets Arrow’s [1962] learning by
doing model by positing *Ȧ /A = θk *+* λ*, where* θ* > 0 is a learning coefficient and *λ* is
an exogenous labor-augmenting productivity/technical change term.

Subsequent contributions constructed increasingly complex models. Romer
[1986; 1990] posited a knowledge-producing sector, alongside a goods-producing
sector. The stock of knowledge is a non-rival good—its use in one sector does not
preclude its use in the other sector. Lucas [1988] proposed models emphasizing
human capital accumulation through schooling and learning by doing, but he
abstracted from the economics of demography.^{14} Grossman and Helpman [1991]

focused on innovation financed by investments in industrial research. Rebelo’s [1991] AK model assumed that all productive inputs, including human capital,

12 For an engaging history of endogenous growth theory, see Warsh [2007]. Solow [1991] has been critical of endogenous growth models with their emphasis on endogenous technical change and increasing returns.

See Chapter 1 of Villanueva et al. [2023] and the third paragraph of the present introductory section.

13 This is my interpretation of the Arrow [1962] model.

14 Lucas [1988:6] admits that this is a serious omission.

*K*

̇

*K* *L*

̇

*L* *Y*

̇

* = ^{*} = *Y*^{*} = *g*^{*}* *= (*n *+ *λ* ) / (1 −

∅

^{).}

^{(8)}

are reproducible.^{15} Aghion and Howitt [1998] highlighted imperfect markets in the
research and development (R&D) sector and Schumpeterian creative destruction.

The knowledge-innovation-R&D sector is assumed to be subject to increasing
returns so that growth does not vanish. Conlisk [1967] had shown that increasing
returns to capital yield explosive growth, which rarely or temporarily happens
in the real world.^{16} He had demonstrated that a growth model with endogenous
labor-augmenting technical change and an aggregate production function that is
subject to diminishing returns to capital is consistent with positive and permanent
growth effects of an increased saving rate (or of any change in the other model
parameters with expected signs).

In all the above growth models, the labor participation rate *P* is an exogenous
constant fraction or percentage by assumption.^{17} Another solution to the knife-
edge problem, besides the S-S model’s variable capital-output ratio implicit in
a well-behaved neoclassical production function with smooth factor substitution
and wage-rice flexibility and endogenous labor-augmenting productivity
multiplier [Villanueva 1994], is a fully-adjusting natural rate via an endogenously-
determined labor participation rate *P* [Villanueva 2020]. In a carefully researched
IMF empirical study, Grigoli et al. [2018:18] found robust results indicating that,
among others, an increasingly educated^{18} labor force influenced significantly and
positively the labor participation rates in 36 advanced economies. Referring to the
US in particular, the Congressional Budget Office (CBO) [2018] issued a working
paper on labor participation, containing similar explanatory variables included in
the IMF study, and arriving at similar statistical results. The study noted that the
US labor participation rate began an uninterrupted decline in the latter half of the
1990s, coinciding with the aging and retirement of baby boomers. In 2007, the
labor participation rate stood at 66 percent. A decade later, in 2017 Q4, it fell to
63.2 percent. The CBO projects that the US labor participation rate will continue to
decline and will be 60.2 percent in 2028 Q4. The projected increase in educational
attainment, which has a positive effect on the labor participation rate, will not be
enough to offset the continued decline attributed to aging and retirements and to
the stagnation in real wages, among other factors.

Motivated by the empirical findings of Grigoli et al. [2018] and the CBO [2018], Villanueva [2020] postulated that the proportionate change in labor

15 Output *Y = AK*, where *Y* is constant returns to capital* K*, implying that* Y* always grows at the same rate as
*K*, and is equal to *s*^{*}*A*, where *s** is the fraction of income saved for investment in physical *and* human capital
(*s*^{*} > *s*; *s* is income saved for investment in physical capital) and *A* is a technological constant. This property
is in sharp contrast to the transitional growth dynamics in the S-S model.

16 However, see Conlisk [1968] and the discussion in the third paragraph of the present introductory section.

17 Whether it is 70 percent or any other percentage, the rate of change in *P* is assumed to be zero. The labor
participation rate and unemployment rate are metrics used to gauge the health of the labor market. The key
difference between the two indicators is that the participation rate measures the percentage of people who
are in the labor force, while the unemployment rate measures the percentage within the labor force that is
currently unemployed.

18 Workers with secondary and tertiary degrees.

participation consists of exogenous components including aging and retirements and endogenous components including aggregate income per man-hour and real wages. The objective was to generalize the equilibrium property of the S-S model by making the natural rate fully adjusting through endogenous labor participation [Villanueva 2020], additional to endogenous learning by doing that improves labor productivity [Villanueva 1994].

The rest of the paper is organized as follows. Section 2 presents and discusses the general neoclassical growth model, followed by the analytics of the temporary and permanent growth effects of changes in the structural parameters, notably, the saving rate, learning coefficient, labor-augmenting productivity/technical change, and components of labor participation. Section 3 concludes.

## 2. A general neoclassical growth model

Equations (9)-(16) below comprise the general neoclassical growth or
extended (*e*) model:

* Y* =* K*^{ α} *L*^{(1-α)} (9)
* L* = *APN* (10)

where *Y* = *GDP*; *K* = capital; *L* = effective labor; *A* = Harrod-neutral productivity
or technical change multiplier; *P* = labor participation rate; *N* = population base;

*RW* = real wage rate; *k *= capital/labor ratio ; *α* = output elasticity with respect
to capital; (1-*α*) = output elasticity with respect to labor; *s* = gross saving rate;

*δ* = depreciation rate; *θ *= learning coefficient; *λ* = constant rate of exogenous
Harrod-neutral technical change; *β* = exogenous, noneconomic determinants of
labor participation; and *ρ*, *ω*, *n* = constant parameters.

*N*

̇

*N*= *n * (14)

*∂Y ∂L*

*RW *= (15)

*KL*

*k *= (16)

*P*

̇

*P* *Y*

= *β + ρ*

( )

*L*+

*ωRW*(13)

*A*̇

*A*= *θk *+ *λ* (12)

*K*

̇

*K* *Y*

= *s K* − *δ* (11)

Equation (9) repeats (1) with the same properties. Equation (10) defines
effective labor *L* as the product term *APN*. Equations (11)-(14) are the dynamic
relationships governing rates of change in *K*, *A*, *P*, and *N*. Equation (15) is a
profit maximization condition that the real wage be set equal to labor’s marginal
product. Finally, (16) defines capital intensity as the ratio of *K* to *L*.

Equation (11), the warranted rate, repeats Equation (3). The derivation
of the natural rate *L*

̇

*/L*is as follows: Villanueva [1994] interprets Arrow’s [1962] learning by doing model by positing

*A*

̇

*/A*=

*θk*+

*λ*, as in (12), where θ > 0 is a learning coefficient. The idea is that as the per capita stock of capital

*K*/

*N*with embodied advanced technology gets larger, the learning experience makes workers more productive.

^{19}

Reflecting the empirical findings of Grigoli et al. [2018] and the CBO
[2018], (13) states that the proportionate change in labor participation *P* is the
sum of an exogenous component *β* and endogenous components *ρ*(*Y/L*) and
*ωRW*, while (14) expresses the standard rate of exogenous population growth.

The exogenous term *β* includes aging and retirements; changes in labor market
policies and institutions, e.g., tax-benefits (tax credits and unemployment
benefits); and a host of non-economic variables identified in the aforementioned
empirical studies.^{20} The endogenous terms are: (a) the portion of aggregate
income per man-hour (*Y/L*) spent on secondary and tertiary education and its
effect on the number of graduates, and the latter’s effect on the labor participation
rate;^{21} and (b) the real wage *RW* that, under profit maximization, is equal to
labor’s marginal product* ∂Y/ ∂L* = (1−*α*)*k*^{α}. Equation (16) defines *k* as the capital/

labor ratio. Increases in the percentage *ρ* of aggregate income per man-hour
(*Y/L*) spent on secondary and tertiary education and in the real wage (*ω* > 0) are
expected to raise the rate of labor participation. There are eight equations with
eight endogenous variables—*Y*, *K*, *L*, *A*, *P*, *N*, *RW*, and *k* (time *t* is suppressed).

*2.1. Reduced model*

Using (9) and (16), (11) becomes

Differentiating (9) with respect to *L* and substituting (15)-(16) yield

19 Using *L* = *AN*, letting *P* = 1, rewrite the above equation as *A**̇ *= *θ*(*K/N)* + *λA*, or (*A** /A*) =* θk **̇* + *λ*,
where *k* = *K/L*.

20 If *β *denotes aging and retirements, then the growth effect is negative; if *β* denotes tax credits and
unemployment benefits, then the growth effect is positive.

21 The coefficient *ρ* > 0 is a composite parameter reflecting the fraction of aggregate income spent on secondary
and tertiary education and its effect on the number of graduates, and the latter’s effect on labor participation.

*K*

̇

*K*= *sk*^{(α-1)} − *δ.* (17)

*RW* = (1 − *α*)*k*^{α}*.* (18)

Using (9) and (16) yields

Equations (18)-(19) into (13) yield

Time differentiating (10), using (12), (14), and (20), yields

The equilibrium growth rate of per capita income is

where *k*^{*} = equilibrium capital intensity, and *e* refers to the extended model.

Time differentiating (16) and substituting (17) and (21) into the result yield the rate of change of capital intensity at any time,

Time differentiating (19) and substituting (22) into the result yield the growth rate of per capita income at any moment of time,

where *k*̇/*k*(*e*) is given by (23). In equilibrium, *k*̇/*k*(*e*) = 0 and (24) reduces to (22).

The reduced models in (*k*

̇

_{/k) , (K}

̇

/*K*) , and (

*L*

̇

/*L*) in the S-S (denoted by

*s*) and extended (denoted by

*e*) models are shown in Figure 1.

^{22}The upper part shows the proportionate rate of change in the capital-labor ratio and the lower part shows the growth rate of output. In both parts, the horizontal axis shows the level of capital intensity. Given the Inada [1963] conditions, the

*k*̇/

*k*line in either model is downward sloping and intersects the

*k*-axis at some positive

*k*, such as

*k*

^{*}

*s*or

*k*

^{*}

*e*

*.*When the

*k*̇/

*k*line intersects the

*k*-axis, the equilibrium capital intensity is

*k*

^{*}

*s*in the S-S model and

*k*

^{*}

*e*in the extended model. In either model, for

*k*<

*k*

^{*},

*k*̇/

*k*>

0, *k* increases until it reaches *k*^{*} at which it becomes constant. For *k* > *k*^{*}, *k*̇/*k* < 0,
and *k* decreases until it goes back to *k*^{*} at which it becomes constant. As capital
intensity changes, diminishing marginal and average productivity of capital (in
either model) and, in the extended model, positive dependence of labor

22 Figures 1 to 4 are phase diagrams. Phase diagramming is a powerful tool in analyzing growth models not explicitly involving time.

*YL*= *k*^{α-1}. (19)

*P*̇

*P*= *ꞵ + *[*ρ* + *ω* (1-*α*)]*k*^{α}. (20)

*L*̇

*L*= *θk + *[*ρ* + *ω* (1-*α*)]*k*^{α} + *λ*+ *n* + *β.* (21)
*g*^{*} − *n* = *θk**e*** + *[*ρ* + *ω* (1-*α*)]*k**e***α* + *λ*+ *ꞵ*, (22)

*k̇ *

*k*(*e*) = *sk*^{(α-1)} − *θk + *[*ρ* + *ω* (1-*α*)]*k*^{α} + (*λ*+ *n* + *β *+ *δ*). (23)

*L*

̇

*L* *k*

̇

– *n* = *g* – *n* = *θk**e*** + *[*ρ* + *ω* (1-*α*)]*k**e***α* + *λ *+ *ꞵ *+ *α * (*e*).* k* (24)

productivity on learning by doing (a positive function of capital intensity) and of labor participation on capital intensity, provide the economic rationale behind the proportionate changes in capital intensity and in the warranted and natural rates.

Specifically, with reference to the lower part of Figure 1, the downward sloping
warranted rate line in either model owes to diminishing marginal and average
capital productivities as *k* increases. The horizontal natural rate line in the S-S
model (*L*

̇

*/L*(

*s*)) reflects the full exogeneity of labor-augmenting productivity. The upward sloping natural rate line in the extended model (

*L*

̇

*/L*(

*e*)) reflects the positive dependence on capital intensity of efficient labor growth via learning by doing and of labor participation via the real wage (a positive function of capital intensity). As capital intensity rises, more intensive learning leads to greater labor productivity. Higher aggregate income per man-hour translates into higher spending on secondary and tertiary education, higher number of graduates, and higher labor participation. As capital intensity rises, labor’s marginal product (real wage) goes up, encouraging higher labor participation and, hence, a larger natural rate. In the lower part of Figure 1, equilibrium growth rates of output

*g*

^{*}

*s*

and *g*^{*}*e *respectively, in the S-S and extended models, are indicated when the
warranted and natural rates are equal at points *k*^{*}*s * and *k*^{*}*e *. Note that *g*^{*}*e * > *g*^{*}*s * because
of endogenous learning by doing and endogenous labor participation in the
extended model.

*g*^{*}*e*

*g*^{*}*s* *L̇* (*s*)* = λ* + *n*

*L*

**FIGURE 1. Extended (e) and Solow-Swan (s) models**

*k*^{*}*s*

*k*^{*}*e*

k

(*e*)* = *s*k*^{(α-1)}* *−* θk *− [*ρ + ω*(1 − *α*)]* k*^{α }−(*λ + n + β + δ*)
*k̇*

*k*

(*s*)* = sk*^{(α-1)}* *−* *(*λ + n + δ*)
*k̇*

*k*

k

(*e*)* = θk +* [*ρ + ω*(1 − *α*)]* k*^{α}
+ *λ + n + β*

*L̇*
*L*

(*e*, *s*)* = sk*^{(α-1)}* *−* δ*
*K̇*

*K*

### 0

### 0

*2.2. Temporary and permanent growth*

Table 1 shows the permanent effects of changes in the S-S and extended model’s parameters on capital intensity and growth rate of income. In the S-S model, only the rates of exogenous labor-augmenting change and population growth have permanent growth effects. In the extended model, higher values for the saving rate, learning coefficient, expenditures on secondary and tertiary education, real wage, exogenous labor-augmenting productivity/technical change, and population growth have growth effects, while aging/retirements and physical capital depreciation impact negatively on growth. On the balanced growth path of the extended model, a constant equilibrium capital/labor ratio means that the warranted rate is equal to the natural rate, and by the constant returns assumption, to the growth rate of per capita output/income as well,

= *g*^{*} − *n* = *θk**e**

* * *+ *[*ρ* + *ω* (1-*α*)]*k**e***α* + *λ *+ *ꞵ*,
which is (22). At any moment of time, the output growth rate is a weighted
average of the warranted and the natural rates,

* = α*(*sk*^{(α-1)} − *δ*) + (1 - *α*){*θk *+ [*ρ *+ *ω*(1-*α*)] *k*^{α} +* λ *+ *n* +*β*}.

There is a divergence between the warranted and natural rates in the short-
run transition to equilibrium. In the S-S model, in equilibrium, the natural rate
is equal to a constant term *λ* + *n*. In the transition to equilibrium, output growth
adjustment falls only on the warranted rate as capital intensity adjusts to its
equilibrium value. In the extended model, the equilibrium natural rate adjusts as
well to a moving capital/labor ratio.

**TABLE 1. Permanent growth**

*s* *θ* *ρ* *ω* *β* *λ* *n* *δ*

*k** S-S + 0 0 0 0 - - -

Extended + - - - - - - -

*g** S-S 0 0 0 0 0 + + 0

Extended + + + + + + + -

Notes:

a. *s *= gross saving rate;

*θ*= learning coefficient;

*ρ*= the portion of aggregate income per man-hour (

*Y/L*) spent on secondary and tertiary education and its effect on the number of graduates, and the latter’s effect on the labor participation rate;

*ω*= effect of real wage on labor participation;

*β*= effects on labor participation of aging and retirements, changes in labor market policies and institutions, e.g., tax-benefits (tax credits and unemployment benefits), and a host of non-economic variables;

*λ*

**= effect of rate of exogenous labor-augmenting productivity/technical change;**

*n*= rate of exogenous population growth;

and** ***δ* = rate of depreciation; *k*^{*} = equilibrium capital intensity; *g*^{*} = equilibrium output or income growth.

b. The equilibrium capital intensity *k** is the root of *k**̇**/k* (*e*) = *sk**^{(α-1)} - *θk*^{*}- [*ρ *+ *ω*(1-*α*)] *k*^{*}^{α }- (*λ* + *n* + *β* + *δ*)

= ψ(*k*^{*}*; s, θ, ρ, ω, β, λ, n, δ*) = 0.

c. In the above table, for the extended model *∂k*^{*}/*∂s *= -ψ'_{s}* */ ψ'_{k *} > 0, since ψ'_{s} >0, and ψ'_{k }_{*} < 0. The
equilibrium output growth rate is *g** = *sk*^{*}^{(α-1)} - *δ *= *H*(*k**; *s*, *δ*), or *g*^{*} = *θk** + [*ρ* + *ω*(1-*α*)] *k*^{*α} + *λ*+ *n *+ *β*

=*J*(*k*^{*}*;θ,ρ,ω,λ,n,β*). Taking partial derivatives, *∂g*^{*}/*∂s *= *H'*_{k *} *H'*_{s} > 0; *∂g*^{*}/*∂δ* = *H'*_{k *} - *H'*_{θ} = *H'*_{k *} -1 < 0; *∂g*^{*}/*∂θ *=
*J'*_{k }_{*} *J'*_{θ *} > 0. The same procedure was used to derive the signs of the other parameters in the above table.

*Y*

̇

*L*

̇

*Y*

*K*

̇

*L*

*K*^{ *} = ^{*} = ^{*}
*Y*

̇

*Y*

*2.2.1. Growth effects of higher saving rate*

Figure 2 shows the effects of an increase in the saving rate on equilibrium capital intensity and equilibrium output growth in the S-S and extended models.

The starting equilibrium positions are points *B*(*k*^{*}*s *, *g*^{*}*s *) for the S-S model and
*A*(*k*^{*}*e *, *g*^{*}*e *) for the extended model. A higher saving rate shifts the warranted rate
line to the right in either model. The new equilibrium positions are indicated by
point *C* in the S-S model and point *D* in the extended model. In both models, the
capital/labor ratio goes up, albeit the new ratio remains lower in the extended
model than in the S-S model, owing to positive labor participation in the extended
model. The key difference is that the new equilibrium output growth increases in
the extended model but remains unchanged in the S-S model.

The short-run dynamics of the S-S model is taken up first, followed by that of
the extended model. During the transition between equilibrium points *B* and *C*,
the S-S output growth rate is momentarily higher than the natural rate *g*^{*}*s * at point
*E* because of a higher warranted rate.^{23} As noted, Figure 2 repeats the lower panel
of Figure 1 (see (17) and (21)). The capital/labor ratio begins to rise from *k*^{*}*s * to
*k*^{*}*s**' * , which slows the warranted rate. Since the natural rate is independent of the
capital/labor ratio, only the warranted rate adjusts downward along the segment
*EC*.^{24} Over time, labor becomes a bottleneck, and the output growth rate slows to
the constant natural rate *g*^{*}*s * = *λ* + *n* at *C*. At this point, the capital/labor ratio stops
rising and stabilizes at a new and higher-level *k*^{*}*s**' * . Thus, the output growth rate
effect of a higher saving rate is temporary, and a higher equilibrium capital/labor
ratio is the only permanent effect.

In the extended model, following the increase in the saving rate, equilibrium
shifts from *A* to *D*. At the starting position *A*, capital grows faster than labor (by
the segment *AF*), and the capital/labor ratio rises from *k*^{*}*e * to *k*^{*}*e**'** *. The marginal and
average products of capital fall, lowering the level of saving per unit of capital,
thus slowing the warranted rate downward along the segment *FD*. On the other
hand, the natural rate, instead of remaining constant as in the S-S model, rises
because of enhanced learning by doing and higher labor participation associated
with a rising capital/labor ratio.^{25} This process continues until the warranted
and natural rates are again equal via a continuous increase in capital intensity at
the new long-run equilibrium *D*, at which point the warranted rate would have
fallen to the new and higher value of the natural rate, equal to the new and higher
equilibrium output growth rate *g*^{*}*e**'** *( > *g*^{*}*e *). Thus, the higher output growth effect
of a larger saving rate is both temporary (like in the S-S model) and permanent
(unlike in the S-S model), the latter owing to the existence of endogenous learning
by doing and endogenous labor participation, making the natural rate respond
positively to an increase in capital intensity.

23 The output growth rate at *E = αg*_{s}* + *(1*-α*)*g*^{*}_{s }.

24 The output growth rate adjustment is traced by the segment *BEC* in terms of the weighted average
of the warranted and natural rates.

25 The natural rate adjustment is traced by the segment *AD*.

*2.2.2. Growth effects of higher labor-augmenting productivity, enhanced *
*learning by doing*

Figure 3 illustrates the temporary and permanent growth effects of enhanced
learning by doing and higher exogenous labor-augmenting labor productivity in
the S-S and extended models. The growth effects of higher labor-augmenting
productivity are taken up first, followed by the growth effects of enhanced
learning by doing. The starting equilibrium positions are points *D*( *k*^{*}*s *, *g*^{*}*s*) for
the S-S model and *A*( *k*^{*}*e *, *g*^{*}*e *) for the extended model. Higher exogenous labor-
augmenting productivity *λ* shifts the natural rate of the S-S model upward to the
*L*

̇

_{ /L = λ}

_{1}+

*n*line, a parallel shift from the previous line. The extended model’s natural rate shifts upward to the left. The new equilibrium positions are indicated by point

*F*in the S-S model and point

*C*in the extended model. In either model, the capital/labor ratio goes down, albeit the new ratio remains lower in the extended model than in the S-S model, owing to positive labor participation and learning by doing in the former. The key difference is that, while the new equilibrium output growth increases to the higher rate of

*g*

^{*}

*s*

*'*=

*λ*1 +

*n*in the S-S model, in the extended model the new equilibrium output growth increases to an even higher rate equal to

*g*

^{*}

*e*=

*g*

*'*

^{*}

*s*

*'*+

*θk*

^{'}

*e** + [

*ρ*+

*ω*(1-

*α*)]

*k*

^{*}

*e*

*'*

*α*+

*λ*1 +

*n*+

*β*(point

*C*).

The short-run (temporary) dynamics of the S-S model is taken up first, followed
by that of the extended model. Note that at any moment of time the output growth
rate is given by *Y*

̇

/*Y*=

*λ*+

*n*+

*α*

*k*

̇

/*k*. Begin with enhanced learning by doing or higher labor-augmenting productivity. Before the steady-state transition between equilibrium points

*D*and

*F*begins, starting from

*k*

^{*}

*s*the output growth rate jumps

**FIGURE 2. Temporary and permanent growth effects of higher saving rate**
(*e*, *s*)* = sk*^{(α-1)}* - δ*

*K̇*
*K*

(*e*)* = θk +* [*ρ + ω*(1-*α*)]* k*^{α}* *
* + λ + n + β*

*L̇*
*L*

(*s*)* = λ + n*
*L̇*

*L*
*g**s*

*g*^{*}*e**'*

*g*^{*}*e*

*g*^{*}*s*

0 *k*^{*}*e* *k*^{*}*s* *k*^{* }*e**'* *k*^{* }*s**'* * k*
*C*

* B*
* A*

* D*
* F*

* E*

Growth Rate of Output

to *Y*

̇

_{/Y = αg}

^{*}

_{s }+ (1 –

*α*)

*g*

^{*}

*s*

*'*=

*λ*1 +

*n*+

*α*

*k*

̇

_{/k < g}

^{*}

_{s}

^{'}because

*k*

̇

/*k*< 0 at

*k*

^{*}

*s*—the natural rate is higher than the warranted rate by segment

*ED*. Capital intensity begins to fall from

*k*

^{*}

*s*to

*k*

^{*}

*s*

*'*, resulting in

*k*

̇

/*k*turning less and less negative, thus raising output growth until

*k*

̇

/*k*= 0 at point

*F*, wherein the new permanently higher S-S growth rate

*λ*1 +

*n*is reached.

In the extended model, following the increase in *λ* from *λ*0 to *λ*1, equilibrium
shifts from *A* to *C*. At the starting position A, labor grows faster than capital, and
the capital/labor ratio declines from *k*^{*}*e* toward *k*^{*}*e**'*

* *. The extended model’s output
growth rate adjustment is traced by the weighted average of segments *BC* and *AC*,
as capital intensity falls from *k*^{*}*e* to *k*^{*}*e**'*

* *. The marginal and average products of
capital rise, raising the level of saving per unit of capital, accelerating the
warranted rate upward along the segment *AC*. On the other hand, the natural rate,
instead of remaining constant at *λ*1 + *n* as in the S-S model, slows from *B* to *C*
because of a lower labor participation rate associated with a declining capital/

labor ratio. This process continues until the warranted and natural rates are again
equal via a continuous fall in the capital/labor ratio at the new long-run equilibrium
*C*, at which point the warranted rate would have risen to the new value of the
natural rate, equal to the new and higher equilibrium output growth rate *g*^{*}*e**'* ( > *g*^{*}*e* ).

Thus, whereas in the S-S model the higher output growth is temporary, it is permanent in the extended model because of the existence of endogenous learning by doing and endogenous labor participation.

**FIGURE 3. Temporary and permanent growth effects of enhanced learning by **
**doing (****θ**_{1}** > ****θ**_{0}**) or higher exogenous labor-augmenting productivity (****λ**_{1}** > ****λ**_{0}**)**

* C*
* B*

* D*

* F* * E* *λ*1

*λ*0

* A*

0
*g*^{*}*e**'*

*g**e*

*g*^{*}*e*

*g*^{*}*s*

*g*^{*}*s**'*

*k*^{*}*e* *k*^{*}*s*

*k*^{*}*e**'* *k*^{*}*s**'*

k

(*s*)* = λ + n*
*L̇*

*L*
(*e*, *s*)* = sk*^{(α-1)}* *−* δ*
*K̇*

*K*
*θ*0 *λ*0

(*e*)* = θk +* [*ρ + ω*(1 − *α*)]* k*^{α}* + λ + n + β*
*L̇*

*L*

*θ*1 *λ*1

Growth Rate of Output

*2.2.3. Growth effects of lower labor participation*

Figure 4 illustrates the temporary and permanent growth effects of a decline in
labor participation from the CBO [2018] forecast—a lower *ρ*, *ω*, or *β*. The initial
equilibrium is at point *A*, with capital/labor ratio *k*^{*}*e* and output growth *g*^{*}*e *.
Following the fall in labor participation, the natural rate line shifts downward to
the right. Equilibrium shifts from *A* to *D*. The capital/labor ratio goes up from *k*^{*}*e*
to *k*^{*}*e**'*

* *, and the equilibrium output growth goes down from *g*^{*}*e* to *g*^{*}*e**' *. The increase in
the equilibrium capital/labor ratio owes to lower effective labor induced by a
lower rate of labor participation. The fall in equilibrium output growth is the
result of a lower natural rate line along with an unchanged warranted rate line.

Notice that there is temporary overshooting of the lower output growth rate at *F*
(in relation to the new permanent growth rate at *D*). At the starting capital intensity
*k*^{*}*e* the natural rate has a precipitous drop to *g**e* (< *g*^{*}*e**'* ) at *F*, following the decline in
labor participation.^{26} As the capital-labor ratio begins to rise from *k*^{*}*e* to *k*^{*}*e**'*

* *, the
natural rate recovers along the segment *FD*, while the warranted rate falls along
the segment *AD* because of diminishing marginal and average products of capital.

26 In Figure 4, before *k* has time to adjust, the output growth rate drops to *g*_{e }*. *Thus, a decline in labor
participation results in a short-term contractionary overshooting (a sharp drop of short-run output growth,
temporarily even lower than the permanently lower output growth at *g*^{*}*e* (< *g**'* ^{*}*e*)).

**FIGURE 4. Temporary and permanent growth effects of a decline in labor **
**participation (lower ****ρ****, **

𝜔

𝜔

**, or β****)**

Growth Rate of Output

0
*g*^{*}*e**'*

*g*^{*}*e*

*g**e*

*k*^{*}*e* *k*^{*}*e**'* * k*

*K̇*

*K = sk*^{(α-1)}* *−* δ*
*L̇*

*L* *= *[*ρ + ω*(1 − *α*)]* k*^{α}* + λ + n + β*

* F* * D*
* A*

This process continues until the warranted and natural rates are again equal via
a continuous increase in the capital/labor ratio at the new long- run equilibrium *D*,
where the permanent output growth *g*^{*}*e**'* is lower than the initial prevent the decline
in, and to encourage higher labor participation through vigorous implementation
of public policies on education, on-the-job training, upgrading skills for a
digital economy, real wage increases in line with labor productivity, and other
labor market participation initiatives identified by Grigoli et al. [2018] and the
CBO [2018].

## 3. Conclusion

This paper’s main conclusion is that endogenous learning by doing and endogenous labor participation ensure a fully adjusting natural rate. Together with a fully adjusting warranted rate of the S-S model, the equilibrium growth rate of output and capital intensity are functions of all the structural parameters reflecting saving, learning by doing, and labor participation. There are temporary and permanent growth effects of increases in the saving rate and in the coefficient of learning by doing. A growth-oriented policy includes measures to raise public and private saving rates, as well as expenditures on education and health care aimed at raising labor productivity.

The CBO [2018] projected decline in the labor participation rate over the next
decade will result in a permanently lower per capita output growth path, with
temporary recessionary overshooting. Policies to restore the previous growth path
or to achieve a permanently higher growth path involve avoiding the projected
fall in labor participation by aggressive and calibrated spending on secondary
and tertiary education, on-the-job training, and skills upgrade to a full-fledged
digital economy (a higher *ρ*), steady increases in real wages in line with labor
productivity (a higher *ω*), vigorous labor market participation activities and more
generous tax-benefits (higher *β*), in order to offset the negative effects of aging
and retirements (lower *β*).

*Acknowledgments*. I am indebted to my mentor, the late John Conlisk who pioneered
endogenous growth theory, and to two anonymous referees for valuable comments. The usual
disclaimer applies.

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