Neoclassical properties.

The Richard function has interesting properties that fit with the production process. However, it does not properly satisfy the axioms of the neoclassical production function. Assuming the function

(4)

where K and L are variables denoting capital and labor respectively. Notice that the above function has constant returns to scale and by the homogeneity we get

where and f(k) is the Richard function. The partial derivatives of Y = L f (k) and the inequalities (3) follows the next proposition

Proposition 2. Let α, λ, A and B, be positive parameters of the Richard function, k ^⋄ = the inflection point of 𝑓(𝑘), then for any k ≥ 0 it satisfies the following

(5)

(6)

where

(7)

Proposition 2 shows that the Richard production function has increasing and decreasing marginal returns for k < k^⬦ and k > k^⬦, respectively. As ^{Sachs et al. (2004)} stressed, this feature is feasible for any firm, “is needed a minimum threshold of capital before the modern pro-duction processes started.” They explained that factory production requires roads, ports access, and literate and numerate workers. Without these basic conditions, at a low level of k, the marginal productivity will be negligible, even negative, under some assumptions. Nevertheless, when capital has reached basic infrastructure and human capital, the marginal productivity of capital and labor might become high and grow faster in a low-income country until it achieves the saturation level. Finally, the marginal productivity will become slower and decrease. Indeed, growth theory literature names the variability of marginal returns of the production function as “non-convexities in the production function” cf. ^{Azariadis and Stachurski (2005).}

From equations (2) and (5), follows that the marginal product of labor depends on the parameters B and λ, due to . Therefore, we have the next proposition.

Proposition 3. If the parameters λ and B satisfies λeλ+1 > B, then for any k > 0 we get

(8)

At this moment is not easy to see the conditions over the parameters B and λ satisfying (8); however, keep in mind that in a perfect competitive economy the wage is defined by , the traditional assumption in the neoclassical model is Now, proposition 3 points out the inequality (8) may reverse for some conditions of B and λ, as we will show in theorem 1. Finally, the Richard function does not satisfy the first Inada condition because

^{Inada (1963)} introduced two mathematical conditions in the production function to ensure the existence and unicity of the steady-state in a neoclassical growth model. These conditions do not represent any natural state of production. Therefore, the production function may be well-founded without Inada conditions.

Elementary assumptions.

Assume a close economy with a Richard production function F(K, L) as defined in (4), with constant returns to scale in the two inputs, capital K and labor L. Let 0 < s < 1 be marginal propensity to save, assume a constant fraction of the net production sY(t) allocated to the creation of new capital, it depreciates at a fixed rate δ, in addition, the labor force grows at a constant rate η. Therefore, the trajectories of L(t) and K(t) are described by

(9)

The production factors assume a perfectly competitive assignment, thus wage and capital rate are equal to the marginal product of its factors

(10)

From the equations (10) and Euler’s theorem for homogeneous functions, follows that

(11)

This model also assumes that firms maximize their income and market factors are exhausted, i.e., the demand for labor and capital are equal to the supply of the factors. Furthermore, we assume the classical hypothesis “the workers do not save, and the capitalist does not consume” cf. ^{Wray (1991);} therefore, we have the equality sY(t) = rK(t). All variables depend on time; therefore. we omit the notation of t. Now we introduce the per capita variables, let and be the aggregates capital and product per capita, respectively. Notices that y = f (k); hence, in per capita terms, we have the following equality

(12)

It is useful to express the identities (10) as follow

(13)

(14)

Competitive market wage curve.

In proposition 3, the inequality over the parameters λ and B was not clear and gave conditions that the wage could be negative. We will first prove the inequality, and then explain the economic meaning of the parameters λ and β

From inequality (7), follows the next result.

Lemma 1 (Slope of the CMW-curve). Let w be the CMW-curve in the industrial sector, then

is decreasing for k < k ^⋄ , increasing for k > k ^⋄ , and reach the minimum when k = k ^⋄ .

Evaluating the identity (14) in the inflection point , we get

If , then ; therefore . Finally, reversing the previous inequalities we get the next theorem.

Theorem 1 (CMW-curve). Assume the Richard production function 𝑓(𝑘), its derivative f `(k), and k⋄ is the inflection point of 𝑓(𝑘).

If , then exists an interval U centered in k⋄, such that for all satisfies

(15)

If , then for all 𝑘 ≥0 satisfies

(16)

Pandemic and infectious diseases.

Representative studies about infectious diseases like ^{Goenka and Liu (2020), Agénor (2015)}, and ^{Acemoglu and Johnson (2007)} show that diseases have a negative impact on the labor force, productivity, and GDP. ^{Acemoglu and Johnson (2007)} pointed out that affectation in low-income countries is higher than in high-income countries. On the other hand, ^{Goenka and Liu (2020)} claim that developed countries have control of diseases or are disease-free, therefore grow at a faster rate, while developing countries have disease-endemic cases and either grow at a slower rate or are in a poverty trap. Theorem 1-(a) shows that infectious diseases that affect a considerable part of the labor population in a country with low initial capital will fall into a poverty trap. Meanwhile, countries with a high initial capital level after a high-impact shock will recover faster. On the other hand, ^{Chou et al. (2004), Hai et al. (2004),} and ^{Siu and Wong (2004)} showed that the 2003 SARS outbreak had a significant GDP decline in the service and manufacturing sectors. ^{Bell and Blanchflower (2020), Bick and Blandin (2020)}, and ^{Coibion et al. (2020),} centered on the Covid-19 outbreak, pointed out that household income had a sharp decrease, the unemployment rate had significantly increased, and the consumer spending fell. Theorem 1 perfectly depicts the diminution of the wage after a negative shock that affected the supply side; hence it matches with the empirical studies mentioned above. The fast recovery of the developed countries after exogenous shocks has many explanations in empirical studies. One of them is the human capital and technological advances, distinctive of the developed countries cf. ^{Acemoglu and Zilibotti (2001), Acemoglu (2002)}, and ^{Acemoglu and Johnson (2007)}. In this model we emphasize that the Richard function describes the different stages of industrialization. In this context, any economy with an initial capital k > k^⬦, describes an economy with a high level of technology. The Richard function curve increases rapidly for these values of k, which means that a reduced labor force makes efficient use of available technology.

Cyclicality of labor Productivity.

The neoclassical production function assumes diminishing marginal product in both inputs capital and labor; this implies that average labor productivity has countercyclical behavior, which contradicts the empirical findings that point out the labor productivity is procyclical (rising in recessions, falling in booms). ^{Biddle (2014)} shows a broad summary of the cyclical behavior of labor productivity in the early 1920s to 1960s and analyzes relevant studies over this period. Indeed, Robert ^{Solow (1964)} called “perverse behavior of productivity in the short run” to procyclical behavior of the productivity because it does not match with the standard neoclassical theory. The shape of the Richard function curve implies decreasing and increasing marginal product of labor cf. Lemma 1; as a result, Theorem 1, states that after a negative exogenous shock, the aggregate productivity reduces, for values k < k^⬦ the marginal product of labor decreases sharply, and the marginal product of labor increases faster for values k > k^⬦, as shown in figure1. In this context, an economy with initial capital k < k^⬦, represent a country with a low level of technology or an underdeveloped country, in consequence, the productivity falls due to lag in technology. Conversely, in countries with a high level of technology, productivity grows faster, even in times of recession. The cyclicality of productivity is a controversial subject until now. ^{Dossche et al. (2021), Fermand and Wang (2016),} and ^{Biddle (2014)}, bring up several empirical explanations about it.

Overcoming the poverty trap.

Theorem 1 claims that a country with a low level of technology may fall into a poverty trap after an exogenous shock because the shock surpasses the productive capacity. Indeed, the shock impacts the marginal product of labor; in consequence, the household income decreases in the proportion of the shock, thus explaining poverty traps and other factors avowed in the literature cf. ^{Azariadis and Stachurski (2005)} and ^{Azariadis (2006)}. On the other hand, ^{Romer (1986),} and ^{Lucas (1988),} stressed the importance of human capital and technological improvements in productivity and economic growth. In several works, Daron Acemoglu claims that investment in technology and human capital will allow developing countries to attain better income and welfare in the long term. “Encouraging the development of technologies more appropriate to the LDCs could therefore reduce the output gap” ^{Acemoglu and Zilibotti (2001)}. In Theorem 1, the inequality 𝐵≥𝜆 𝑒 𝜆+1 , induces a poverty trap. Note that adequate increases in λ changes the inequality, thus representing improvements in productive capacity, corresponding with the implementation of better technologies and human capital, as suggested in the literature above. Figure 2, shows the CMW-curve after the exogenous shock denoted by B = 45 and parameter λ = 2. Note the CMW-curve reach very low values, even negative for 𝑘∈(4,6). The negative values correspond with the exogenous shock that surpassed the productive capacity, leaving the economy with almost all economic sectors down and only a small number of operating firms, with deplorable physical and human capital, and using lagged technologies. These are the features of the poverty trap. Increases in the parameter λ allows to overcome the poverty trap as shows in figures 2-(a) and (b).

a) Overcoming poverty trap by increasing λ₀=2 to λ₁=2.2

Figure 2.a Function and CMW-curve with initial λ₀=2 , and a perturbation λ₁=2.2. 

b) Parameters λ₀=2 and λ₁=2.2 .

Figure 2.b and CMW-curve with initial λ₀=2, and a perturbation λ₁=2.2. 

c) Parameter λ₁ =2.2

Figure 2.c and CMW-curve with initial λ₀=2, and a perturbation λ₁=2.2