Growth theory under heterogeneous heuristic behavior

Abstract

Over the last decades, conceptual frameworks formulated to address the dynamics of economic growth have hypothesized, discussed and tested a large number of different assumptions concerning the role of capital accumulation, labor productivity, learning-by-doing, formal education, innovation, and diffusion of ideas. Underlying all these theoretical contributions is, on the demand side, a pervasive and apparently unshakable structure of analysis: economic agents invariably set an optimal intertemporal consumption plan which allows then to maximize utility over an infinite horizon. Such behavior, however, suggests a planning ability that agents often lack. In fact, household decisions are frequently designed on the basis of heuristics or rules-of-thumb that, although not optimal, are reachable under the cognitive constraints typically faced by human beings. This paper revisits some of the most prominent models of the mainstream growth theory, taking a specific heuristic to account for consumption-savings decisions. The heuristic, which allows for the consideration of distinct profiles of saving behavior across individual agents, is a static rule, which might suggest a return to a Solow-like growth analysis. Notwithstanding, the adopted rule-of-thumb encloses a series of novel and relevant implications for growth theory. Such implications are duly highlighted and discussed in this study.

Appendices

Appendix A: Proof of proposition 1

For x ≥ 0, the process of capital accumulation may initiate under one of two dynamic rules, depending on whether $x<\ln \left [ 1+\frac {K(0)}{Y(0)} \right ]$ or $x\geq \ln \left [ 1+\frac {K(0)}{Y(0)}\right ]$. In the first of these two cases, the agent adopts the middle capital accumulation rule in heuristic (3); otherwise, the first rule, such that capital accumulation grows at the constant negative rate 1 + δ, is followed. Observe, as well, that, in both cases, households will not invest and the level of capital will progressively fall to zero. Given the properties of the neoclassical production function, the value of the ratio $\frac {K(t)}{ Y(t)}$ will also decline over time and, as a result, someone starting at the second capital accumulation dynamic rule will necessarily pass onto the first one at a given point in time (a point that is as much further away in time as the smaller is the value of x). Therefore, in the long-term, as long as x ≥ 0, capital depletion will invariably occur at rate 1 + δ, which obviously implies zero steady-state levels of income and consumption, given the production technology and the consumption heuristic.

For x < 0, we are faced with a Solow-like capital accumulation constraint, with the constant savings rate given by $(1-\zeta )\left (1-e^{x}\right )$. In this case, it is trivial to determine a unique zero-growth steady-state point by solving $\overset {\text {{\Large .}}}{K}(t)=0$. The solution is

$$ K^{\ast }:\frac{Y^{\ast }}{K^{\ast }}=\frac{\delta }{(1-\zeta )\left( 1-e^{x}\right) } $$

(62)

The steady-state output-capital ratio, $\frac {Y^{\ast }}{K^{\ast }}$, is, under the neoclassical specification, a continuous, twice-differentiable, decreasing function, which diverges to infinity for K^∗ = 0 and converges to zero whenever $K^{\ast }\rightarrow \infty$. Therefore, it crosses the constant value in the right hand side of (62) once and only once. If K^∗ is unique, positive and constant, then Y^∗ and C^∗ are also unique positive and constant values, given the production function and the consumption rule.

Appendix : B: Proof of proposition 2

Start by observing that income in a given sentiment class is presentable as in equation (5). The main point to highlight concerning this equation is that all agents, regardless of their sentiment, will receive the same wage, w(t), and the same return rate from capital, r(t), because they are identical to one another except, eventually, for the value of x. This argument is precisely what makes the analysis under heterogeneous sentiments relevant: everyone will always get an income regardless of the trajectory followed by the capital variable, as long as there is at least one ant among grasshoppers. Agents who suffer a complete depletion of their capital stock will continue to be able to consume, because they continue to receive a wage for the participation in the productive activity.

If x ≥ 0, then $K(t,x)\rightarrow 0$, and the income of the agents in the specific sentiment class converges to the steady-state outcome

$$ Y^{\ast }(x)=w^{\ast }L(x) $$

(63)

The steady-state wage level is

$$ w^{\ast }=F_{L}\left( \int\limits_{-\infty }^{0}K^{\ast }(x)dx,L\right) $$

(64)

In the Cobb-Douglas specification,

$$ w^{\ast }=(1-\alpha )A\left( \frac{\int\limits_{-\infty }^{0}K^{\ast }(x)dx}{ L}\right)^{\alpha } $$

(65)

Under the heuristic, the consumption equality that subsists in the steady-state will be, as long as x ≥ 0, C^∗(x) = K^∗(x) + Y^∗(x). Because K^∗(x) = 0, the long-term level of consumption of any class of grasshoppers in this economy is identical to the respective income, as is expressed in (63). Grasshoppers will take advantage, in the long-run, of the presence of ants in the economy; they can work for them (ants will be those who hold all capital in the steady-state) and continue to consume their wage income level at each period.

With respect to x < 0, the capital accumulation dynamic equation of households in some sentiment class x is

$$ \overset{\text{{\Large .}}}{K}(t,x)=(1-\zeta )\left( 1-e^{x}\right) \left[ w(t)L(x)+r(t)K(t,x)\right] -\delta K(t,x) $$

(66)

The steady-state value of the capital stock is the value of K^∗(x) such that

$$ \frac{w^{\ast }L(x)+r^{\ast }K^{\ast }(x)}{K^{\ast }(x)}=\frac{\delta }{ (1-\zeta )\left( 1-e^{x}\right) } $$

(67)

with w^∗ defined in (64) or (65), and

$$ r^{\ast }=F_{K}\left( \int\limits_{-\infty }^{0}K^{\ast }(x)dx,L\right) =\alpha A\left( \frac{L}{\int\limits_{-\infty }^{0}K^{\ast }(x)dx}\right)^{1-\alpha } $$

(68)

To determine K^∗(x) explicitly, it would be necessary to solve a system of dimension equal to the number of sentiment classes that possibly exists for the ant type of households. Nevertheless, the simple observation of expression (67) indicates that the right-hand side of the equation is a constant value, while the left-hand side is, as is for (62), a continuous, twice-differentiable, decreasing function that falls from infinity to zero as we make K^∗(x) to vary from zero to infinity. Hence, the two sides of (67) cross once and only once at some positive K^∗(x) value, which signifies that we guarantee the existence of a unique steady-state amount of K^∗(x), and, consequently, also unique steady-state levels of income and consumption.

Appendix C: Proof of Proposition 3

The examination of differential equation (27) directly suggests the result in the proposition: the two first rules in the capital accumulation equation yield negative growth and a convergence to a zero capital long-term state. Note that unlike what one has observed in the neoclassical growth model, now the threshold between the two first dynamic rules in the heuristic is a constant value, meaning that there is no transition between rules as the stock of capital falls. This, however does not change the fact that, in both cases, the absence of savings and investment throws the economy to an extinction state. For x < 0, there is a constant growth rate $\gamma =\frac {\overset {\text {{\Large .}}}{K}(t,x)}{ K(t,x)}$ that is perpetuated over time. For this growth rate to be positive, the sentiment level must be not only negative but sufficiently strong, in absolute value, in order to guarantee γ > 0; this condition is equivalent to the one in the proposition setting an upper bound on the value of x.

Appendix D: Derivation of the differential equation representing the motion of the human capital share in the Uzawa-Lucas model (x < 0

Let the current-value Hamiltonian function for the intertemporal Uzawa-Lucas growth problem be written as follows, with p(t) and q(t) the co-state variables associated, respectively, to physical capital and human capital,

$$ \begin{array}{@{}rcl@{}} \mathbf{H}\left[ K(t),H(t),u(t),p(t),q(t)\right] &=&p(t)\left\{ F\left[ K(t),u(t)H(t)\right] -C(t)-\delta K(t)\right\} \\ &&+q(t)\left( G\left\{ \left[ 1-u(t)\right] H(t)\right\} -\delta H(t)\right) \end{array} $$

(69)

First-order optimality conditions are as follows:

$$ \frac{\partial \mathbf{H}(t)}{\partial u(t)}=0\Rightarrow \frac{F_{u}}{-G_{u} }=\frac{q(t)}{p(t)} $$

(70)

$$ \overset{\text{{\Large .}}}{p}(t)=\rho p(t)-\frac{\partial \mathbf{H}(t)}{ \partial K(t)}\Rightarrow \overset{\text{{\Large .}}}{p}(t)=\left( \rho +\delta -F_{K}\right) p(t) $$

(71)

$$ \overset{\text{{\Large .}}}{q}(t)=-\frac{\partial \mathbf{H}(t)}{\partial H(t)}\Rightarrow \overset{\text{{\Large .}}}{q}(t)=\left( \rho +\delta -G_{H}\right) q(t)-F_{H}p(t) $$

(72)

and the transversality conditions come

$$ \underset{t\rightarrow \infty }{\lim }K(t)e^{-\rho t}p(t)=\underset{ t\rightarrow \infty }{\lim }H(t)e^{-\rho t}q(t)=0 $$

(73)

where ρ > 0 is the intertemporal discount rate. Defining $Q(t)\equiv \frac {p(t)}{q(t)}$, equations (71) and (72) can be condensed in a unique equation for the dynamics of ratio Q(t),

$$ \frac{\overset{\text{{\Large .}}}{Q}(t)}{Q(t)}=G_{H}-F_{K}+F_{H}Q(t) $$

(74)

Equation (74) might be further simplified by resorting to (70), i.e.,

$$ \frac{\overset{\text{{\Large .}}}{Q}(t)}{Q(t)}=G_{H}-F_{K}-F_{H}\frac{G_{u}}{ F_{u}} $$

(75)

Further insights require specifying functional forms for the production functions. Consider a Cobb-Douglas production function for the goods sector and a linear production function for the human capital sector. The second is already displayed in (34). The first one is

$$ F\left[ K(t),u(t)H(t)\right] =AK(t)^{\alpha }\left[ u(t)H(t)\right]^{1-\alpha }\text{, \ }A>0\text{, }\alpha \in (0,1) $$

(76)

Under (34) and (76), we rewrite equation (75) as

$$ \frac{\overset{\text{{\Large .}}}{Q}(t)}{Q(t)}=B-\alpha A\left[ \frac{ u(t)H(t)}{K(t)}\right]^{1-\alpha } $$

(77)

and optimality condition (70) becomes,

$$ Q(t)=(1-\alpha )\frac{A}{B}\left[ \frac{K(t)}{u(t)H(t)}\right]^{\alpha } $$

(78)

From (78) one draws the following relation between growth rates,

$$ \frac{\overset{\text{{\Large .}}}{u}(t)}{u(t)}=\frac{1}{\alpha }\frac{ \overset{\text{{\Large .}}}{Q}(t)}{Q(t)}+\frac{\overset{\text{{\Large .}}}{K} (t)}{K(t)}-\frac{\overset{\text{{\Large .}}}{H}(t)}{H(t)} $$

(79)

Replacing (77), (32), and (33), into (79 ), an equation of motion for the optimal allocation of human capital across sectors is obtained, which is the equation in rule (35) for x < 0.

Appendix E: Proof of Proposition 4

Under systems (38) or (39), it is straightforward to notice that the ratio Ω(t) declines towards zero with the passage of time. This observation implies that variables income and consumption will both converge to a steady-state of complete exhaustion, as mentioned in the proposition.

Only system (40) involves no trivial dynamics. For this system, we can compute steady-state values and analyze transitional dynamics in the vicinity of the steady-state, given a Cobb-Douglas technology. Straightforward algebra allows for calculating the steady-state value of the human capital share allocated to the goods sector. To derive such result, just consider $\overset {\text {{\Large .}}}{\Omega }(t)=\overset {\text { {\Large .}}}{u}(t)=0$,

$$ u^{\ast }=1-\frac{(1-\zeta )\left( 1-e^{x}\right) }{\alpha } $$

(80)

Given (80), in order to exist an interior solution it is required that the following condition holds: $(1-\zeta )\left (e^{x}-1\right ) <\alpha$. The physical capital – human capital ratio will be, in the long-term equilibrium,

$$ {\Omega}^{\ast }=\left( \frac{1}{\alpha }\frac{B}{A}\right)^{1/(1-\alpha )}u^{\ast } $$

(81)

Replacing the human capital share (80) into the human capital accumulation constraint, (33), one immediately derives the steady-state growth rate of human capital, which is also the balanced growth rate of physical capital, since Ω^∗ is constant. Furthermore, given the shape of the production function and the consumption heuristic, the mentioned growth rate is also the steady-state growth rate of these aggregates.

To prove that dynamic system (40) is saddle-path stable, we linearize it in the vicinity of the steady-state, thus computing the following matricial system:

$$ \left[ \begin{array}{c} \overset{\text{{\Large .}}}{u}(t) \\ \overset{\text{{\Large .}}}{\Omega }(t) \end{array} \right] = B\!\left[ \begin{array}{cc} \alpha u^{\ast }-\frac{\left( 1-\alpha \right)^{2}}{\alpha } & \frac{ 1-\alpha }{\alpha }\left[ 1-\alpha \left( 1-u^{\ast }\right) \right] \frac{ u^{\ast }}{{\Omega}^{\ast }} \\ {\Omega}^{\ast }+\left( 1 - \alpha \right) \left( 1 - u^{\ast }\right) \frac{ {\Omega}^{\ast }}{u^{\ast }} & -\left( 1-\alpha \right) \left( 1-u^{\ast }\right) \end{array} \right] \left[ \begin{array}{c} u(t)-u^{\ast } \\ {\Omega} (t)-{\Omega}^{\ast } \end{array} \right] $$

(82)

The determinant and the trace of the Jacobian matrix in system (82) are, respectively,

$$ Det(J)=-\frac{1-\alpha }{\alpha }B^{2}u^{\ast }\text{; }Tr(J)=B\left( u^{\ast }-\frac{1-\alpha }{\alpha }\right) $$

(83)

what implies that the eigenvalues of J are λ₁ = Bu^∗ and $\lambda _{2}=-\frac {1-\alpha }{\alpha }B$. With one negative and one positive eigenvalues, one confirms that the system is saddle-path stable.⁶

Appendix F: Proof of proposition 5

For every x ≥ 0, the share of human capital allocated to the production of physical goods is the constant value $u(t,x)=1-\frac {\delta }{B}$ and the stock of physical capital declines over time to zero. Hence, consumption of grasshoppers will be, in the steady-state,

$$ C^{\ast }(x)=F_{H}^{\ast }u^{\ast }(x)H^{\ast }(x)=(1-\alpha )A\left( \frac{ {\Omega}^{\ast }}{u^{\ast }}\right)^{\alpha }\left( 1-\frac{\delta }{B} \right) H_{0}(x)\text{, \ }\forall x\geq 0 $$

(84)

where

$$ \frac{{\Omega}^{\ast }}{u^{\ast }}=\frac{\int\limits_{-\infty }^{0}K^{\ast }(x)dx}{(1-\vartheta )L\left( 1-\frac{\delta }{B}\right) H_{0}(x)+\int\limits_{-\infty }^{0}u^{\ast }(x)H^{\ast }(x)dx} $$

(85)

In (85), (1 − 𝜗)L represents the number of grasshoppers. Examining result (84), one realizes that C^∗(x) is positive and constant. As in the neoclassical growth model, grasshoppers take advantage of the fact that there are ants who save and apply their capital in production, so that grasshoppers have the possibility of working and receiving a wage, which they integrally apply, at each period, in consumption.

Let us now explore the steady-state result for the ants’ classes. Under x < 0, the relevant dynamic system for agents in class x is, adapting from system (40),

$$ \left\{ \begin{array}{c} \frac{\overset{\text{{\Large .}}}{u}(t,x)}{u(t,x)}=\frac{1-\alpha }{\alpha } B+Bu(t,x) - [e^{x}+\zeta (1 - e^{x})]\left[ w(t)u(t,x)H(t,x)+r(t)K(t,x)\right] \\ \frac{\overset{\text{{\Large .}}}{\Omega }(t,x)}{\Omega (t,x)}=(1-\zeta )\left( 1-e^{x}\right) \left[ w(t)u(t,x)H(t,x)+r(t)K(t,x)\right] -B\left[ 1-u(t,x)\right] \end{array} \right. $$

(86)

Solving $\overset {\text {{\Large .}}}{\Omega }(t,x)=\overset {\text {{\Large .}} }{u}(t,x)=0$, one gets expressions for the share of human capital u^∗(x) and for the long-term growth rate γ^∗(x) that mimic (80) and (41), respectively. Growth rate γ^∗(x) is the growth rate of both forms of capital and, given the income expression and the consumption function, also the steady-state growth rate of these two variables (income and consumption).

Appendix G: Proof of proposition 6

The observation of differential equations (55) and (56 ) indicates that, under x ≥ 0, there is no capital accumulation in the steady-state. This is true for both types of capital. Therefore, no income and no consumption will subsist in the steady-state in an economy of grasshoppers that give priority to the short-run.

The third rule in the consumption heuristic introduces interesting dynamics into the growth process and allows us to perceive that, in this case, endogenous growth emerges. Defining ratio ${\Phi } (t)\equiv \frac {K(t)}{R(t)}$, the following equation of motion is derived,

$$ \frac{\overset{\text{{\Large .}}}{\Phi }(t)}{\Phi (t)}=(1-\zeta )\left( 1-e^{x}\right) A\left[ \frac{L}{K(t)}+\frac{1}{\Phi (t)}\right]^{1-\alpha } \left[ \sigma -(1-\sigma ){\Phi} (t)\right] $$

(87)

In this case, for x < 0, the capital accumulation constraint is

$$ \frac{\overset{\text{{\Large .}}}{K}(t)}{K(t)}=\sigma (1-\zeta )\left( 1-e^{x}\right) A\left[ \frac{L}{K(t)}+\frac{1}{\Phi (t)}\right]^{1-\alpha }-\delta $$

(88)

From the two above equations, it is possible to derive the steady-state growth of physical capital. Start by noticing that a constant steady-state ratio Φ^∗ corresponds, given (87), to:

$$ {\Phi}^{\ast }=\frac{\sigma }{1-\sigma } $$

(89)

From differential equation (88) one may, then, write the steady-state growth rate of physical capital as

$$ \gamma_{K}=\sigma (1-\zeta )\left( 1-e^{x}\right) A\left( \frac{L}{K^{\ast } }+\frac{1-\sigma }{\sigma }\right)^{1-\alpha }-\delta $$

(90)

Because K^∗ grows at a constant rate in the steady-state and L remains constant, ratio $\frac {L}{K^{\ast }}$ falls to zero, and this observation directs us to the explicit growth rate of physical capital, which is (57). This is also the steady-state growth rate of robots, as one understands by noticing that under x < 0,

$$ \frac{\overset{\text{{\Large .}}}{R}(t)}{R(t)}=(1-\sigma )(1-\zeta )\left( 1-e^{x}\right) A{\Phi} (t)\left[ \frac{L}{K(t)}+\frac{1}{\Phi (t)}\right]^{1-\alpha }-\delta $$

(91)

Replacing Φ(t) in (91) by its steady-state value and remarking once again that the labor-capital ratio falls to zero, the steady-state evaluation of the equation of motion for robotic capital allows us to unveil, once again, growth rate (57). Given that the production technology is Cobb-Douglas, then income also grows, at the balanced growth path, at the same rate, which is also the growth rate of consumption, once we take consumption heuristic (54).

Appendix H: Proof of Proposition 7

From differential equations (49) and (50), it is straightforward to observe that K(t,x) and R(t,x) converge to zero for every non-negative x. Thus, the income and consumption levels of grasshoppers will tend to

$$ Y^{\ast }(x)=C^{\ast }(x)=F_{L}^{\ast }L(x)=(1-\alpha )A\left[ \frac{ \int\limits_{-\infty }^{0}K^{\ast }(x)dx}{L+\int\limits_{-\infty }^{0}R^{\ast }(x)dx}\right]^{\alpha }L(x) $$

(92)

Because L is constant and K and R grow at a same constant rate in the steady-state, for every x < 0, grasshoppers consume a positive constant amount in the steady-state (as long as labor is not entirely replaced by robotic capital). Furthermore, notice that $\frac {\int \limits _{-\infty }^{0}K^{\ast }(x)dx}{L+\int \limits _{-\infty }^{0}R^{\ast }(x)dx}={\Phi } ^{\ast }$ and, thus, expression (92) might in a simpler way be displayed as: $Y^{\ast }(x)=C^{\ast }(x)=(1-\alpha )A\left (\frac {\sigma }{ 1-\sigma }\right )^{\alpha }L(x)$.

For the ants’ classes, a positive endogenous growth rate is determinable. The growth rate of physical capital held by agents in class x, ∀x < 0, is

$$ \begin{array}{@{}rcl@{}} \frac{\overset{\text{{\Large .}}}{K}(t,x)}{K(t,x)} &=&\sigma (1-\zeta )\left( 1-e^{x}\right) \frac{Y(t,x)}{K(t,x)}-\delta \Leftrightarrow \\ \frac{\overset{\text{{\Large .}}}{K}(t,x)}{K(t,x)} &=&\sigma (1-\zeta )\left( 1-e^{x}\right) \left[ F_{L}\frac{L(x)+R(t,x)}{K(t,x)}+F_{K}\right] -\delta \Leftrightarrow \\ \frac{\overset{\text{{\Large .}}}{K}(t,x)}{K(t,x)} &=&\sigma (1-\zeta )\left( 1-e^{x}\right) \left\{ F_{L}\left[ \frac{L(x)}{K(t,x)}+\frac{1}{\Phi (t,x)}\right] +F_{K}\right\} -\delta \end{array} $$

(93)

In the steady-state, term $\frac {L(x)}{K(t,x)}$ converges to zero and ${\Phi }^{\ast }(x)=\frac {\sigma }{1-\sigma }$; the wage rate and the rate of return on capital are, respectively, $w^{\ast }=F_{L}^{\ast }=(1-\alpha )A\left (\frac {\sigma }{1-\sigma }\right )^{\alpha }$ and $r^{\ast }=F_{K}^{\ast }=\alpha A\left (\frac {1-\sigma }{\sigma }\right )^{1-\alpha }$. Thus, the evaluation of (93) in the steady-state locus conduct directly to growth rate expression (57).

A similar reasoning can be taken for capital variable R(t,x), leading exactly to the same outcome,

$$ \begin{array}{@{}rcl@{}} \frac{\overset{\text{{\Large .}}}{R}(t,x)}{R(t,x)} &=&(1-\sigma )(1-\zeta )\left( 1-e^{x}\right) \frac{Y(t,x)}{R(t,x)}-\delta \Leftrightarrow \\ \frac{\overset{\text{{\Large .}}}{R}(t,x)}{R(t,x)} &=&(1-\sigma )(1-\zeta )\left( 1-e^{x}\right) \left[ F_{L}\frac{L(x)+R(t,x)}{R(t,x)}+F_{K}\frac{ K(t,x)}{R(t,x)}\right] -\delta \Leftrightarrow \\ \frac{\overset{\text{{\Large .}}}{R}(t,x)}{R(t,x)} &=&(1-\sigma )(1-\zeta )\left( 1-e^{x}\right) \left\{ F_{L}\left[ \frac{L(x)}{R(t,x)}+1\right] +F_{K}{\Phi} (t,x)\right\} -\delta \end{array} $$

(94)

To confirm the identical outcome, observe that $\underset {t\rightarrow \infty }{lim}\left \{ F_{L}\left [ \frac {L(x)}{R(t,x)}+1\right ] +F_{K}\right \} =F_{L}^{\ast }+F_{K}^{\ast }\frac {\sigma }{1-\sigma }=A\left (\frac {\sigma }{ 1-\sigma }\right )^{\alpha }$. Replacing this steady-state value into (94), we get (57). If both forms of capital follow a same balanced growth path, then it is straightforward to infer, as before, that income and consumption share the same long-term growth rate.