Economic History

Towards rethinking the saving rate in Solow-Swan Model.

The saving rate in the neoclassical growth model forces a Trend Stationary Process (TSP) which determines the path of economic growth. By looking at the behavior of the Saving Rate in Consumer Optimization Models, I argued that it is possible to transform the saving rate from a parameter exogenously determined into a variable endogenously dependent of the labor-share with a Cobb-Douglass production function. This transformation is achievable by dropping the presumption of the so-called “inter-temporal substitution effect” (Theta) pertaining the behavior of the saving rate in the model of growth with consumer optimization, which leads to the following inequality, α (0)< α. Note to the reader: please forgive format in which equations are present in. Please also request them in case you can not follow the ideas.

Introduction:

The saving rate in the neoclassical growth model is a Trend Stationary Process (TSP) which determines the path of economic growth. The neoclassical model and its applied regressions handle the saving rate as a parameter which usually is positive -at least for developed countries. Gregory Mankiw for instance argues that higher levels of saving rates lead to a higher income level in the steady-state (Mankiw, G. et al. 1992). However, neither have saving rates always been positive for all periods, nor for all regions –at least in the United States (Dolfman, M. & McSweeny, 2006). Two very interesting historic cases in which the US saving rate has fallen negative are 1901 and 1984, -2.53% and -3.48% respectively. So, if it is true that the saving rate is a Deterministic Trend with a Trend Stationary Process, why economies are able to get in and out of debt in the reality?

In this blog post I will argue that it is possible to transform the saving rate from a parameter exogenously determined to a variable endogenously dependent of the labor-share with a Cobb-Douglass production function. That is, α (0)< α. I argue that I can achieve such a transformation by dropping the presumption of the so-called “inter-temporal substitution effect” (Theta) pertaining the behavior of the saving rate in the model of growth with consumer optimization. I also understand this transformation would avoid violating Ponzi-game and comply with transversality condition.

The article is organized as follows, the second section after this introduction takes a brief look at the neoclassical growth model with a short mention to its Human Capital extension in the third section. The fourth section looks at the transitional dynamics of the neoclassical model. In the fifth section I take a look at the behavior of the Saving Rate in Consumer Optimization Models, and attempt to provide a cogent argument for the transformation of the parameter into a variable. Finally I present conclusions and left the research open for further development.

Theories of economic growth:

The fundamental equation of the Solow-Swan model states that the net increase in capital accumulation over time equals gross investment less depreciation. It is basically a statement in which capital accumulation changes over time as a function of investment and depreciation of capital ,

k ̇(t)=I(t)-δk(t) (1)

Thus, investment as long as it is understood as a difference between income and consumption/spending can be interpreted as saving rate. Then, the fundamental equation of the Solow-Swan model is rewritten as follows,

k ̇=s⋅f(k)-(n+δ)⋅k (2)

Where k ̇ denotes differentiation of capital with respect to time; s, saving rate equals investment rates; f(k) represents some production function in which physical capital plays a role; and (n+δ) denotes depreciation rate of capital itself and depreciation rate given by increase in population. The model assumes depreciation is constant and always greater than zero, n+δ>0. To some extent, the model assumes that in the process of capital reproduction there are economic losses for which no agent can capitalize any economic gain. Also, the model exhibits diminishing returns to capital. On the other hand, what the basic model does not include constitutes its main assumptions. First, technological progress is exogenous and fixed coefficient of a given production function. Second, the neoclassical model does not account neither for foreign trade effects nor Government purchases. Third, since depreciation is constant, expectations are suppressed from the model.

Generally speaking, the neoclassical model stresses the fact that higher rates of savings foster economic growth, whereas higher rates of population growth halt economic growth. Clearly two vector point out in opposite directions.

s⋅f(k) (3)

Which is called the “saving curve” and,

(n+δ) (4)

Known as the depreciation curve.

Empirical studies of the Solow-Swan model have found that those two vector forces actually affect income in the predicted way (Mankiw, et. al. 1992). In other words, “the steady-state capital-labor ratio is related positively to the rate of savings and negatively related to the rate of population growth (Mankin, et. al. 1992. Page 410).

Modifications to the Solow-Swan model:

Generally speaking, the literature on economic growth recognizes three modifications to the neoclassical model. The first one and perhaps the most cogent one is the model that includes human capital as a complementary input for the production function. The second modification concerns consumer optimization problems, and the third considers poverty traps. For now I will focus on the modifications that include human capital and technology given its relevance for explaining cross-country data.

Endogenous technological change models like Paul Romer’s are based on three premises: First, technological change drives economic growth. Second, technological change is an outcome of market incentives. Firms respond to market incentives through innovations in technology. Third, developing new technology basically counts within fixed costs for firms (Romer, P. 1990). When these set of premises are considered, the stock of human capital determines economic growth and changes in technology becomes a strategy for accumulation of capital (Romer, P. 1990). Paul Romer supports its argument by discerning the nature of human capital, which stems from the reproduction and acquisition of knowledge. On one side, knowledge is understood as a rival good with incomplete excludability, which means that knowledge acquired by people is applied by individuals, and at the same time it gets shared by societies. An analogy that would explain Romer’s point is the following: a student and whatever she learns at the library makes up the rivalry aspect of knowledge. On the other hand, whatever content she learned from the repository of the library becomes the shared aspect of knowledge. In Romer’s words, “the owner of a design has property rights over its use in the production of a new product, but not over its use in research” (Romer, 1990, page 84).

On the empirics of the model, it is precisely the human capital extension that better fits data for cross-country samples. Gregory Mankiw has stressed the fact that an augmented Solow-Swan model with human capital provides a good description in comparative economics perspective. In Mankin words, “the evidence indicates that, holding population growth and capital accumulation constant, countries converge at about the same rate the augmented Solow-Swan Model predicts” (Mankiw et al. 1992). In addition, Mankiw claims that saving rates and population growth affect income in the direction that Solow predicts. Then, by considering human capital, Mankin asserts a production function that captures the dynamics of economic growth would look like Y=K^(1/3) H^(1/3) L^(1/3).

Some of the most relevant implications of the augmented Solow-Swan models are that in spite of not having substantial externalities from the accumulation of physical capital, “a higher saving rate leads to a higher income in the steady-state, which in turn leads to a higher level of human capital” (Mankiw et al., 1992. Page 433). Thus, the question about a lower rate, or even a negative rate of savings becomes interesting.

Dynamics of the neoclassical model:

The neoclassical model starts by considering a maximizing agent who has access to certain technology for production. Usually, such production function takes the form of Cobb-Douglass,

y=Ak^α L^(1-α) (5)

The agent maximizes its capital with the available technology and available stock of knowledge. Then, the first part of the model is rewritten as,

max_c(T) v(0) ∫_0^T〖[f(k),c(t),t]⋅ⅆt〗 (6)

Where V(0) is the value of the objective function.

The idea of having a depreciation rate slowing down the pace of economic growth translates onto the first constraint imposed to whatever function of production is chosen. Equation number (6) has a constraint that represents the depreciation curve. It is called the accumulation constraint or transition equation. Also, it is assumed that the initial endowment of the agent is positive and cannot be negative at the end of the lifetime. This fact imposes two more conditions for a total of three constraints,

k ̇(t)=g[k(t),c(t),t] (7)

k(0)=k_0>0

k(t)⋅ⅇ^(-r ̅(T)⋅T)≥0

Thus, the challenge for solving the neoclassical model stems from finding vector fields that expand simultaneously the matrix within the boundaries imposed by the initial conditions. For achieving such a goal, the strategy sets up two Lagrange multipliers on the transition equation and on the discount equation. Therefore, the model can be solved by integrating by parts the following expression,

L=∫_0^T〖v[k(t),c(t),t]⋅ⅆt〗+∫_0^T〖{μ(t)⋅(g[k(t),c(t),t]-k ̇(t))}⋅ⅆt〗+v⋅k(t)⋅ⅇ^(r ̅(T)⋅T) (8)

Where μ(t), is the first Lagrange multiplier, and v is the simultaneous multiplier that expands the vector field of the discount equation without falling below zero. So, the objective of solving the neoclassical model resides in finding the associated eigenvector for those two multipliers. Likewise, the Lagrange can be rewritten as a Hamiltonian in order to discern amongst the equations,

L=∫_0^T〖{H[k(t),c(t),t]+u ̇(t)⋅k(t)}⋅ⅆt〗+μ(0)⋅k_0-μ(T)⋅k(T)+v⋅k(T)⋅ⅇ^(-r ̅(T)⋅T) (9)

Altogether, the Hamiltonian can be interpreted as the instantaneous total economic contribution of capital whenever a different choice of consumption is made. This is basically the difference between the control variable and the state variable respectively.

Behavior of the Saving Rate in Consumer Optimization Models: the trap and the way out.

Thus far we have seen that the saving rate in the framework of the neoclassical growth model is no more than a constant which leads the growth dynamics towards a deterministic trend. This very fact stems from the linear relation between change in capital stock and whatever the production function might be. In other words, the saving rate in the neoclassical growth model is a Trend Stationary Process (TSP) which determines the path of growth,

c ̂=(1-s)⋅f(k ̂ ) (9)

Depending upon the sign of the first term, c ̂ goes either up or down. Contrariwise, the flip side of such a dynamic must represent the dynamics of the saving rate. In spite of the reality being different, the neoclassical model does not allow for any changes in the behavior of the saving rate, when in fact people and countries actually get in and out of debt. Models with consumer optimization –the Ramsey Model- advanced an answer on this matter. However, the Ramsey model shows an ambiguity that stems from the assumption that the saving rate can either increase or decline from two sources: either income effect or inter-temporal substitution effect. I would argue that such a dilemma is false since consumer optimization perspective double-accounts for depreciation of capital; which also means that the only way to get in or out of debt is by adding or subtracting labor hours. In other terms, increasing or decreasing the labor-share within Cobb-Douglas production function. Let’s start by looking at the behavior of the saving rate in the growth model with consumer optimization.

The behavior of the Saving Rate in the Ramsey Model:

The behavior of the saving rate in the Ramsey model redefines the savings rate in terms of consumption,

s = (1-c) (10)

Or dynamically as,

c ̂=(1-s)⋅f(k ̇ ) (9)

Which is the same as,

s=1-c ̂/f(k ̇ ) (11)

This redefinition of the saving rate in the neoclassical model aims at both avoiding to have s as a constant and redefine it in terms of the actual variables present in the First Order Conditions. Since the model is dynamic, inter-temporal consumption decisions have to be made by the optimizing agents. Agents face a decision between consuming today and/or saving for tomorrow. This aims at demonstrating that the transversality conditions must hold and that any debt must be offset in the future given that in the Ramsey model agents prefer to consume today rather than tomorrow. In such a scenario consumption, and therefore saving rate becomes a function of the rate of return of saving (say market saving rate), and a function of consuming time preference. Regardless of the algebraic complication, the model concludes that agents prefer to consume sooner than later.

In other words, change in consumption will reflect the saving rate. Then, consumer optimization model redefines the optimal saving rate conversely to consumption as,

s^*=α⋅((x+n+δ))/((δ+p+θx) ) (12)

Where alpha represents the labor share ratio, p a time preference term, and theta the inter-temporal substitution parameter, which is nothing more than a responsiveness measure of consumption to interest rate. Still, regardless of the boundaries of such an elasticity, people prefer to consume today rather than tomorrow.

Barro and Sala-i-Martin (2004) state that “heuristically, the behavior of the saving rate is ambiguous because it involves the offsetting impacts from a substitution effect and income effect” (Barro and Sala-i-Martin, 2004. Page 107). They claim that “as k ̇ rises, the gap between current and permanent income diminishes; hence, consumption tends to fall in relation to current income, and the saving rate tends to rise. This force –income effect- tends to raise the saving rate as the economy develops” (Barro and Sala-i-Martin, 2004. Page 107). Contrariwise, “as k ̇ rises, the decline in f'(k ̇ ) lowers the rate of return, r, on savings. The reduced incentive –an inter-temporal substitution effect- tends to lower the saving rate as the economy develops” (Barro and Sala-i-Martin, 2004. Page 107). To put differently, the dynamics of the saving rate depends on whether income increases/decreases or interest rates go up or down.

Thus, the dynamic of the saving rate under consumer optimization depends upon the forces mentioned above. However, those factors end up being parameter exogenously determined. Therefore, the logic goes back to where it was, namely, the saving rate as a constant.

The way out of the constant trap: to assume n as marginal labor always greater than labor share in Cobb-Douglass.

The way out of ending up with values exogenously determined is to drop the step in which the Ramsey model enters the dilemma between substitution effect and income effect. That is, to assume that substitution effect equals depreciation of capital in terms of inflation/deflation, which is something that the neoclassical model already accounts for by the depreciation curve. Thus, there is no need to account for depreciation twice. If we drop these terms we will be left out with a single variable n, and the parameter alpha.

If we take the optimal s equation and further simplify it, we end up with the variable n. That is basically the following,

s^*=α⋅((x+n+δ))/((δ+p+θx) ) (13)

we simplify it to get,

s^*=α⋅n/(θ+ρ) (14)

Now recall that theta and p are the inter-temporal substitution effect of the rate of return, and p is the time preference of consuming, respectively; alpha is the labor share in the production function and n is population growth. Regardless of the value of the latter two parameter, they both will leave the equation looking like the following,

s^*=α⋅n (15)

Simply put, saving rate hinges on both the labor-capital ratio and population. Then the question becomes, what is the economic link that tides population and the neoclassical growth model? The answer is labor productivity. Therefore, the variable n needs to enter the model by its labor connotation: marginal labor hours added.

So, the only new thing we have is a constraint that requires labor to be greater than the initial labor share.

s=v[α(0)>α] (16)

Our initial maximization problem becomes,

max_c(T) v(0) ∫_0^T〖[f(k),c(t),t]⋅ⅆt〗

Subject to, k ̇(t)=g[k(t),c(t),t]

k(0)=k_0>0

k(t)⋅ⅇ^(-r ̅(T)⋅T)≥0

α (0)< α

Conclusion:

In this post I wanted to argue that it is possible to transform the saving rate from a parameter exogenously determined into a variable endogenously dependent of the labor-share with a Cobb-Douglass production function. This transformation is achievable by dropping the presumption of the so-called “inter-temporal substitution effect” (Theta) pertaining the behavior of the saving rate in the model of growth with consumer optimization, which leads to the following inequality, α (0)< α. By doing so, the neoclassical growth model may overcome a negative saving rate by adding labor or by increasing labor productivity, and inversely the economy may get indebted. Essentially, a fourth constraint pertaining to the labor-share evolution will allow the model to drop the deterministic trend –positive or negative- dictated by a constant value of the saving rate. In other words, this approach would allow for variation of the saving rate.

Reference list:

Barro, R. & Sala-i-Martin, X. (2004). Economic Growth.MIT. Second Edition. Mankiw, G., Romer, D. & Weil, D. (1992). A contribution to the empirics of Economic Growth. The Quarterly Journal of Economics. Romer, P. (1990). Endogenous Technological Change. Journal of Political Economy. Vol. 98. No. 5. Part 2: the problem of development. A conference of the Institute for the Study of Free Enterprise. Dolfman, M. & McSweeny. (2006). 100 years of US Consumer Spending. US Department of Labor.

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