Production Function as a Growth Model

Cobb-Douglas_Production_Function

Any science is tempted by the naive attitude of describing its object of enquiry by means of input-output representations, regardless of state. Typically, microeconomics describes the behavior of firms by means of a production function:

y = f(x) —– (1)

where x ∈ R is a p×1 vector of production factors (the input) and y ∈ R is a q × 1 vector of products (the output).

Both y and x are flows expressed in terms of physical magnitudes per unit time. Thus, they may refer to both goods and services.

Clearly, (1) is independent of state. Economics knows state variables as capital, which may take the form of financial capital (the financial assets owned by a firm), physical capital (the machinery owned by a firm) and human capital (the skills of its employees). These variables should appear as arguments in (1).

This is done in the Georgescu-Roegen production function, which may be expressed as follows:

y= f(k,x) —– (2)

where k ∈ R is a m × 1 vector of capital endowments, measured in physical magnitudes. Without loss of generality, we may assume that the first mp elements represent physical capital, the subsequent mh elements represent human capital and the last mf elements represent financial capital, with mp + mh + mf = m.

Contrary to input and output flows, capital is a stock. Physical capital is measured by physical magnitudes such as the number of machines of a given type. Human capital is generally proxied by educational degrees. Financial capital is measured in monetary terms.

Georgescu-Roegen called the stocks of capital funds, to be contrasted to the flows of products and production factors. Thus, Georgescu-Roegen’s production function is also known as the flows-funds model.

Georgescu-Roegen’s production function is little known and seldom used, but macroeconomics often employs aggregate production functions of the following form:

Y = f(K,L) —– (3)

where Y ∈ R is aggregate income, K ∈ R is aggregate capital and L ∈ R is aggregate labor. Though this connection is never made, (3) is a special case of (2).

The examination of (3) highlighted a fundamental difficulty. In fact, general equilibrium theory requires that the remunerations of production factors are proportional to the corresponding partial derivatives of the production function. In particular, the wage must be proportional to ∂f/∂L and the interest rate must be proportional to ∂f/∂K. These partial derivatives are uniquely determined if df is an exact differential.

If the production function is (1), this translates into requiring that:

2f/∂xi∂xj = ∂2f/∂xj∂xi ∀i, j —– (4)

which are surely satisfied because all xi are flows so they can be easily reverted. If the production function is expressed by (2), but m = 1 the following conditions must be added to (4):

2f/∂k∂xi2f/∂xi∂k ∀i —– (5)

Conditions 5 are still surely satisfied because there is only one capital good. However, if m > 1 the following conditions must be added to conditions 4:

2f/∂ki∂xj = ∂2f/∂xj∂ki ∀i, j —– (6)

2f/∂ki∂kj = ∂2f/∂kj∂ki ∀i, j —– (7)

Conditions 6 and 7 are not necessarily satisfied because each derivative depends on all stocks of capital ki. In particular, conditions 6 and 7 do not hold if, after capital ki has been accumulated in order to use the technique i, capital kj is accumulated in order to use the technique j but, subsequently, production reverts to technique i. This possibility, known as reswitching of techniques, undermines the validity of general equilibrium theory.

For many years, the reswitching of techniques has been regarded as a theoretical curiosum. However, the recent resurgence of coal as a source of energy may be regarded as instances of reswitching.

Finally, it should be noted that as any input-state-output representation, (2) must be complemented by the dynamics of the state variables:

k ̇ = g ( k , x , y ) —– ( 8 )

which updates the vector k in (2) making it dependent on time. In the case of aggregate production function (3), (8) combines with (3) to constitute a growth model.

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