RDP 8501: Neoclassical Theory and Australian Business Investment: A Reappraisal 3. An Integrated Model of Investment Behaviour

In this section we discuss a fairly simple model of investment behaviour based on neoclassical theory and which seems broadly consistent with the facts, particularly with the developments of the seventies and early eighties. We give special attention to three major decisions linked with investment and capital: the decision to own capital goods, the decision to use capital goods, and the decision to produce capital goods. By the same token, we are led to distinguish between three important variables related to the decisions listed above: the price of capital, the rental price of capital, and the return on capital. The discussion follows the broad lines of Foley and Sldrauski's (1970) work, and it can be set in a two-input, two-output framework. Generalisation can be undertaken at a later stage.

The paper adopts the assumption that the stock of any commodity remains equal to its beginning-of-period value until the last instant of the period. It then rises to its recorded end-of-period value. This assumption is also applied to interest rates. Implicit price deflators are averages for the period but are assumed to be constant for the whole of each period in order to be consistent with the treatment of other variables. Investment and output are flows over the period.

It might be preferable to adopt the approach whereby variables are “centred” in each period. Further development of the model will explore such an alternative construction.

We assume an economy that uses two factors of production – capital and labour – to produce two goods – investment goods and consumption goods. We assume that capital and labour are homogeneous and mobile between firms. Let x_N be the input of labour services, and let w_N be the rental price of labour. We denote the outputs of investment goods and consumption goods by y_I and y_C respectively; p_I and p_C are the corresponding prices. Let T be the production possibility set, i.e. the set of all feasible input and output combinations. We assume that T is a convex cone. Assuming that profit maximisation takes place, the aggregate technology can be represented by a gross domestic product (GDP) function defined as follows:^[12]

for p_I. p_C ≥ O and x_K, x_N > 0. Given the assumptions about T, π(.) is linearly homogeneous, nondecreasing and convex in output prices, and linearly homogeneous, increasing, and concave in input quantities.

The description of the technology by a GDP function makes it easy to derive the profit maximising supply of output and inverse demand for input functions. Of particular interest to us are the supply of investment goods and the inverse demand for capital services. Hotelling's (1932) lemma implies that:^[13]

and similarly:

The supply of consumption goods and the inverse demand for labour services could be obtained in the same way.^[14] The homogeneity of π(.) implies that y_I(.) is homogeneous of degree zero in prices and linearly homogeneous in quantities, while the reverse is true for w_K(.). Furthermore, the curvature properties of π(.) imply that ∂y_I/∂p_I ≥ 0 and ∂w_K(.)/∂x_K < 0, i.e. the investment good supply schedule is upward sloping (or at least not downward sloping), and the inverse demand for capital services is negatively sloped.

The description of the technology by a GDP function is very convenient whenever one views input quantities and output prices as exogenous.^[15] We will indeed assume that the capital stock and employment are given at any point in time, and the price of consumption goods will be assumed exogenous as well.^[16] As to the price of investment goods, we assume that it is determined outside the production model by a process yet to be described. Note that we do not assume that production is non-joint in input quantities, i.e. that the two outputs are produced by separate production functions.^[17] Non-jointness plays an important role in many areas of economics, for instance in growth theory and international trade theory; it leads to a number of remarkable results, such as the Stolper-Samuelson and the Rybczynski theorems, but it need not be invoked to derive (4)–(5), and it is not needed for our empirical work.^[18]

It is worth pointing out that (5) is very similar to (2) above. The main differences are that we have chosen here to treat employment as a fixed input,^[19] and, of course, w_K(.) is now a function of two price variables. The inclusion of labour as a fixed input, and the fact that our data are uncorrected for technological change may lead to some difficulties in our empirical work. A convenient way of handling these is to include a time trend in the estimating equations.^[20]

Equations (4) and (5) determine the supply of investment goods and the rental price of capital, given factor endowments and output prices. The flow of new capital goods will, of course, bring about changes in the capital stock over time as

where x_K(+1) is the end-of-period stock of capital (the stock at the beginning of the following period), and δ is the rate of depreciation of capital.

In this model investment is viewed as being supply determined. Given the production possibility set, the investment flow simply depends on the relative price of capital goods. A similar view, albeit in a one-sector model context, is held by Tobin (1969).

An important question that must now be answered is what determines the price of investment goods? So far we have looked at two decisions related to capital: the decision to utilise existing capital services, and the decision to produce additional capital goods. Me must now look at a third important decision: the decision to own existing capital goods. This question can best be examined within a portfolio framework.

We assume three assets: capital goods, money, and bonds.^[21] New capital goods are assumed to be perfect substitutes for existing ones, hence the price of existing capital goods is P_I. Let W be beginning-of-period wealth:

where M and B are the beginning-of-period stocks of money and (unit-price) bonds respectively.

Standard portfolio theory^[22] suggests the following beginning-of-period demand for capital:

where y is real income. The demand for the ownership of capital is expressed in real terms; p is a general price index. (Alternatively P_C could be used as a deflator). h(.) is a function of the rates of return on capital (r_K) and on the alternative assets (r_M, r_B), as well as on income and real wealth. The risk elements are assumed constant and imbedded in h(.). The demand for the ownership of capital is assumed to be non-decreasing in r_K; furthermore, it is reasonable to assume that h(.) is homogeneous of degree zero in interest rates, and linearly homogeneous in income and wealth.

The demands for the other assets can be expressed in a similar way.^[23] In the case of money, for instance, we have:

Naturally one would expect ∂k(.)/∂r_M ≥ 0.

The rate of return on capital (r_K) is closely related to the rental price of capital. It can be calculated as follows:

where π_I is the expected change in the price of investment goods. For simplicity, we use the actual change in the price of investment goods to proxy π_I. For given w_K and x_K, equations (8) and (10) simultaneously determine r_K and p_I, i.e. portfolio equilibrium can be viewed as determining the price of capital goods.^[24]

The full model of investment behaviour that we propose thus consists of three behavioural relationships – equations (4), (5), and (8) – and three technical relationships – equations (6). (7), and (10). Together these six equations can be used to determine p_I, w_K, r_K, y_I, x_K(+1) and W.

In view of (6) it is a dynamic system. (9) can be added to the model and used to endogenise M, r_M or r_B.

The model of equations (4)–(10) can be used to calculate the short-run and long-run effects of changes in the exogenous variables. The formal mathematical derivation of short-run and long-run multipliers in a dynamic system is rather tedious, however, and we prefer to address these issues and the question of stability with the help of dynamic simulations once that the model has been estimated. This is undertaken in Section 5 below.

Footnotes

See Diewert (1974) and Kohli (1978). [12]

Diewert (1974). [13]

The linear homogeneity of π(.) implies that π(.) = P_IY_I + PcYc = w_Kx_k + w_Nx_N [14]

Thus the GDP function is particularly useful for international trade theory; see Kohli (1978, 1983c) and Woodland (1982). [15]

The price of consumption goods would become endogenous if the model were closed by addition of a consumption function. [16]

For a discussion of non-jointness, see Kohli (1983a). Note that Foley and Sidrauski (1970) do assume non-joint production. [17]

A production model similar to the one considered here, but assuming non-jointless, has been estimated for the United States by Kohli (1981). [18]

Alternatively, one could assume that w_N is exogenous and that x_N is endogenous; see Kohli (1983b). [19]

This specification is consistent with the assumption of Harrod-neutral technological change; see Kohli (1981, 1983b). [20]

In a world of many assets (and liabilities), one could assume that capital, money, and bonds are separable from all other items. [21]

See Tobin (1958), Foley (1975). [22]

Note that all three asset demand functions must add up to beginning-of-period wealth; Foley (1975). [23]

See Sharpe (1964) for a theory of the pricing of assets. Note, however, that w_K is itself a function of p_I; see equation (5). [24]