The contributions of productivity, price changes and firm size to profitability

Abstract

Sources of profit change for Telstra, Australia’s largest telecommunications firm, are examined. A new method allows for changes, in a firm’s profits to be broken down into separate effects due to productivity change, price changes, and growth in the firm’s size. This in turn allows us to calculate the distribution of the benefits of productivity improvements between consumers, labor, and shareholders. The results show that around half the benefits from Telstra’s productivity improvements from 1984 to 1994 were passed on to consumers in the form of real price reductions.

Appendix

Here we provide a justification from economic theory for the profit decomposition method that was presented in the methodology section. Note that this is an optional justification, in that the method was justified above solely from accounting algebra and the axiomatic approach to index numbers.

In using an economic justification, an important issue that needs to be considered is the exogeneity of prices. For infrastructure firms, the regulator typically sets output prices so that price taking behavior is an acceptable approximation to what would prevail under deregulation. That is, whether the prices are determined by a perfectly competitive market or by a regulator, the effect is the same in that prices are taken as exogenously given by the firm(s). For inputs, we assume that the prices are exogenously determined in a competitive market. Also, the quantity of capital available to the firm at time t, k ^t, is taken as exogenously given (“quasi-fixed”) in every period.

Consider a general representation of a restricted profit function for a firm, π^t, as follows:

$$ \pi (p^t,k^t, t) = \max_{y^t} \left\{ p^t \cdot y^t: (y^t,k^t,t) \in S^t \right\}, $$

(12)

where S ^t is the production possibility set for the firm. Hence, profit is maximized by the choice of y ^t, subject to the constraint that k ^t is exogenously given in each period (Samuelson 1953–4; Gorman 1968). The conditions which define a restricted profit function with constant returns to scale are that it is (1) a nonnegative function, (2) positive homogeneous of degree one in p ^t, (3) convex and continuous in p ^t for every fixed k ^t, (4) positive homogeneous of degree one in k ^t, (5) nondecreasing in k ^t for every fixed p ^t, and (6) concave and continuous in k ^t for every fixed p ^t.

We consider the case where the log of π(·) in (12) has the translog form (Christensen et al. 1973; Diewert 1974; Russell and Boyce 1974), such that for each period t

$$ \ln \pi(p^t,k^t,t) \equiv \alpha^{t}_0 + \sum^N_{i=1} \alpha^{t}_i \ln p_i +\frac{1}{2} \sum^N_{i=1} \sum^N_{j=1} \alpha_{ij} \ln p^t_i \ln p^t_j + \ln k^t, $$

(13)

where α _ij  =  α _ji , for i,j  =  1,... ,N, and the following restrictions hold so that the functional form in (13) exhibits constant returns to scale: ∑ α^t _i  = 1 and ∑ α _ij  = 0. Note that only the “second-order” coefficients in (13) are restricted to be constant across time. This translog profit function is “flexible” in the sense that it can approximate an arbitrary, twice continuously differentiable function to the second order (Diewert 1974, p113).

Diewert and Morrison (1986) exploited the translog identity of Caves et al. (1982) to prove a relationship between the translog functional form and the Törnqvist index formula and which they use for decomposing the growth in domestic product for a trading economy. In the current context we have the following theorem.

Theorem 1

If the functional form for a firm’s profit function, π, is translog as defined by (13) in periodst − 1 andt, firms are price takers and there is profit maximising behavior in both periods, then a theoretical productivity index of the form.

$$ R^t \equiv \left[\frac{\pi(p^{t-1}, k^{t-1}, t)}{\pi(p^{t-1}, k^{t-1}, t-1)} \frac{\pi(p^{t}, k^{t}, t)}{\pi(p^{t}, k^{t}, t-1)} \right]^{1/2} $$

(14)

can be written as

$$ R^t = \frac{G^t / P^t}{K^t} $$

as in Eq. (2), whereP^tandK^thave the Törnqvist form of Eqs. (4) and (9), respectively.

Note that first ratio in the brackets of the theoretical productivity index in (14) is an index of productivity difference using period t − 1 reference netput prices and capital quantities, and the second ratio is a competing index of productivity change which uses t reference netput prices and input quantities. In each case, the only thing changing in going from the denominator to the numerator is technology. In this way, technological change (which is equal to “productivity change” in this context) is captured by both ratios. Because it is unclear which of these two possible theoretical indexes is preferred, a geometric mean of the two is used in (14).

Proof of the Theorem If producers are price-taking profit maximisers, then from Hotelling’s Lemma,

$$ y^{t} = \bigtriangledown_p \pi^t(p^t,k^t,t) $$

(15)

using vector notation, where ▽ _p denotes the vector of first order derivatives with respect to each element of the price vector p^t. Then,

$$ \begin{aligned}R^{t} & = \left[\frac{\pi(p^{t-1}, k^{t-1}, t)/k^{t-1}}{\pi(p^{t-1}, k^{t-1}, t-1)/k^{t-1}} \frac{\pi(p^{t}, k^{t}, t)/k^t}{\pi(p^{t}, k^{t}, t-1)/k^t} \right]^{1/2}\\ & = \frac{\pi(p^t,k^t,t)/k^t}{\pi^{t-1}(p^{t-1},k^{t-1},t-1)/k^{t-1})} \left[\frac{\pi(p^{t-1}, k^{t-1}, t)/k^{t-1}}{\pi(p^{t}, k^{t}, t-1)/k^{t}} \frac{\pi(p^{t-1}, k^{t-1}, t-1)/k^{t-1}}{\pi(p^{t}, k^{t}, t-1)/k^t} \right] ^{1/2}\\ & = \frac{p^t \cdot y^t}{p^{t-1} \cdot y^{t-1}} \exp \left\{ \frac{1}{2} \left[\bigtriangledown_{\ln p} \ln \frac{\pi(p^t,k^t,t)}{k^t} +\bigtriangledown_{\ln p} \ln \frac{\pi(p^{t-1},k^{t-1},t-1)}{k^{t-1}} \right] \ln \left(\frac{p^{t-1}}{p^t}\right)\right\}\frac{k^{t-1}}{k^t},\end{aligned} $$

(16)

where we have used the translog identity, π(p^t,k^t,t)  = p^t·y^t, and the notation p^t·y^t  =  ∑ p^t _i y^t _i . Using (15) to re-express this last line of (16) yields

$$ R^{t} = \frac{p^t \cdot y^t}{p^{t-1} \cdot y^{t-1}} \exp \left\{ \frac{1}{2} \sum_{i=1}^N \left[\frac{(p_i^t y_i^t)/k^t} {p^t \cdot y^t/k^t} + \frac{\left(p_i^{t-1} y_i^{t-1}\right)/k^{t-1}} {p^{t-1} \cdot y^{t-1}/k^{t-1}} \right] \ln \left(\frac{p_i^{t-1}}{p_i^t}\right)\right\}\frac{k^{t-1}}{k^t}, $$

which can easily be simplified to prove the theorem.

As shown by Diewert and Morrison (1986), similar theorems can be proven to provide a theoretical link between price and quantity and economic theory. In fact, every sub-index of Eqs. (6) and (7) can be similarly derived from theoretical price and quantity indexes which are functions of profit functions. This provides a firm micro-theoretic foundation for every component of the decomposition of profit growth represented by Eqs. (3), (6), and (7).