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Synopsis – A budget is an operating tool by which Firms organise their resources and plan their actions for a following market date with the objectives of expanding their Telecommunication plant and getting economical return. The program is built up through the co-operation of different sectors of the Firm and takes a long work based upon detailed statistics, consumption forecast, market analysis. It would be, then, of great help if the management, that co-ordinates the budget preparatory activity, might rely on a simplified model, based upon the fundamental variables of the production process, to check or refine its perception of the business. The objective of present paper is just to introduce a simplified scheme of telecommunication production process by which two fundamental input resources, X₁ (lines) and X₂ (personnel) are jointly used to produce a given quantity (Q) of commodity. The activity is described by a set of three equations which account for quantity to produce (production function), for the definition of objectives (path function) and for cost involved (budget available). By this approach the Service Provider not only may verify the results of past strategies adopted but, as well, may explore alternative strategies to support his own personal judgement.

1. The general model
2. The production process
2.1 - The production function
2.2 - The cost function
2.3 - The path function
3. Optimising Producer’s behaviour
3.1- Cost minimisation
3.2 - Profit maximisation
4. The main indicators of production process
4.1 - Average and marginal products
4.2 - The elasticity of production factors
5. The productivity of input variables
5.1 – Measuring the productivity of labour
5.2 – Impact of technology upon labour recruitment
6. Experiencing the model of production process
6.1 - The cost function
6.2 – The production function
6.3 - The input variables
7. Applications: the productivity approach
8. Application: the path approach
9. Conclusions

1. The general model
From a very general point of view, the process developed by a Telecommunication Firm, when growing up its business, may be seen as dependent by two main activities: a technical one and an operating one.

The technical activity concerns the management and the maintenance of plant necessary to provide the service and to expand the system facing the market expressed needs. The plant expands by increasing the number of lines (X₁) operated to provide greater traffic capacity, under the condition that network infrastructures are, in the same time, adapted to the new expected consumption. The number of lines (X₁) is a variable proportionate to the traffic capacity of plant and marks its growth: as such, it is the suitable reference indicator to measure the technical aspect of production process.

The personnel, engaged in a Telecommunication Firm, cover many functions which are directly (planning, maintenance i.e.) and indirectly (marketing, accounting, management) concerned with the running of the system. Labour is engaged either to keep full availability of network and to provide adequate operation of the system: it cannot vary suddenly because of political, social and economical constraints which the Firm has to satisfy. The number of employees (X₂) is taken as the appropriate reference indicator to measure operating activity.

By the scheme above, we assume that the Provider uses two main external resources X₁ (lines) and X₂ (personnel) to produce a given quantity (Q) of commodity. The matching of market demand depends upon the size (X₁) of plant provided, but the availability of network (removal of faults) and the volume of consumption (marketing) depend upon proper and adequate operation (X₂). The two resources are used jointly to produce the output Q: by no mean labour can replace physical network capacity or viceversa.
The variables are, then, complement not substitute.

2. The production process
The activity of the Provider, when performing the production process, can be described by three different functions. The “production function” which links the two input resources, X₁ and X₂ , to the quantity Q provided; the “cost function” which gives the cost of plant as a function of the two input variables; and the “path function” which accounts with the expansion strategies.

2.1 - The production function

The production function states the quantity of traffic capacity Q produced as a function of input variables, X₁ and X₂, used:

Q = f(X₁;X₂) (1)

Function (1) is assumed to be continuous and differentiable for nonnegative values of the input and output levels. In the X₁X₂ quadrant the production function is concave and represents all possible combinations of X₁ and X₂ that yield the output level Q₀. Over the same production curve (isoquant), the increase of one factor (X₁ for example), would correspond to an equivalent decrease of the other factor (X₂ for example): such a move in the variables does not alter the final product “Q₀”. The rate of substitution of the two variables is the ratio of partial derivatives of Q with respect to X₁ and X₂:

f₁/f₂ = dX₂/dX₁ (2)

2.2 - The cost function

The resources X₁ and X₂ are purchased in the market at unit prices that, in the short run, can be considered constant. The total cost of production is given by the linear equation:

C = R₁X₁ +R₂X₂ (3)

where R₁ and R₂ are the costs of X₁ and X₂ and include the cost of fixed input.

In the reference plane X₁X₂, the isocost line is the locus of all input combinations that may be purchased for a specific total cost C₀ (budget):

X₁ = C₀/R₁ – R₂X₂/R₁

The slopes of the isocost line equals the negative of the input prices ratio. The intercept of an isocost line on the X₁ axis, C₀/R₁, is the amount of X₁ that could be purchased if the entire budget were expended upon X₁ and the intercept on the X₂ axis is the amount of X₂ that could be purchased if entire budget were expended on X₂.

2.3 - The path function

The point of tangency between an isoquant and the relevant isocost line is the solution of the associated production problem: it provides the optimum combination (X₀₁ and X₀₂) of input variables necessary to produce the output quantity Q₀. The locus of tangency points is the path function and represents the planning choices available to the provider. The explicit curve is derived with reference to (2) and (3) above, that is:

f₁/f₂ = dX₂/dX₁
dC/dX₁ = R₁ ; dC/dX₂ = R₂

dCdX₂/dX₁dC = dX₂/dX₁ = R₁/R₂ (4)

In the reference quadrant X₁, X₂, the path function is a line passing by the origin of the axis and may have constant (straight line) or variable (curve) slope. The (4) states that the optimum ratio of input variables is proportionate to the inverse ratio of their cost.

3. Optimising Producer’s behaviour

The Service Producer is engaged with the choice of strategies as to optimise final outputs provided by his activity. In the following we consider two main objectives:

1. minimisation of cost (budget C₀) subject to a given output (Q₀);
   
2. maximisation of profit “P”
   

3.1- Cost minimisation

The minimisation of cost, C = R₁X₁+ R₂X₂ subject to the production of a given amount of output Q = f(X₁;X₂), is expressed by the following function:

V = R₁X₁+ R₂X₂ + _ (Q - f(X₁;X₂))

To find the minimum, set the partial derivatives with respect to X₁, X₂, _ equal to zero:

dV/dX₁ = R₁ – _f₁ = 0
dV/dX₂ = R₂ – _f₂ = 0
dV/d_ = Q - f(X₁;X₂) = 0

that gives:

f₁/f₂ = dX₂/dX₁ = R₁/R₂ (5)
_dQ = R₁dX₁ = R₂dX₂ (6)

Condition (5) defines the path function and states that, when the objective of the Firm is the production of a given output, the available budget is minimised when the combination of input variables is proportionate to the inverse ratio of their own cost. In its inverse form dX₁/dX₂ = R₂/R₁ the (5) gives the operating efficiency in terms of lines per employee: it might serve as a reference indicator for variables apportionment.

Condition (6) defines the multiplier “_” as the unit price that lets produce the additional output that balances every additional money unit spent upon each variable. On practical ground, if profit is the main economical objective, the following inequalities hold:

pdQ > R₁dX₁; pdQ > R₂dX₂

with “_” replaced by “p”, price of commodity.

3.2 - Profit maximisation

If the Service Provider wants to maximise its profit he compares total revenue (number of units sold, Q, multiplied by the unit selling price p), to the total cost (annual cost of plant and labour) necessary to produce Q units of service. The function to maximise is:

P = p*f(X₁;X₂) - R₁X₁ - R₂X₂

Setting the partial derivatives of P with respect to X₁ and X₂ equal to zero, gives:

dP/dX₁= pdQ – R₁dX₁= 0 (7)
dP/dX₂= pdQ – R₂dX₂= 0 (8)

The terms “pdQ” under conditions (7) and (8) are the product of price by partial derivative of the production function with respect to input variables: they represent the marginal product, MP, of input variables. The marginal product is the rate at which the Service Producer increases its revenue by increasing input variables: profit increases as long as the addition to revenue relevant to an additional unit of X₁ (X₂) exceeds its cost. The input variables can be used up to a point where the value of their MP equal their price.
Finally, as the ratio (7) to (8) takes back to the (5), it can be assumed that the maximum profit-input combination lies upon the path function.

4. The main indicators of production process

The growth of telecommunication business depends upon harmonic growth of labour, of capital invested and of revenue. If the three rates were equal the system would be perfectly balanced: when, in fact, no external factor affects the system, the ratios revenue/labour and revenue/capital would be constant over time.

4.1 - Average and marginal products

As total product of X₁ is intended the quantity of Q that can be secured from the input of X₁ if X₂ is assigned a fixed value X₂₀ and is treated as a parameter .
The average product (AP) for X₁ is total product divided by X₁:

AP₁ = Q/ X₁ = f(X₁;X₂₀)/X₁

The marginal product (MP) of X₁ is the rate of change of total product with respect to variations in the quantity of X₁:

MP₁ = dQ/dX₁ = f₁(X₁;X₂₀)

The same applies to X₂ factor. Its AP and MP are, respectively:

AP₂ = Q/X₂ = f(X₁₀;X₂)/X₂
MP₂ = dQ/dX₂ = f₂(X₁₀;X₂)

4.2 - The elasticity of production factors

Again, a possible tool to decide the appropriate apportionment of X₁ and X₂ is the estimate of the proportionate change of Q with respect to X₁ and X₂ (elasticity).

W₁ = X₁dQ/QdX₁ = MP₁/AP₁ (9)
W₂ = X₂dQ/QdX₂ = MP₂/AP₂ (10)

The sum of elasticity for X₁ and X₂ equals the degree of homogeneity of the production function f(X₁,X₂). From Euler’s theorem the following condition holds:

X₁f₁ + X₂f₂ = Q

where f₁, f₂ are the derivative of f(X₁; X₂) with respect to X₁, X₂. Dividing by Q:

W₁ + W₂ = 1

In case of homogeneity of degree one, the elasticity account, as well, for the weights of the distribution of total output between capital and labour:

Q = W₁Q + W₂Q

5. The productivity of input variables

The productivity of a variable (lines; labour) is defined as the quantity of revenue that it can provide: a first approach to measure the productivity of a variable is the ratio between total revenue and the value of variable (gross productivity = pQ/X₁ = pQ/X₂). The indicator, nevertheless, does not account for the fact that an input resource can produce output only when it is used jointly with other resources. A better approach is, then, given by the ratio between revenue, assigned to a factor (lines, labour), and the value of that factor (net productivity = (pQ-hR₂X₂)/X₁ = (pQ-kR₁X₁)/X₂).
Given the difficulty of separating revenue as a function of a single variable, we may assume, for sake of simplification, that it is legitimate to regard labour as the only factor producing revenue. In this way the difference between total revenue and capital expenses is assumed as the net revenue attributable to labour.

5.1 – Measuring the productivity of labour

The productivity of labour is measured by the following:

% productivity = (R-S)/S

where “R” is the net revenue per employee and “S” is his average salary.
As a function of labour (X₂) productivity is assumed to increase, firstly, up to a maximum and, then, to decline. Such an expectation is based upon the fact that consumption can increase as long as the operated plant is far from congestion; but, once the maximum capacity of network is engaged, additional traffic cannot be routed irrespective of growing of personnel. Consumption and, consequently, revenue do not increase any further: revenue per employee decreases. The minimum (zero productivity) is reached when revenue/employee equals salary/employee.

5.2 – Impact of technology upon labour recruitment

It is important to remark that the apportionment of labour is neither a pure nor an easy mathematical choice. Personnel has to cover different sectors in technical and administrative Provider’s structure and is engaged to work under current production methods. But, when technical progress (greater capacity per line, digital transmission systems) lets increase revenue without requiring proportionate increase in labour, then existing number of employees may turn redundant. If new technology allows for a better management of commercial and technical process, the ratio X₁/X₂ can be increased.

6. Experiencing the model of production process

For practical applications the model developed so far should be made easier to handle. In particular, the production function Q = f(X₁;X₂) must be made more explicit: the simplification arrived at is the following.

6.1 - The cost function

The cost may be expressed as a function of R₁, R₂, Q: the mathematical process is shown in Appendix 1. In case of a + b =1, the final function arrived at is:

C = (R₁/a)^a(R₂/b)^b(X₁)^a(X₂)^b

6.2 – The production function
Production function is derived from cost function. In Appendix 2 the mathematical process is shown. The general form of function arrived at is:

Q = A(X₁)^a(X₂)^b

which is a Cobb-Douglas function where “a” and “b” are determined once R₁X₁ and R₂X₂ are calculated or estimated.

6.3 - The input variables

The input demand functions derive from the system of equations shown in Appendix 3 which includes production function, cost function and path function. Input variables are obtained as a function of R₁, R₂, A, Q. In our case (a + b = 1) we obtain:

X₁ = _(aR₂)^b(Q/A)_/(bR₁)^b
X₂ = _(bR₁)^a(Q/A)_/(aR₂)^a

7. Applications: the productivity approach

The reference to productivity of labour represents one possible mean to estimate the value of input variables X₁ and X₂ . Assume that, at a given year, the Provider has to expand its existing plant to match the consumption expected in the next market date. The preliminary items to serve as reference, when preparing the provisional budget, are:

X₁ = the wanted size of plant
t₀ = minutes per line and per year (final data of previous year)
Q = total consumption expected
a₀ = exponent of production function (final data of previous year)
R₁ = the unit capital cost
R₂ = the salary per employee
P₀ = labour productivity (final data of previous year)

Data listed above are supposed to be available. They are necessary and sufficient to estimate remaining items that serve to complete the provisional budget:

X₂ = the number of employees
a,b = exponents in the production function
A = the constant “A” in the production function
c = expected unit cost
p = expected unit price
pQ = total expected revenue
P = expected labour productivity

The most delicate item to calculate is the number of employees. From theoretical point of view, the choice is constrained by two indicators: the employment level and the labour productivity (lines/employee) that the Provider has to satisfy. The budget assumes that both indicators will not be lower than the values recorded in previous year.

Once labour is estimated, the net labour productivity is plotted as a function of labour (variable X₂) within a restrict domain which includes the maximum of the function. The approach of actual net productivity by a theoretical curve better describes the indicator and lets adjust the previous estimate of X₂.

Before releasing the provisional budget, the process above should be repeated, either to investigate alternative solutions and to verify that production function, budget available and path function develop in a sound way.

8. Application: the path approach

An alternative way of estimate the value of input variables X₁ and X₂ is to derive both as the point of tangency between the isoquant and the isocost curves relevant to the year under study. Since:

C₀ = R₁X₁/a = R₂X₂/b

The wanted point of tangency is calculated from the following equalities:

(Q/A)^1/a/( X₂)^b/a = C₀/ R₁ – R₂ X₂/R₁ = aC₀/R₁
(Q/A)^1/b/( X₁)^a/b = C₀/ R₂ – R₁ X₁/R₂ = bC₀/R₂

That gives the variables X₁ and X₂ :

X₁ = (Q/A)^1/a(R₂/bC₀)^b/a
X₂ = (Q/A)^1/b(R₁/aC₀)^a/b

The concept upon which this method is based is to compare product “Q” to total annual cost “C₀” (capital + operating): both variables X₁ and X₂ are derived under the condition that production curve and cost line have a common point. The solution only accounts for the optimum use of the budget (C₀) and for the theoretical optimisation of the ratio between the apportioned variables. Further investigations are necessary to verify that the input resources, so calculated, well fit into the Provider’s plan. In particular, X₁ should be consistent with the plant expansion to match the expected consumption and X₂ should meet the socio-economic constraints that the Firm must satisfy.

9. Conclusions

A provisional budget involves strategies to be defined, under uncertainty, for future market date. When the final objective is the profit, the decisions to be taken concern, mainly, the forecast of revenue and the estimate of costs. That is: the Provider fixes the price “p” consistent with the expected consumption “Q”; and tries to keep at the possible lowest level technical (plant) and operating (personnel) expenses.

The model presented in this paper can be used on a case-by-case basis as it should be supported with a number of information only available to the Firm. When designing the production function and its parameters, attention should be paid to basic data such as: plant fault rate, structure of personnel, consumption forecast, load per circuit, budget available, objectives, regulation environment.

itc/production 27/2/2002