Statistics for business decision making + Formulario

J. PRODUCTIVITY INDICES: PARTIAL AND TOTAL

The concept of productivity has three components:

1. the result of the productive activity;

2. the inputs of the productive activity;

3. the technological progress that connects the previous two.

Productivity measures can be divided in measures of

1. Partial productivity of labour or capital, given by the relationship between the value of obtained production (in a

given interval) and the quantity of labour or capital used in the production process.

2. Total productivity or global factor productivity, given by the value of the production and the value of the factors

employed in the productive process.

Notation: We will call respectively:

1. Y the value of obtained production

2. L the value or quantity of labour employed in production

3. K the value or quantity of capital employed in production

Labour partial productivity index

=

The labour partial productivity index (LPI) is given by:

The time comparison between the present year t and a base years (generally the previous year) can be obtained by

calculating the variation of the partial productivity labour index

Capital partial productivity index

=

The partial capital productivity index (CPI) is given by:

The time comparison between the present year t and a base year s (generally the previous year) can be obtained by

calculating the variation index of capital partial productivity

-> same idea, but capital (K) at the denominator instead of Labour

Example

Suppose that a business process X produces a given product and that the distance between s and t is of 2 years.

We proceed to calculating the change in labor and capital productivity in the 2 years.

24.6

(, ) = = = 1.442

21.5

So between the year s and t production increased of 44.2%.

Then proceed to the calculation of the variation in productivity work from s to t.

140800

(, )

= = = 1.0594 -> increased by nearly 6%

132900

The variation of labour productivity from s to t is given by:

(, ) 1.442

= = 1.36

(, )

1.0594

It has therefore been, within the two years, an increase in the labor productivity by 36%.

In a similar way it can be calculated the change in the capital productivity (+ 34.7%).

For single-product company, productivity indices are expressed by relationships in which the numerator is Y

expressed in terms of the quantity of product. This allows us to express the productivity in physical terms, this

measure is particularly accurate and thus enables you to make meaningful comparisons in time or with other

companies that produce the same product.

Multi-output processes

Multi-output processes require a preliminary aggregation of the outputs with quantity complex indices (e.g.

Laysperes quantity index), before the partial productivity calculation.

The most direct aggregation method is to consider the monetary value, i.e. measure output as value of production.

∈

If we index with h the h-th output (where h {1, ..., N}).

That can be expressed also as the weighted mean of the simple indices of quantities.

Where

Empirical example

Suppose that the production process X has two outputs A and B. The variation of labour productivity is calculates as:

First we must aggregate outputs to obtain (,) =

(,)

1.0817 ℎ ⅈ ℎ ⅈ 8.17%

The change in labor input is given by

= (80 ∗ 1760)/(75 ∗ 1772) = 1.0594.

(,) (,) 1.0817

= = 1.021

The variation of labour productivity from s to t is given by: (,) 1.0594

Hence in the generic time interval t − s labour productivity has increased by 2.1%.

(,) 1.0817

= = 1.0792

Analogously the variation of capital productivity from s to t is given by: (,) 1.0706

(, )

= 9100/8500 = 1.0706

where

Hence in the reference time interval there has been an increase of capital productivity of 7.9%

Total factor productivity index

=

A generic total factor productivity index (TFP) is expressed as: )

(

∈

where is the generic production factor (input) where i {1, ..., M} and g(.) is a generic input aggregation function.

= /

By using the notation for the index of variation of input i from period s to period t, the corresponding

index of variation of global productivity can be written as:

(,)

=

(,)

(,)

where g ( ) = is a measure of the variation of the whole of inputs.

(,)

- The chosen aggregation function impacts the productivity measure.

- The aggregation function generally refers to the production function that links inputs and outputs.

- In practice a weighted arithmetic mean is used, weighted proportionally to the overall input cost at time s.

Multi-period comparison

It is important to note that in the discussion we implicitly assumed that the two sets were in two different

subsequent times.

In the case of multi-period comparisons, the main limit of the fixed-base indices is the loss of representativeness of

the weighting system, which becomes increasingly important as we move away from the base period (changes that

occur in both the quantity produced and in prices).

To overcome this loss of representativeness or the so-called attrition of the base, you can use the chain indices.

In practice we calculate the indices of change in quantity (input and output) of each year compared to the previous

and to combine them in a suitable manner to obtain the measures of variation for the entire period.

The definition of a chain index is: I(0,t)= I(0,1) I(1,2)… I(t-1,t) -> separate them in yearly variation and multiply them

Where I is any complex index number.

Usually, the approach still uses indexes as Laspeyres ones.

=

So, for example, the chain index of Laspeyres output from time 0 to 2 is given by (0,1) (1,2)

(0,2)

=

while the corresponding index of the change of the input is: (0,1) (1,2)

(0,2)

Therefore the concatenated index of the variation in total productivity in period 0-2, is:

Comparisons of productivity between companies

Some partial and total productivity indices can be used for comparisons of productivity in space, or for measure the

productivity gaps between different companies observed with reference to the same period of time. If we denote

with s and t two companies that in a given period produce the same types of output, employing the same types of

input, the previous indices provide us with measures of the differences in partial and total productivity of the

company t compared to the company s chosen as the base.

However to ensure that the comparisons are independent from the chosen base, it is needed that the used indices

have appropriate formal properties. For binary comparisons the property required is the reversibility of the bases.

Only if such property is worth comparing will be unique in the sense that, if the productivity of the t is twice that of

the company s (chosen as the base), using the same index, but taking as basis t will get a productivity index of s with

respect to t equal to 0.5. This necessity excludes the use of the Laspeyres index for binary comparisons between

companies. We have to use other indices, as the Tornqvist index, which has the property of reversibility of the bases.

In the case of multilateral comparisons it is also required that the indices satisfying, in addition that the property of

reversibility of the bases, also the transitivity of the bases. This property is not satisfied by any of the most common

index numbers, therefore it should be used a complex construction methodology that goes beyond the program of

this course.

K. EXTERNAL EFFICIENCY

With the study of the side of external efficiency, we aim to determine the correct volume of production, given the

complex expectations of the current and potential customers that are derived from the needs, experiences and

environmental influences. In particular we focus on statistical methods that can be used in those areas of activities in

which the firms are focusing their efforts to increase and/or retain the number of customers (growing in this way the

production volumes), i.e. studies of strategic marketing.

BSC, the four dimensions: equation (2) ).

= (

Formally we refer to the second of relations introduced in the balanced scorecard system;

In fact this equation, the correct volume of production ( ), is a function of the complex expectations of current and

potential customer ( ).

Statistical tools for marketing decisions strategic

Statistical methods in support of strategic marketing are all those quantitative methods that enable the company to:

- measure the size of the market on which decisions and investment will have an impact;

- identify a market segment;

- build a new positioning for a product;

- invest on the design and launch of a new Product/Service.

Quantitative methods for prediction of markets evolution

A first class of quantitative methods of analysis can be detected in those statistical methods that allow to obtain

forecasts on market size.

In particular, the quantitative methods for forecasting purposes can be divided into two main categories:

- regression methods;

- methods based on time series (not treat this argument).

Regression methods

The regression methods are based on the assumption that the variable object of the analysis (for example, the level

of sales or the decision of a customer to buy or not to buy a product), called the dependent variable can be

explained by a set of other variables (for example, the price of the product, the advertising and promotional

expenditure, investment competitors, the characteristics of the customers), called explanatory or independent

variables.

Regression methods: linear regression model

We have already discussed the theoretical point of view the topic. Here, we only underline that the dependent

variable Y is a quantitative variable, so we focus on an application on the external efficiency.

The problem

Suppose you want to open a restaurant-pizzeria on the territory of the province of Arezzo. We want to be able to

answer to the question: what would be the most suitable place to open a new business? The analysis can be divided

in five phases:

1. Study of the macro-place; 4. Comparison of the analysis previously carried out;

2. Analysis of the supply; 5. Analysis of customer’s behaviour.

3. Analysis of the demand;

1. Study of the macro-place

It means that it is necessary to define the area in which we want to act.

You choose a criteri