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Empirical estimation of a theoretical multi-output production model that uses multiple inputs is difficult because of the complexities of its functional form. By using proper parameterization to linearize theoretical model’s functional form, this paper develops an empirical estimation for multi-output production decision using multiple inputs in the profit maximizing firm, namely, multi-output production decision model. The model aligns with the dual approach of cost minimization and revenue maximization for the profit maximizing multi-product firm while keeping jointness in production structurally intact.

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Recently, [

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The empirical model suggested that multi-output production decision was a function of the Divisia output volume index and the relative changes in the individual input and output prices. The model was derived in such a way that the theoretical adding-up conditions held for all parameters. The restrictions like homogeneity, symmetry, input-output separability and output independence cpuld be imposed and tested statistically.

The paper was organized in the following way: Section 2 developed the multi-output production decision model for profit maximizing firm. Section 3 linearized and reformulated the model empirically, and Section 4 concluded the paper.

Profit maximizing multiproduct firm implies the following multi-output production decision equation for the rth product [

where _{r} is the quantity of the rth output; p_{r} is the price of the rth output_{r} is the price of the rth output;

_{i} is the price of the ith input

pure substitution effect between the rth and sth products. Therefore, we define

scribed as the rth product marginal revenue for the ith input.

Finally, the revenue-cost ratio is

Note that when the firm is input-output separable, the multi-output production decision model becomes independent of the input price changes. This restriction implies that the individual input price indexes are the same

for each output. Hence, the Frisch input price indexes are equal to each other,

This section simplifies Equation (1) to a linear form. We can show Equation (1) as the three-term summation:

When we decompose output price terms into two terms using the above definition,

Collecting under output price term, the two terms can be written as

where

Then, we rewrite the last term of Equation (2) as,

By using

We substitute

By distributive property, we rewrite expression (7) as

Substituting simplified terms into expression (8) yields

Define

The properties of the parameters are demonstrated in Appendix B. In summary, all parameters automatically hold the adding-up condition,

output price parameters

We parameterize Equation (10) by assuming

To estimate parameter, we calculate arithmetic means of output shares,

The model naturally maintains the adding-up conditions. Symmetry conditions on

After dropping one equation we estimate the remaining m-1 equations simultaneously [

where input prices disappear from the linear form. Further output separability restriction simplifies Equation (12) to

Both restrictions can be tested with an LRT.

This study makes it possible to empirically estimate the decision model that optimizes the production process with multiple inputs being used across multiple outputs. The model aligns with the dual approach of cost minimization and revenue maximization for the profit maximizing multi-product firm while keeping jointness in production structurally intact.

We reformulate multi-output production decision model by using proper parameterization to linearize theoretical model’s functional form. Homogeneity property for the individual input price parameter for each output is formally proven, which is never done before.

EkaterinaVorotnikova,SerhatAsci, (2015) An Empirical Multi-Output Production Decision Model for the Profit Maximizing Multiproduct Firm. Theoretical Economics Letters,05,555-560. doi: 10.4236/tel.2015.54065

The expression

Using

In explicit form,

Further rearrangement and simplification, it yields

and we can write this simplified expression as

Define

and hence,

Lastly, symmetry holds for m × m matrix, where

Define

Individually summing over r and i, respectively, yields

and

for the parameter with m × n dimension.

Define

and hence,

Lastly, symmetry does not hold for m × n matrix.