For Economists, More Formally

Economists do know, of course, than we won’t get around the optimality condition ‘marginal productivity equals, say, lambda. But maybe, there is a totally different story behind.

And economists also intuitively know: if we have a production function with decreasing marginal productivity, and want to maximize the output of a society, for a given input, we will want to maximize the number of firms: get as close as possible to the point where marginal productivity tends to infinite.

However, we don’t define the second derivation of our production function at this point.

Maximize the Output

We maximize the output of an economy, for a given input, by maximizing the output of a firm i for all firms I = 1,…n. under the additional condition that the production technology is efficient.

Meaning, there is no production technology that allows a firm to produce the same amount of output with lesser input, or vice versa.

We’ll get the following necessary optimality conditions:

First: ‘marginal productivity equals lambda for all i’. Naturally.

Second: ‘average productivity equals lambda, or my, for all i’.

Then let us call the integrated optimality condition ‘marginal productivity equals average productivity equals lambda for all i’.

Cost Minimi-zation

And now we remember from cost minimization, that this inte-grated optimality condition (first order) is fulfilled for the minimum of the average cost function, if, and only if the cost function is the inverse of an s-shaped production function. Meaning: cost minimization is not confirming profit maximization, since the first one presupposes an negative u-shaped production function, and the latter an s-shaped production function.

We are not so much interested in any specific value of the marginal productivity per se. We are interested in the marginal productivity because the moment the marginal productivity coincides with the average productivity will be the moment of a maximum, or minimum of the average productivity, or costs.

Second Order Conditions

What, of course, has to be checked, using the second deriva-tion of our production function.

For globally in- or decreasing marginal productivity of our production function we get boundary solutions.

For globally decreasing marginal productivity we get the optimality condition ‘number of firms go to infinity, or a natural boundary that has to be defined’.

For globally increasing marginal productivity we get the optimality condition ‘number of firms go to one, meaning monopoly’.

Marginal Equals Average Productivity

And for any s-curved production function follows maximization of average productivity as optimal.

In words: a society that is driven by efficiency will maximize its average productivity! The average productivity, by the way, is perfectly correlated with the total productivity of a society.

Incentive Compatability

We already know from classical profit maximization that adjusting the average productivity to the marginal productivity means zero profits for firms.

We also know that firms prefer positive profits and thus this optimality condition will not be incentive compatible.

And that’s just fine. That’s why we define competition in the first place: we define the number of firms n a variable of choice. Not a constant.

Competiton

Firms, or the threat of new firms will force firms in their profit minimum. As long as there are profits, call them producers’ surpluses if you want to, there will be firms, or market participants rather, who by starting up new firms will arbitrage these profits. So there can be no profits unless competition is restricted.

Market clearing is analogous to the classical market clearing, only this time it will include the whole cake, incl. producers’ surpluses, which will be zero.