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In microeconomics, we usually 'allow' utility functions with negative partial derivatives, indicating a 'bad' commodity, such as $u(x,y)=x^2-y$. Naturally, a utility-maximising consumer with a usual budget constraint $px+qy=m$ will choose $y=0$ at the optimal. I understand these do not represent well-behaved preferences, but we still allow them in theoretical analysis.

However, we usually don't see production functions with negative marginal products, such as $f(k,l)=k-l$. Most textbooks impose the positive marginal product condition on production functions.

Why this difference?

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Actually, even if in the textbooks in most cases the marginal product is always positive, it is not unusual to have a production function with negative marginal product, that is a total product function that, beyond a certain level, decreases with an increase of the input under consideration, say labor.

This kind of production function is a short period production function, where some input is fixed.

The graphs of the production function and of the marginal product, in this case, usually look as in the following picture, where the relationship among Total, Marginal and Average product function is shown:

enter image description here

As you can see, beyond the level $L=24$, the total production decreases and the marginal product becomes negative.

As an example of a microeconomic textbook where a similar production function is illustrated, you can see for instance Frank, Microeconomcs and Behavior.$^1$ But if you search on Google there are plenty of images of production function with negative marginal product

A reason why this can happen is the usual argument of the flower pot, that is an example of scarcity of an input in the short run, that illustrates the law of diminishing returns: it is not possible to grow the world’s food supply in a flower pot.

With an input, land, fixed at a low level (the flower pot) the increase of other inputs, say labor, rapidly cease to have a positive effect on the production of wheat: the efficiency of the production will decrease quickly (decreasing marginal product) and we can imagine a situation in which the production stops to grow and a level at which too many workers on the land can damage, instead of increasing, the production.

The reason why most textbooks present the positive part of marginal product only are formal reason, as pointed out in the answer by @1muflon1, and also economic reasons: a rational manager, as the wage is positive, will never decide to employ the input, labor, beyond the level $L=24$ in the picture.

So, it is reasonable to consider the part of the production function with positive marginal product only.$^2$

$^1$ Unfortunately, I have the Italian edition of Frank's book only, so I can't cite the pages of the English edition. For the Italian edition, see Frank, Microeconomia, Fourth Edition, Mac Graw-Hill, 2006, pp. 265-266.

$^2$ See Ibid., pp. 262-263.

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  • $\begingroup$ My question is: if it is reasonable to only consider the positive MP portion of the production function, why do we allow negative marginal utility in consumer theory? $\endgroup$
    – AcadEconInd
    Commented Jan 13 at 17:41
  • $\begingroup$ Because it is equally reasonable also to allow the negative part, it reasonable to neglect it but it is also reasonable to consider it. It is a possible type of production function with reasonable caracteristic. The consideration of the positive part only can come when we consider the process of maximization of the producer, but in the theory of production is perfectly reasonable the negative part. On the contrary, it can be more 'complete' as illustrated in the flower pot example. And what I said is that it is not true that the textbooks consider the positive part only. $\endgroup$
    – BakerStreet
    Commented Jan 13 at 17:57
  • $\begingroup$ In consumer theory we can allow the existence of bads, instead of goods, so we can have negative utility, and in the theory of production we can think of a 'bad' , inefficient, use of the inputs. $\endgroup$
    – BakerStreet
    Commented Jan 13 at 18:03
  • $\begingroup$ It is ‘reasonable’ to consider only the positive part here means ‘we can, it is convenient for simplification’, it does not mean ‘it is wrong, unreasonable, to consider also the negative part’, it is pefectly justified. In the theory of the consumer too we can dispose of the negative utility of the ‘bads’, setting Good =Absence of the bad, so we have only goods, for convenience. $\endgroup$
    – BakerStreet
    Commented Jan 13 at 18:29
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Negative marginal product is allowed theoretically. In fact law of diminishing marginal returns requires for marginal product to be negative at some point.

Most textbooks work with well-behaved production functions because it makes complex problems easier to solve.

For example, if you use Cobb-Douglass function that does not have negative marginal returns, you are virtually always guaranteed that your fundamental equation will reduce to the so called Bernoulli differential equation, and that is one of the few types of differential equations we know how to solve explicitly using simple analytic methods. Proving uniqueness and stability of the equilibrium is also extremely simple.

Hence the reason why you don’t see negative marginal product in textbooks is that it’s simply too difficult for average Economics student (even at graduate level) to solve economic problems without well-behaved production function. Moreover, such problems can’t often be solved explicitly, and it can be difficult to prove the equilibrium you found is unique and stable.

This is why most student oriented literature will use very limited pool of production functions. Mostly you will see just Cobb-Douglass with few exceptions.

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  • $\begingroup$ I think he means production functions with strictly negative marginal producs for each positive level of the input considered $\endgroup$
    – Matteo Bulgarelli
    Commented Jan 13 at 15:36
  • $\begingroup$ @1muflon1 But the mathematical structure of utility and production functions is same. If we can solve consumers problem with x-y, why is it difficult to solve a producers problem with k-l? $\endgroup$
    – AcadEconInd
    Commented Jan 13 at 17:16
  • $\begingroup$ @AcadEconInd it wouldn’t be difficult but k-l production function make 0 sense. Then you just use 0 labor input so that’s not theoretically interesting or realistic. The kind of production function that makes sense is one where production increases up to some point in both inputs and then declines at some point. $\endgroup$
    – 1muflon1 ♦
    Commented Jan 13 at 17:26
  • $\begingroup$ I get that. But similarly, x-y utility function also should not make sense. Why do we still allow its possibility? $\endgroup$
    – AcadEconInd
    Commented Jan 13 at 17:43
  • $\begingroup$ @AcadEconInd utility x-y could make sense. You could argue x is good and y is bad (eg pollution), next you can have budget constraint where you pay p for acquiring x and pay q for removing y and it could still be interesting problem. I don’t believe any textbook has just simple problem where you have x-y st px + qy = m because that would also be uninteresting trivial problem as it would just reduce to max U(c) st px=m. However, when it comes to production it does not make sense to say that any amount standard factor of production (labor, capital, land) has always declining marginal product $\endgroup$
    – 1muflon1 ♦
    Commented Jan 13 at 21:45

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