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by Valentino Piana (2006)




This free software to draw isoquants and isocosts will allow you to understand the mechanics of a key building block of the neoclassical theory of production as taught in ECON 101 first year classes. The software computes the optimal choice of the firm that minimizes costs, given a certain target level of production, responsive to input prices. Students will find diagrams and numerical solutions to exercises related to Cobb-Douglas isoquants, as well as for the cases of perfect substitutes and fixed proportions.

This theory is, however, fundamentally false, as this short introduction will try to explain, after some advice about how to use the software.

1. Definitions

An isoquant is defined as the set of all possible bundles of productive inputs exactly sufficient to produce a given quantity of output.

Productive inputs are the necessary ingredients for obtaining a good (as milk and coffee to obtain milk&coffee) as well as the work and the machines involved.

Isoquants are easy to draw in diagrams when productive inputs are just two, whereas when there is only one productive input the production function we describe here is more relevant.

The isocost curve is defined as the set of all possible bundles of productive inputs whose cost is the same. Depending on the price of inputs (say wages for labour), the slope of the isocost curve will be steeper or flatter.

2. How to use this software

The software considers the case of 2 productive inputs, whose quantities are represented as axes in a bi-dimensional Cartesian space.

Productive inputs diagram


A point in the space is the bundle of the quantity of the 2 inputs used in production.

One combination of two productive inputs

The theory goes as every point in the diagram were technically feasible, so that for each point there is a production level of the output. Every point has also a cost (the product of the prices of the input for the respective quantities).

The curve made up of all points corresponding to the same level of production is the isoquant (just click on Draw button to see an example).

Isoquant as the set of all combination of inputs to obtain a given quantity of output

In the boxes you see while using the software, you can modify the target level of production, click again the Draw button and obtain the graph of the new isoquant.

The optimal choice is the intersection between the lowest possible isocost curve with the isoquant corresponding to the target production level[1].

This theoretical sequence perfectly mirrors the neoclassical theory of consumer.


3. The objections about quality invariance

In all this, the key assumption is that it is possible to obtain exactly the same output by several (indeed infinite) combinations of productive inputs. This is false.

Productive inputs can be of four main kinds:

1. the physical ingredients (i.e. raw material) that will constitute the good;

2. the components to be assembled that will constitute the good;

3. labour and capital used during the production process;

4. ingredients that are used in the process but that do not remain in the final good.

3.1. The case of ingredients

Let's take the example of the production of a cup of milk and coffee at home.

Production ingredients - why isoquants fail

You mix the milk and the coffee and you get it. If you change the proportion of the two ingredients, you get a different taste because one made up of 90% milk + 10% coffee is not at all the same as another which is made by 10% milk + 90% coffee. It is not the same homogeneous product: they are two variants.

In other words, significant changes in the proportions of ingredients results in product differentiation, which is one major reason for product differentiation being the rule and not the exception, which in turn implies the ubiquitous phenomenon of imperfect competition against the much more limited situation of perfect competition.

To repeat the basic fact: isoquants as continuous curves have infinite points but each point would produce a qualitatively different good, even if the quantity (e.g. measured in grams) remains the same.

If you consider sweet milk&coffee, you might easily figure out that different quantities of sugar results in widely different output. This is not, however, the only objection to the isoquant concept usefulness.

Since the quantity has to remain the same, every gram of further sugar is compensated by one gram less of (say) milk. This means that the case is one of perfect substitution (isoquants as straight lines, as you see in the software while choosing this option).

With perfect substitution the optimal choice lays always on the extreme (left or right) of the isocost, as you can see clicking the Draw button. Experiment with different prices of inputs and you shall see that the solution is always there, jumping from one extreme to the other depending on which is the cheapest input.

The cost minimization procedure selects one point on the axis of the cheapest input: 100% milk if one gram of milk is cheaper than one gram of coffee and than one gram of sugar. Thus the optimal milk&coffee production is... just milk!

The optimal choice is a wrong choice: consumers are not stupid and they will not pay milk as it were milk&coffee.

Please note the this example is by no means extreme: in every case a good is made up of "ingredients", the same problem appears.

3.2. The case of components

Let's take the example of the production of a car in a factory. Cars are made by assembling a huge number of components, each one having a role and being compatible with the others. The isoquants approach emphases the choice of the right "quantity" of input, whereas the true problem is the "quality" of the input. Carmakers do not have doubts about "how many engines" and "how many wheels" they insert in the car, but on "which kind of engine" and "which kind of wheels".

In other words, the demand of carmakers is one about features of differentiated goods, confirming - on the demand side - the importance of product differentiation.

3.3. The case of capital and labour

In the production of hot milk&coffee, you need a moka machine:

Capital and labour: a further example of isoquants failure

This is a piece of capital. You also need some gas kitchen like this for a while:


These machines cannot work by their own: it's a person that works with them so as to produce the coffee and the hot milk. In other words, in most production processes, the "action" is performed by labour and capital together.

Now, the isoquant approach states that it is possible to substitute capital with labour and viceversa. Can your hands substitute fire? Can they substitute the moka machine? Until a totally automatic milk&coffee machine is not employed, can you full substitute the supervisor role of a human? Would the resulting product be exactly the same?

The isoquant is a continuous function, so it implies that it is possible to reduce say 1% of labour time if capital - however measured - is increased by x%. This ignores the indivisibility in technology.You cannot increase moka machines by 1.56945%. Even if we were to interpret this as an increase in size of the machine, the idea that "the same level of output" can be obtained with slightly less labour and slightly more capital is false most of the time. It can be true in discrete jumps, not a continuous function.

In other words, handcraftmanship and automatization produce usually widely different results (e.g. artistic objects vs. mass production), generating further product differentiation and price premiums.

The isoquant approach assuming that a product made with large amounts of labour + small amount of capital is exactly the same as another one made with small amount of labour + large amount of capital is a very special case, not the general one.

3.4. Ingredients that are used in the process but that do not remain in the final good

During the production processes certain chemicals or other inputs are used but leave the process without entering the final composition of the good. For instance, in the cooking process of boiling rice, water is such an input. Any increase or decrease of water will have no impact on the quantity of boiled rice produced. So cost minimization (and productivity maximization) leads to eliminate water from the process. Needless to say, by doing so, the quantity of final rice remains the same, but its quality is completely different now (and is totally un-eatable and useless!).


Isoquants: an example of quality degradation


The rational cook is a stupid! Changes in such inputs do have impact on quality of the final product.

How differently should we have analysed this case? One should have undelined that a recipe exists that inspires a skilled cook who will perform a number of steps and cooking techniques to obtain a boiled rice matching certain quality requirements. This would be an example of production routine in the bounded rational tradition.

4. Objections about price elasticity

A key effect of the isoquant approach is that every change in the price of inputs provokes a change in input usage (a change in technology). By experimenting with the software, you'll see that any change in price of input results in a new optimal choice. In reality, this does not happen.

4.1. The case of ingredients and components

If the price of milk on the shelves of your supermaket changes (e.g. because of a promotion), do you change the composition of your milk&coffee? Is the proportion of sugar, coffee and milk reactive to small changes in prices? This would make the supply of a producer completely uncoherent over time in terms of tastes, calories, etc. On the contrary, firms strive to keep constant the quality of their products, because changes would disappoint their customers and generate a wide movement against brand loyalty.

4.2. The case of capital and labour

Any rise of the price of capital implies, according to the isoquants mechanics, a rise in labour employment. What's the price of capital is an open question, because capital has several conflicting definitions. However, mainstream economics sometimes says that the price of capital is the interest rate. If so, any change decided by the Central bank (and they happen quite often!) would imply in every factory of the country a change of machines and technology, with significant effects on employment: an increase of interest rates leads to a boom in employment. Difficult to accept this as a realistic statement: rather the opposite happens, since the rise in interest rate - if anything - rather cools down the economy with falling emploment, as you can see by comparing our datasets.

5. Your experiments

Well, we are not sure we convinced you of the fundamental flaws of the neoclassical approach. Please make your own mental and practical experiments with products and production processes you know or can observe.

Is smooth substitution of productive inputs possible? Does it lead to perfectly identical output? Can ingredients be substituted? Can capital be substituted with labour?

Ask managers: do they know isoquants? Do they care about them? Is their work mainly oriented to cost minimization obtained by substituting given inputs?

6. Consequences for policy-making

The failure of isoquants to adequately represent the economic reality weakens the exclusive reliance on price signal to push technological innovation. In the isoquant setting, any change in e.g. electricity would provoke the immediate adoption of labour-intensive capital-saving techniques. Instead, it might be possible that labour cannot substitute in any way a piece of machine, so there will be no change at all in techology used (for a wide range of small price increases).

In the particularly urgent case of spreading clean technologies to cope with climate change, our book on "Innovative Economic Policies for Climate Change Mitigation" puts forth several proposals, such as a special tax scheme.

7. Conclusions

If you firmly believe in neoclassical theory of production, fine. If you are disappointed with it and you are worried about the real world, be happy because there is an alternative: the evolutionary economics paradigm has widely discussed technology, its contribution to production, the ways technical change starts and spread out, how production is decided and, more broadly, the main decision-making processes of real managers.


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[1] A change in input price does not change the shape of isoquant but the optimal (chosen) proportion of inputs: an higher proportion of the input costing less than before and a lower proportion of the input costing more than before.


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