## Competitive Strategy: How much should a small firm produce?

###### (by Hon Sing Lee, April 24 2003)

In neoclassical economics, every firm is a price taker.  That is, none of the firms can affect the price regardless how many goods they bring to the market.  In an economy with a big consumer market, and very small firms, that is each firm can only produce a relatively small amount of goods, this would be true.  Suppose a particular small firm wants to maximize profit.  Then how many goods should it bring to the market?  If the firm has no cost constrains, then the answer would be infinitely many!  Now suppose the firm's production cost is proportional to the number of goods, then the answer will depend on the unit cost.  If the unit cost is lower than the price, then the answer would again be infinitely many; if the unit cost is higher than the price, then the answer would be zero.

Now suppose the cost function is convex, that is, the unit cost gets more and more expensive as the total number of goods produced increases.  For example the price for bringing 100 goods to market costs \$1 each, 200 goods to market costs \$2 each, and 300 goods to market costs \$5 each.  The unit cost gets more expensive because the machinery maintenance and labor gets more expensive as it produces near to or beyond its production capacity.  Eventually there would be a quantity where it is impossible to bring to market, perhaps bringing 1 million goods to market costs \$1 million each.  How many goods then should the small firm bring to the market?

One obvious suggestion might be that the small firm brings exactly that number of goods, such that the unit cost would equal the price.  That is, if the price is \$5, then in our previous example the small firm should bring 300 goods.  However on deeper thought, this would turn out to be unfavorable, because the small firm would then earn nothing.  To earn a profit, the small firm should bring somehow less number of goods to the market.  How much less should the firm bring?  In the extreme where the firm only bring 1 good to the market, then no doubt the firm would earn a profit, but the firm may not be earning the maximum that it can possibly earn.

### A Calculus Analysis

Hence we use some Mathematics to help us.  Let the price be p, and the quantity the firm brings to the market be q.  Let C(q) be the cost function, thus as mentioned above, C(q) is convex.  Now the firm profit is
Y(q) = p*q - C(q).
The firm wants to maximize this profit.  Hence we do what most Mathematical methods would do, perform calculus on the formula and find its maximum.  Differentiating by q, we have
Y'(q) = p - C'(q).
The first order conditions thus give
p = C'(q).
Hence calculus gives us a suggestion to the solution.  It suggests that we should bring q many goods to the market where q is such that the marginal cost of production exactly equals the market price.  Let us consider what this means intuitively.

The marginal cost is the increment in cost to produce the next good.  For example say it costs \$100 to produce 100 goods.  If it costs \$101.01 to produce 101 goods, then the marginal cost at the 100th good is \$1.01.  If it costs \$102.03 to produce 102 goods, then the marginal cost at the 101st good is \$1.02.  Since the production cost function is convex, the marginal cost will always be increasing.  The calculus solution suggests that the firm should bring exactly the quantity q whose marginal cost is equal to the market price p.  Why does this maximize the firm's profit?

The criterion maximizes the firm's profit in the following way.  Suppose the price is \$2 and the quantity whose marginal cost exactly equals \$2 is 140.  That means the production of the 141st good would cost \$2.  Hence this 141st good would earn zero profit even if sold in the market.  Therefore this 141st good need not be brought to market.  If the firm produces more than 140 goods, then all those goods in excess of 141, would earn negative profit, that is these goods will incur loss when sold at the market at \$2 each.  Therefore the small firm should definitely not bring more than 140 goods to the market.  Should the firm bring less than 140 goods?  Since the production cost function is convex, we know that the marginal cost of each good before and at the 140 quantity is less than \$2 each.  Hence the trade of each one would earn a positive profit.  Hence the firm should attempt to sell every one of these goods.  Therefore the firm should not bring less than 140 goods to the market.  Thus the calculus solution is verified to be indeed the optimal quantity to earn maximum profit.

The above thought experiment teaches us many things.  First, we must consider the cost of production when deciding on how much quantity to bring to the market.  Otherwise, we would not be able to optimize our profit.  Second, the conventional concept of 'unit cost' is only useful when the production cost is proportional with production quantity.  That is, the marginal cost of goods do not increase or decrease no matter how many goods is brought to market.  In this case, we have the simplistic solution that if the price is favorable, we would sell infinitely many, else we do not sell any at all.  Third, when the marginal cost of goods do change with the number of goods brought to market, then the marginal cost becomes our main decision factor.  In the real world, it is intuitively agreeable that the production cost function is convex.

For example suppose I have a pencil factory and I would like to decide on how many pencils to produce.  I assume that all pencils produced will be brought to the market, and the market would pay me a fixed price for all my pencils.  The factory has a human operated machine, which produces pencils at a constant rate.  As I increase the number of pencils produced, the machine would need more frequent repairs and maintenance, then the production workers need higher compensation pay.  Thus the production cost is arguably convex.  Therefore by the calculus solution above, I would do a study on the marginal production cost of each additional pencil, and would produce exactly the number that matches the market price of pencil.  This quantity is not directly dependent on the standard capacity of my factory's production.  For example I may find that it is more optimal for me to under work each day, doing 7 hours days instead of the standard 8 hours days. I may just as likely find that it is more optimal for me to over work each day, doing 20 hours days instead of the standard 8 hours days.  Thus only through the study of marginal cost, can I realize whether my present production is at its optimal.

### Implementation Issues and How to Overcome Them

Having said the above, now I shall discuss some difficulties in implementing the above optimality.  First, marginal cost is extremely difficult to estimate.  The explicit factors such as raw material cost, labor cost, machine running cost and other fixed costs are easy.  The implicit factors however are more difficult to estimate.  For example we cannot measure exactly the increase in machine wear and tear as we increase the production quantity.  Thus we cannot establish a probability of machine damage and its repair cost.  Hence the marginal cost which we base our decision on, would never be as precise as in the calculus problem.  In fact, we may end up implementing a less optimal quantity due to our error in estimating the marginal cost.  Second, the marginal cost of production may change with time.  For example the normal wear and tear of the production machine may increase the marginal cost of production over time.  The increase in workers salary also increases the marginal cost.  Conversely innovations in the production process and the gain in skill and experience of the workers would decrease the marginal cost.  Hence as the marginal cost changes, the firm would often be producing at a sub-optimal level.

Finally I would like to make a suggestion on how to optimize in view of such difficulty in marginal cost estimation.  I propose that the firm should start with a rough estimation and evolve its production decision as outcomes are observed.  For example if the annual review of the production cost is higher than expected, then the firm should likely lower its production.  If the production machines are wearing down, the firm should also lower its production.  Conversely if there are new cost cutting innovations in the production process, or if the financing cost or tax are lowered, then the firm should increase its production.  Thus a lot of sound business judgement is needed to supplement the inaccuracy of marginal cost estimation.  The inaccuracies in estimation do not make the marginal cost concept irrelevant in the business decision.  Instead it helps us identify factors that the firm should pay attention to when making optimal production decisions.