Profit maximising with non-constant prices per unit

Question:

Hi. I studied economics at undergraduate level and have encountered the profit maximising rule of MR = MC; with the standard premise for such calculations involving the production of goods with a fixed price per unit. I currently work in the construction sector (demolition) where each of our projects have a different cost and revenue from other projects. As such, I would like to know if/how the above measure could be applied to the service sector? My current thinking is that the data must be manipulated to achieve the cost per square foot of a site. However, this will involve significant data manipulation just to run the calculations. Could you tell me if I am on the right track with this direction or whether there is an alternative way that I could calculate the profit-maximising output? My goal in answering this question is to determine the appropriate mark-up for our business when submitting a quote to a client. At present, we are charging a 10% mark-up which I currently believe to be too low to be sustainable. Any assistance on this would be greatly appreciated.

Answer:

Hello:

Suppose you have N projects, each of them with possibly different but constant marginal costs (MC). Since the marginal cost for each project is constant, the average variable cost is also constant, and your profits can be expressed as:

Profit = (P₁ – MC₁) Q₁ + … + (P_N – MC_N) Q_N – FixedCosts,

where P_n is the price per unit of project n, Q_n is the number of units of project n, and FixedCosts are your fixed costs. To maximize profits, you need to choose the number of units for each of the projects so that marginal revenue equals marginal cost for each individual project. That is, the “first order conditions” (FOCs) for optimization are

Project 1:             P₁ + Q₁ ¶P₁/¶Q₁ – MC₁ = 0,

…

Project N:            P_N + Q_N ¶P_N/¶Q_N – MCN = 0.

Unless you face a perfectly elastic demand, the term ¶P_n/¶Q_n is a negative number, because it says that you must reduce the price per unit of project n if you want to increase the number of units sold for that project (in the case of perfect competition, ¶P_n/¶Q_n = 0 because you can sell as many units as you want at the going price).

To determine the optimal amounts for each project using a more workable rule than the above FOCs, define the project n’s elasticity of demand as e_n = –P_n/Q_n ¶P_n/¶Q_n. For example, e_n = 0.5 if increasing the price by 1% reduces the number of units sold for project n by 0.5% (and we say that demand for project n is “inelastic”); alternatively, e_n = 2 if increasing the price by 1% reduces the number of units sold for project n by 2% (and we say that demand for project n is “elastic”). Then, using this definition in the above FOCs, we can rewrite them as follows:

Project 1:             P₁ (1 – 1/e₁) – MC₁ = 0,

…

Project N:            P_N (1 – 1/e_N) – MC_N = 0.

Thus, the optimal price for project n must satisfy the condition

                P_n = e_n/(e_n – 1) MC_n.

For example, if demand for project n is elastic at e_n = 3 (so that increasing its price by 1% reduces its number of units sold by 3%), you must set price for project n at 50% above its marginal cost, because e_n/(e_n – 1) = 3/(3 – 1) = 1.5, so that P_n = 1.5 MC_n. In other words, the optimal markup for a project with e_n = 3 is 50% above marginal cost. Similarly, you should be able to verify that the optimal markup for a project m with e_m = 2 is 100%.

It is important to note that the optimal markup rule does not necessarily mean that the firm will be viable in the long run. For the firm to be viable in the long run, the sum of the markups for the individual projects must be sufficient to cover the fixed costs. In other words, if FixedCosts > [(P₁ – MC₁) Q₁ + … + (P_N – MC_N) Q_N], the firm will have negative profits and won’t be viable in the long run.