Optimizing Profit

Overview


One of the goals (if not the primary or only goal) of a firm is to maximize the profits that it earns for its shareholders.

Derivation


Profit is defined to be the total revenue minus the total cost. Here, revenue is the number of products sold times the price of each product.
{% Profit(p) = p \times q(p) - Total \: Costs (q(p))%}
We will abbreviate

  • {% Profit = \pi %}
  • {% Total \, Costs = TC %}


To find the maximum level of profit, we differentiate by the price, and set the result to zero. (see finding a max/min)
{% \frac{d \pi(p)}{d p} = q + p \times \frac{dq}{dp} - \frac{d TC (q(p))}{dq} \times \frac{dq}{dp} %}

Setting the left side to zero, we get
{% \frac{d TC(q(p))}{dq} \times \frac{dq}{dp} = q + p \times \frac{dq}{dp} %}

Solving for p, we get
{% p = \frac{d TC (q(p))}{dq} - q \times \frac{dp}{dq} %}
If we an approximation and take dq = 1, then {% dTC/dq %} is the marginal cost of a single product.

{% p = MarginalCost - q \times \Delta p%}


where now {% \Delta p %} is the change in price required to increase demand by one unit. The change in price is negative, so the above equation is the sum of two terms, the first is the marginal cost. This clearly shows that to optimize price, you first have to hurdle the marginal cost, and then you need a specified amount of revenue above the marginal cost.

In order to measure the marginal contribution to cost, it is common to implement a cost allocation methodology.

Contents