Profit maximization

When talking about the profit maximization problem for a firm we have to distinguish between long run and short. This is because in the short run firms bear also fixed costs while in the long run they do not (did you miss our lecture on costs?).

SHORT RUN PROFIT MAXIMIZATION

Let’s consider a firm using only two inputs of production: input 1 and input 2. Suppose that input 2 is fixed at some level ẍ2. In this notation the x means the quantity of the good considered while the subscript indicates which good we are referring to. The quantity of the goods the firm sells on the market is expressed with y. Let’s denote with p the price the firm asks for its products on the market and with w1 and w2 the prices of the two inputs. The profits a firm is making are calculated as total revenues minus total cost:

π = py – w1x1 – w2 ẍ2

if we transform this equation so that y becomes the dependent variable we get:

This equation is the equation of an isoprofit line, a line where the firm’s profits do not change as we change the quantity of the variable input x1. This line is upward sloping because we assume that the more is used as inputs, the more the firm will produce (even thought in reality this is not always true) and its slope is w/p. The vertical intercept of this line is given by: π/p+ w2 ẍ2/p. By changing the profits the firm wants to obtain, we can shift in a parallel fashion the line. Higher lines will correspond to higher profits. Because of that, the profit maximization problem can be thought of the problem of reaching the highest possible isoprofit line given the available production function.

As always this requirement is satisfied with a tangency condition between the two given by:

MP1 = w1/p

MP1 is the marginal product function of input 1. Consider that, even if the vertical intercept is positive, it does not mean that the firm is making positive profits. We expressed the vertical intercept as: π/p+ w2 ẍ2/p meaning that it does not only depends on profits but also on the amount of fixed costs the firms has to bear. Consider for example a startup firm selling its products on the market at a price of $5having as fixed costs $100 but which is not yet profitable and therefore aims at minimizing costs reaching profits of ($-20). As you can see, even tought the profits are negative, the vertical intercept is positive: -20/5 + 100/5 = 16.

WHAT HAPPENS WHEN PRICES CHANGE

When the price of input 1 or the price of the goods on the market change the slope of the isoprofit lines will change. When the w1 increases or when p decreases the ratio w1/p becomes higher and therefore the isoprofit line will become steeper. When this happens, the profit maximizing quantity of input 1 will decrease. The line will become flatter instead when p increases or when w1 decreases because of the fa ct that the ratio w1/p becomes smaller. When this happens, the profit maximizing quantity of input 1 will increase. When we change the price of the fixed input nothing happens to the slope of the isoprofit lines and therefore the profit maximizing quantity will not change. Also the quantity employed of that input will not change because it is fixed at ẍ2. The only thing that will in the end change is the profit the firm will make.

LONG RUN PROFIT MAXIMIZATION

In the long run the firm does not have to bear any fixed cost. This means that the profits for a firm in the long run are given by:

π = py – w1x1 – w2x2

When we consider two variable factor, we just have to impose that the value of both the marginal increase in production due to the marginal increase in each output is equal to the cost of employing that additional unit of input. Therefore we write:

pMP1(x*1 , x*2) = w1

pMP2(x*1 , x*2) = w2

x*1 and x*2 represent the unknown quantities which respect that requirement. When we apply both these conditions we get two equations having as unknowns x*1 and x*2 and, by solving individually each of these equations, we can determine the condition-satisfying quantities for each set of prices. The equations that express the relationship between the prices and the optimal choices on inputs are called factor demand curves.