Profit Maximization

Competitive Firms

• Competitive firms are defined as suppliers that can only choose the amount of goods to output/the amount of inputs used
• Such firms cannot change the prices of its inputs nor its outputs; thus, these values are fixed

Profit Maximizing in the Short-Run

• In the short-run, firms have one or more fixed costs; in the long-run, all costs are variable
• Assuming that a firm uses two inputs (with the amount of one of the inputs being fixed), we can write the profit function as follows: $\pi = pf(x_1, \bar{x_2}) - w_1 x_1 - w_2 \bar{x_2} \\ \text{where p is output price, w is input price, and f is the production function}$
• Maximizing this function means that we must find the local maximum of π with respect to x₁

$$\text{The first derivative of the profit is given as:} \\ \pi' = \frac{p \partial f(x_1^*, \bar{x_2})}{\partial x_1} - w_1 \\ \text{The maximizing quantity of } x_1 \text{ is found when the first derivative is 0 and the second derivative is negative:} \\ \frac{p \partial f(x_1^*, \bar{x_2})}{\partial x_1} - w_1 = 0 \\ \text{This can be simplified to } p MP_1 = w_1 \text{ which will give the maximizing quantity of } x_1$$

Isoprofit Line

$$\text{Using the equation for profit: } \\ \pi = pf(x_1, \bar{x_2}) - w_1 x_1 - w_2 \bar{x_2} = py - w_1 x_1 - w_2 \bar{x_2}\\ \text{The equation of a short-run isoprofit line is: } \\ y = \frac{w_1}{p} x_1 + \frac{\pi + w_2 \bar{x_2}}{p}$$

• The isoprofit line is useful in order to find the profit-maximizing plan, as the best plan will lie on the highest isoprofit line

Comparative Static Analysis of Profit Maximization

• Keeping x₂ fixed:
  • If the price of the output (y) goes up, then the optimal amounts of y and x₁ go up as well
  • If the price of input 1 goes up, then the slope of the isoprofit line goes up, meaning that the optimal amounts of y and x₁ go down
  • 
  • If the price of input 2 goes up, then the optimal amounts of either inputs are unchanged; the only thing that changes is the y-intercept of the isoprofit line, meaning that the firm simply makes less profit

Long-Run Profit Maximization

• Using the same logic as the short-run, we can find the optimal amount of inputs for any amount of variable inputs

$$\text{Recall that the optimal amount of input 1 in the short-run is given by } p MP_1 = w_1 \text{.} \\ \text{When there are multiple variable costs, this equation MUST hold.}$$

Cobb-Douglas Example

$$y = x_1^{1/3} x_2^{1/3} \\ p \cdot MP_1 = w_1 \rightarrow p \cdot \frac{1}{3} x_1^{-2/3} x_2^{1/3} = w_1 \\ p \cdot MP_2 = w_2 \rightarrow p \cdot \frac{1}{3} x_1^{1/3} x_2^{-2/3} = w_2 \\ \text{Dividing the two above equations gives} \frac{x_1}{x_2} = \frac{w_2}{w_1} \rightarrow x_1 = \frac{w_2}{w_1} x_2\\ \frac{p}{3} ({\frac{w_2}{w_1} x_2})^{-2/3} x_2^{1/3} = w_1 \\ \frac{p}{3} ({\frac{w_2}{w_1}})^{-2/3} x_2^{-1/3} = w_1 \\ \frac{p^3}{27} ({\frac{w_2}{w_1}})^{-2} x_2^{-1} = w_1^3 \\ \frac{p^3}{27} ({\frac{w_2}{w_1}})^{-2} w_1^{-3} = x_2 \\ x_2^{**} = \frac{p^3}{27w_1 w_2^2} \\ x_1^{**} = \frac{p^3}{27w_1^2 w_2} \\ \text{Plugging the optimal amounts of input 1 and 2 into the production function:} \\ y^{**} = \frac{p^2}{9w_1w_2}$$

Returns-to-Scale

• If a competitive firm’s techonology exhibits decreasing RTS, then there is only one “highest” isoprofit line, meaning that there is one, unique production plan
• If a competitive firm’s technology exhibits increasing RTS, then there are infinitely many highest isoprofit lines, meaning that the firm does NOT have a profit-maximizing plan
  • A technology can have increasing RTS in the short-run, but if it is infinitely increasing RTS, then firms would keep getting infinitely bigger, breaking the assumption that the firm is competitive
• If a competitive firm’s technology exhibits constant RTS, then the production function is linear, meaning that the isoprofit line MUST coincide with the production function
  • The profit of this isoprofit line has to be 0, as doubling the inputs should lead to a double in your outputs (2 x 0 = 0)
  • If the profit of the isoprofit lines was not 0, then you could continue to make more and more profit infinitely, leading to the same conclusion as increasing RTS
  • If the slope of the isoprofit line does NOT coincide with the production function, then you can keep taking higher and higher isoprofit lines; thus, they must coincide

Revealed Profitability

Suppose that production bundle (x’, y’) is chosen at (w’, p’). (x’, y’) must therefore be profit-maximizing at (w’, p’).

• Assuming that nothing is known about the production function, all we can do is draw one isoprofit line intersecting (x’, y’)
• The more known points, the more isoprofit lines drawn, and the more is revealed about the technology set

Weak Axiom of Profit Maximization (AKA WAPM)

• Suppose that a production bundle (x’, y’) is chosen at prices (w’, p’) and bundle (x’’, y’’) is chosen at prices (w’’, p’’)
  • This means that p’y’ - w’x’ MUST be greater than p’y’’ - w’x’’, as (x’, y’) is the profit maximizing bundle at (w’, p’)
• This observation gives the Weak Axiom of Profit Maximization

$$\text{The Weak Axiom of Profit Maximization gives the following equation:} \\ \Delta p \Delta y - \Delta w_1 \Delta x_1 - ... - \Delta w_n \Delta x_n \geq 0$$

Proof:

• The WAPM also gives observations regarding other fundamentals of economics
  • Law of Supply: If the change in w is 0, then ΔpΔy >= 0, meaning that a competitive firm’s output supply curve cannot slope downward
    • Both Δp and Δy have to be negative or positive; one can’t be negative while the other is positive
  • Law of Demand: If Δp = 0, then ΔwΔx <= 0, implying that a competitive firm’s input demand curve cannot slope upwards
    • Δw and Δx have to have opposite signs to be less than zero