profit maximisation theory Flashcards
Revenue
R(q)=pq
The company can sell as many units as it wants at the price p , so its total revenue depends on how much it produces
Costs
C=wx
The firm’s costs depend on how much of each input it uses. The total cost is calculated by multiplying the input prices by the quantities used.
Profits
Revenue - Costs → profits= pq-wx
what can the firm decide?
This is a single-output firm using L-1 inputs.
The firm is a “perfect competitor,” meaning it has no control over the prices (of the inputs it buys or the product it sells)
But can decide:
- How much output q to produce
- How much of each input x to use in production
The firm’s main goal is to maximize its profits.
The firm is constrained by the technology it uses, which is represented by the function f(x).
what does the function f(x) tell us?
f(x) function tells us how much output q can be produced given the amount of inputs x used.
How the firm choose q and x?
The firm will choose q and that solve the following problem:
max pq-wx st f(x)≥q
q,x≥0
- q is the firm’s decision about how much output to produce
- f(x) is the output that the firm can actually produce given its inputs and technology.
The constraint ensures that the firm only produces as much output as its input choices allow.
Since p>0 , the constraint will always bind.
the firm does not need to worry about deciding how much output to produce because q is automatically determined by its input choices x.
The only decision the firm needs to make is how much of each input x to use, which maximizes its profit.
what is going to be the solution to the PMP (profit maximising problem)?
We assume that the PMP has an interior solution.
We assume that the profit-maximizing solution occurs when all the input quantities are x*>0 (this is called an “interior solution”)
Given this assumption, the firm’s optimal output level is y=f(x) which is the output produced when the firm uses the input vector.
The input vector x is the optimal combination of inputs that the firm should use to maximize its profits.
The solution to the PMP is the optimal choice of inputs: x* = x(p,w) → this gives us the INPUT DEMAND FUNCTION
What is the MRP Marginal Revenue Product?
MRP= p * (∂f(x*)/∂x)
measures how much additional output is produced when one more unit of input is added.
At the profit-maximizing point, the marginal revenue product of every input must be equal to the price of that input.
p * (∂f(x*)/∂x) = w
MRP given 2 inputs
For any two inputs, say input l and input k if both of them are being used in the optimal input mix the MRP will be:
[∂f(x)/∂x_l]/[∂f(x)/∂x_k] = w_l/w_k
at the optimum, the absolute value of the MRTS between any two inputs must equal their price ratio
SUPPLY FUNCTION
if we plus x* into f(x) we get the output supply function, which shows the maximum output q* that can be produced at given prices and input costs.
PROFIT FUNCTION
π = pq-wx
What properties does the profit function have?
If f(*) is continuous, strictly increasing and quasi-concave, then for p≥0 and w≥0, the profit function has the following properties:
1. Non-decreasing in p and non-increasing in w
Profits respond predictably to prices:
- Profits increase when output prices p go up
- Profits decrease when input prices w go up
2. Homogeneous of degree 1 in: if input and output prices change by α , while the production level is held constant, profits would also change by α.
3. Convex in (p,w)
4. Continuous in p>0 and w≫0
Hotelling’s Lemma
if we have the profit function, we can derive the supply function and the input demand function.
supply function: ∂π/∂p
input demand function: - ∂π/∂w
In essence, Hotelling’s Lemma shows how a firm’s optimal output supply and input demand directly relate to changes in prices.
the production function actually has a maximum
- Concave production functions ensure a clear profit-maximizing point exists, and the first-order conditions are enough to find it.
- If the production function exhibits constant or increasing returns to scale, the firm could theoretically grow profits forever, and no maximum profit point exists.
