Mathematics
Lecture Week 7, Chapter 5
Supply curve (optimization function of one variable)
Example 1: Profit maximization
→ Aim is to maximize profit.
So:
1. You have a revenue and cost function, and you have to make a profit function max 𝛑
(y) by:
- 𝛑 (y) = R(y) – C(y).
2. Take the derivative 𝛑’(y).
3. Find the stationary points (SPs) → 𝛑’(y) = 0.
4. Graph with SPs to find if the parts in-between the points are positive/negative.
5. Find maximum profit by filling in the highest stationary point in 𝛑 (y).
Moreover:
- MR(y) = MC(y)
Meaning: Marginal revenue must be equal to marginal cost.
- That the profit is maximal, does not mean that the profit is positive!!!
This mentioned before, put together:
Marginal output rule
The output quantity y > 0 that maximizes the profit
𝛑(y) = R(y) – C(y) satisfies the equation:
MR(y) = MC(y)
1
,Production rule
If y > 0 is the output quantity that maximizes profit, the producer will produce if:
AR(y) ≥ AC(y)
- AR(y) = average revenue at production y (revenue per unit).
= total revenue / quantity.
= R(y) / y
- AC(y) = Average cost at production y (cost per unit).
= total cost / quantity.
= C(y) / y.
→
Use these rules to check beforehand if we are going to produce or not.
This is done by:
1. Check AR(y) ≥ AC(y), to see if there is a positive profit to be made.
2. Use marginal output rule to check where the profit is maximum.
!!!
2
,As p ≥ min AC(y), and you have to determine for each price p, the output quantity that maximizes
profit:
Calculate minimum of AC(y):
1. AC(y) = C(y) / y
2. To find stationary points SPs: AC’(y) = 0
3. Make the graph to find - / +, and then see if the SP is actually a minimum or a maximum.
4. Fill it in AC(y) = …
5. Meaning: if p ≥ … , positive profits are possible
If p < … , then stop producing.
However, what is the optimal p going to look like? → marginal production rule.
!!!
y(p) = supply function
= tells you for each price which production quantity maximizes the firm’s profit.
3
, Supply function
The supply function of a producer with profit function:
𝛑(y) = p * y – C(y) is given by:
!!!
Note: for 0 ≤ p < min AC(y), the producer does not make any profit.
So:
1. Make the formula 𝛑(y) = p * y – C(y).
2. Calculate 𝛑’(y) = 0.
3. Then, find maximum profit, which is equal to MC-1(p).
4. Then, solve AC’(y), to find minimum average cost.
5. Then:
→ (example).
Exercises week 7
Notes:
- When determining all extrema of a function, with a minimum or maximum boundary, find if
the boundary point is actually a minimum or maximum, besides the stationary points, by
making a sign chart.
If it is, fill the value in y(x) for the x, to find the corresponding values (the exact point), for all
extrema.
- If you have to find a, b in z(x, y), and it is given that a stationary point is for example: (2, 3):
First: find z’x(x, y) and z’y(x, y).
Second: both of these are: .. = 0
Third: fill in (2,3) in both derivatives.
Substitute to find the right a and b.
a:
4
Lecture Week 7, Chapter 5
Supply curve (optimization function of one variable)
Example 1: Profit maximization
→ Aim is to maximize profit.
So:
1. You have a revenue and cost function, and you have to make a profit function max 𝛑
(y) by:
- 𝛑 (y) = R(y) – C(y).
2. Take the derivative 𝛑’(y).
3. Find the stationary points (SPs) → 𝛑’(y) = 0.
4. Graph with SPs to find if the parts in-between the points are positive/negative.
5. Find maximum profit by filling in the highest stationary point in 𝛑 (y).
Moreover:
- MR(y) = MC(y)
Meaning: Marginal revenue must be equal to marginal cost.
- That the profit is maximal, does not mean that the profit is positive!!!
This mentioned before, put together:
Marginal output rule
The output quantity y > 0 that maximizes the profit
𝛑(y) = R(y) – C(y) satisfies the equation:
MR(y) = MC(y)
1
,Production rule
If y > 0 is the output quantity that maximizes profit, the producer will produce if:
AR(y) ≥ AC(y)
- AR(y) = average revenue at production y (revenue per unit).
= total revenue / quantity.
= R(y) / y
- AC(y) = Average cost at production y (cost per unit).
= total cost / quantity.
= C(y) / y.
→
Use these rules to check beforehand if we are going to produce or not.
This is done by:
1. Check AR(y) ≥ AC(y), to see if there is a positive profit to be made.
2. Use marginal output rule to check where the profit is maximum.
!!!
2
,As p ≥ min AC(y), and you have to determine for each price p, the output quantity that maximizes
profit:
Calculate minimum of AC(y):
1. AC(y) = C(y) / y
2. To find stationary points SPs: AC’(y) = 0
3. Make the graph to find - / +, and then see if the SP is actually a minimum or a maximum.
4. Fill it in AC(y) = …
5. Meaning: if p ≥ … , positive profits are possible
If p < … , then stop producing.
However, what is the optimal p going to look like? → marginal production rule.
!!!
y(p) = supply function
= tells you for each price which production quantity maximizes the firm’s profit.
3
, Supply function
The supply function of a producer with profit function:
𝛑(y) = p * y – C(y) is given by:
!!!
Note: for 0 ≤ p < min AC(y), the producer does not make any profit.
So:
1. Make the formula 𝛑(y) = p * y – C(y).
2. Calculate 𝛑’(y) = 0.
3. Then, find maximum profit, which is equal to MC-1(p).
4. Then, solve AC’(y), to find minimum average cost.
5. Then:
→ (example).
Exercises week 7
Notes:
- When determining all extrema of a function, with a minimum or maximum boundary, find if
the boundary point is actually a minimum or maximum, besides the stationary points, by
making a sign chart.
If it is, fill the value in y(x) for the x, to find the corresponding values (the exact point), for all
extrema.
- If you have to find a, b in z(x, y), and it is given that a stationary point is for example: (2, 3):
First: find z’x(x, y) and z’y(x, y).
Second: both of these are: .. = 0
Third: fill in (2,3) in both derivatives.
Substitute to find the right a and b.
a:
4
