1. Economic Applications of Functions and Derivatives
Chapter 8

2. 8.1 Introduction How do costs vary in the short- and long-run?
Short run: a period of time during which certain quantities are fixed (e.g. wages cannot be changed before the end of labor contract)
Long run: a period of time during which any quantities may change
How is demand related to total and marginal revenue?
Perfect competition
Monopoly
Profit maximization

3. 8.2 Total Cost Function
The graph below displays a typical total cost function (TC):
Key features:
Positive intercept: TC(0)=$500. You pay these costs even if you don’t produce anything, this is your fixed costs.
TC is positively sloped. It costs more to produce more output.
The slope of TC is increasing with more output. It costs increasingly more to produce one more unit of output when producing more output.
The properties of the total cost function can be formalized as follows:

4. 8.3 Average Cost Function Fixed costs are $2000 since TC(0)=$2000.
Variable costs are equal to
Average costs are equal to
Definition. Part of the firm’s total costs that does not depend on the amount of output is called fixed costs,
Definition. Part of the firm’s total costs that varies with output q is called variable costs,
Definition. Firm’s costs per unit of output are called average cost,
Average costs are often referred to as unit costs.

5. Minimum Average Cost
Consider a firm’s total cost function to be given by
It makes sense to ask at what level of output the unit costs are the lowest.
We start by computing this firm’s average costs:
The average costs will achieve their minimum where and
(we disregard q=-20).
Let us check the sign of the second derivative:
The minimum unit costs are achieved when 20 units of output are produced. In this case it costs (dollars) to produce one unit of output.

6. Minimum Efficient Scale
Definition. The level of output at which a firm’s long-run average cost is minimized is called minimum efficient scale.

7. Asymptotic Behavior of AC
Consider a firm’s total cost function as a sum of fixed and variable costs:
By definition of the average cost function,
Assuming , so and
As , average fixed costs tend to zero: which implies
that for large values of output average costs are very close to average variable costs AVC(q). This happens because fixed costs get spread out across a large number of units, becoming very small. In other words, is an asymptote of the average cost function as output q tends to infinity.
In the same fashion, as output q tends to zero, the cost axis q=0 is the average cost function’s asymptote.

8. 8.4 Marginal Cost Suppose we are producing at point A.
If we increase output by , our total costs will increase by , moving us to point B.
Consider a difference quotient when .
Definition. The value at of a difference quotient is called marginal cost MC1.
Depending on the units, the same can be equal to 1, 10, 1000 or any number.
Definition. The value at of the derivative of the total cost function is called marginal cost MC2, which we also refer to as MC.

9. Marginal and Fixed Costs
Remember that total costs are a sum of fixed and variable costs:
By definition of the marginal cost MC2,
Marginal costs are independent of the fixed costs!
You can arrive at the same conclusion using the definition of marginal cost MC1.
Example 8.3 Consider a total cost function
Compute MC2
Change the fixed costs to 5000, compute MC2 again
Compare the marginal costs in 1) and 2)
Do the same analysis for MC1
Where does MC curve intersect the TC curve?

10. 8.5 Marginal and Average Costs
When MC>AC, AC is increasing.
When MC<AC, AC is decreasing.
Suppose the number of output units is the number of exams you’ve taken, and the costs are the exam scores.
The marginal cost then will be your score on the next exam.
Average cost will be your grade point average.
What will happen to your grade point average if your next score is lower than the average? Higher than the average? Equal to the average score?
The marginal cost curve is always intersecting the average cost curve at its minimum.

11. 8.6 Worked Examples Example 8.4 Compute the average cost AC
Compute the marginal cost MC
Show that the minimum of the average cost curve coincides with the level of output where MC intersects AC
Solve the inequality MC>AC to show that when MC is above AC, the AC curve is increasing
Notice that the slope of the OK line is equal to the average cost at point K: why?
Slope of TC is flatter to the left of K compared to output levels to the right of K.

12. Worked Examples Example 8.5 Compute the average cost AC
Compute the marginal cost MC
Show that the minimum of the average cost curve coincides with the level of output where MC intersects AC
Solve the inequality MC>AC to show that when MC is above AC, the AC curve is increasing
When total costs are linear, AC is always above the MC.

13. Worked Examples Example 8.6 Compute the average cost AC
Compute the marginal cost MC
Show that the minimum of the average cost curve coincides with the level of output where MC intersects AC
Solve the inequality MC>AC to show that when MC is above AC, the AC curve is increasing
Marginal cost has a decreasing part now.
Producing too much or too little output is technologically inefficient.

14. 8.7, 8.8 Market Demand Function
Definition. The number of output units sold in the market at a particular price is called market quantity demanded corresponding to that price.
Definition. A relationship between prices and market quantities demanded is called a market demand function. A graph of the market demand function is called market demand curve.
Example. Consider a demand function of the form
Key features.
Negative slope: consumers buy cheaper things more
Bottom graph: inverse demand function
Total expenditure (aka total revenue) for each q is equal to the rectangular area below the demand curve
Since p and q are related by the demand function,

15. 8.9 Total Revenue with Monopoly
Definition. A firm that is the only supplier of a particular product in the entire market is called a monopoly.
A monopolist’s ability to charge “whatever price it likes” is constrained by the market demand function: if consumers think the price is too expensive, they switch to close substitutes.
A monopolist is facing the entire market demand.
Total revenue is zero when nothing is sold, or when the market is saturated.
Each point on the TR curve is equal to the rectangular area in the graph above.

16. Maximum Total Revenue
Total revenue reaches its maximum level at q=100.
Indeed,
hence q* is a maximum.
The maximum total revenue is TR(100)=5000.

17. Marginal Revenue and Monopoly
Definition. The value at a particular of the difference quotient is called marginal revenue MR1.
Definition. The value at a particular of the derivative is called marginal revenue MR2.
Marginal revenue according to both definitions refers to the additional revenue obtained by selling an additional small amount of output.
MR1 is equal to in case
However, the value of MR1 depends on the units in which we measure output q.
To get rid of this problem, economists use MR2 which we refer to as MR.

18. Using Marginal Revenue
The conditions for maximizing the total revenue can be written down in terms of the marginal revenue:
In the previous example,
Economic story behind the marginal revenue function
The marginal revenue function is a decreasing function of output: if we produce nothing, our total revenue TR is zero. Producing the first unit results in MR(1)>0, so it is a good idea to produce that unit. Producing the second unit gives us an additional revenue MR(2)>0, so it is also a good idea to produce the second unit. Our total revenue now is TR(2)=MR(1)+MR(2). However, at q=100 producing unit number 101 will result in MR(101)<0, which means we will be decreasing our total revenue, so we stop increasing production volume at q=100.

19. 8.11 Demand, TR, and MR with Monopoly
Consider a monopolistic producer facing the inverse demand function
The marginal revenue function is
Key features.
MR function is linear, and it has the same vertical intercept with demand function
The slope of MR function is twice the slope of the demand function: general property for linear demand functions
The horizontal intercept of the MR function is half that of the demand function
MR is positive when TR is growing, and vice versa
When TR reaches its maximum, MR=0

20. Relationship between MR and TR
TR increases with price reduction when q<100, the level of output at which TR is maximized.
TR decreases with price reduction when q>100.
General rule: if MR is positive at some q, price reduction will result in an increase in total revenue.
In case of the general form demand functions, the following rules apply (see Appendix 8.2):

21. 8.12 Demand, MR, and TR with Perfect Competition
Definition. A firm is said to operate under perfect competition if
There are a large number of small firms
The product is homogeneous
Each firm produces a similar amount of the good
Corollary 1. A firm operating in a perfectly competitive environment is a price taker,
meaning no matter how much it sells, the price it is facing will not change. As a result, the competitive firm’s demand is flat.
Corollary 2. MR=demand for a perfectly competitive firm.
where is the market price given by the intersection of market supply and demand.
Under perfect competition, total revenue TR has no maximum.
since a perfectly competitive firm does not have to reduce the price to sell more units of output

22. 8.13 Worked Examples
Example 8.9 A formal analysis of the linear demand functions.
Consider an inverse demand function
The slope of the MR function is twice the slope of the linear demand function.
The horizontal intercept of the MR function is half that of the demand function.
Total revenue TR is maximized at
This is, indeed, a maximum since

23. Example 8.10 Consider an inverse demand function .
Total revenue is maximized at q=8.

24. 8.14 Profit Maximization
Definition. The difference between total revenue and cost for a specific level of output q is called profits, and is denoted by
Profits under perfect competition: price p does not depend on the quantity:
Profits under imperfect competition, including monopoly: p=p(q)

25. 8.15 Profit Maximization with Monopoly
Example Consider and market demand
The inverse demand function is , and total revenue will be equal to
The monopolist’s profits will be given by:
The profit function has an extremum at
To see whether q* is a maximum, let us check the sign of the second derivative is negative:

26. Example 8.11
Key features
Profits are equal to the vertical difference between the total revenue and total cost lines
Profits are maximized when this distance is at its maximum
This maximum is achieved when MR=MC
Two points where TR=TC so profits are zero are called break-even points due to fixed costs (left one), and pressure on plant capacity (right one).

27. 8.16 Profit Maximization, MC and MR
Consider another way to represent an extremum condition for the profit function:
By definition of the marginal revenue and marginal costs, the extremum condition assumes the form:
Since the MR represents the slope of the total revenue, and MC the slope of the total cost functions, profits are maximized at the level of output where TC and TR have parallel slopes.
MR=MC is necessary, but not sufficient condition for profit maximization! Profits can be actually minimized when MR=MC.

28. Price and Marginal Cost
Profits are maximized at q*=33.
The marginal cost MC=2q*+2=68
The price at which q* is sold can be found on the demand function:
P=-2q*+200=134
A monopolist is charging more than the marginal cost when it maximizes profits!
Under perfect competition, P=MC.

29. 8.17 Profit Maximization under Perfect Competition
Example Consider a total cost function
Under perfect competition, a firm sells anything it wants at market price .
Assuming (arbitrarily) that , the firm’s total revenue is
The profits will be
The extremum condition:
To ensure that q* is a maximum:
The competitive firm’s profits are equal to

30. MR=MC, Perfect Competition
Key features
TR is a straight line so it has a constant slope: a competitive firm does not have to reduce the price to sell more
The slope of TR is equal to the slope of TC where MC=MR
Profits are maximized at that point
The slope of TR is the market price, so
Differently from monopoly, p=MC

31. 8.18 Monopoly and Perfect Competition
Monopolist. Inverse demand function
Perfect competition. Same market demand
. Same industry total cost function
. The MC(q)=2q+2 will be the industry supply curve.
As before, let us denote by the market price faced by a single competitive firm. For a perfectly competitive firm, price must equal the marginal cost of production, so
To ensure that supply equals demand in the market as a whole,
A monopolist produces less and charges more compared to the competitive market.

32. 8.19 Profit Maximization and Minimum Average Cost
In the examples 8.11 and 8.12, the total cost function is , and the profit-maximizing output levels are and for a monopolist and perfect competition, respectively.
Consider the average cost function Its minimum is achieved at
Monopolist and competitive firms maximize profits at a level of output when the average costs are not minimized!

33. Profit Maximization and Total Revenue
Profits are not maximized (in general) when total revenues are maximized, too.
Indeed, the first-order condition for TR maximization is
In the graph to the left, MR=0 at q=50, which is greater than the profit-maximizing output in this case.

34. 8.20 The Second-Order Condition for Profit Maximization
The full set of conditions for profit maximization involves the second derivative of the profit function:
At points J and L MR=MC, however,
profits are minimized at J.