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ECF 510- Advanced Micro Economics
Lecture 3
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Supply side of economy:
-
Composed of productive units (firms).
-
Firms transform inputs into outputs.
Definition of firms:
-
corporations or other legally recognized businesses
-
productive possibilities of individuals or households
-
potential productive units that are not organized
Two important concepts:
1. Production set:
-
Representation of the set of all technologically feasible production plans.
2. Behaviour assumptions:
-
Behavioural assumptions express the guiding principle that the producer uses to make choices. A
general assumption is that the producer seeks to identify and select a production set that is most
profitable.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Specification of Technology
Production vector: (an economy with L commodities)
(input-output, netput vector, or production plan):
- A vector y = (y1, …, yL)  RL that describes the (net) outputs of the L
commodities from a production process.
- Positive numbers denote outputs and negative numbers denote inputs.
- When the production process has no net output of a commodity the
elements of the production vector will be zero.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Example: Suppose that L = 6.
-
Production vector y = (5, 2, 6, 3, 0, 4)
-
2 and 3 units of goods 2 and 4, respectively, are produced
-
5, 6 and 4 units of goods 1, 3 and 6, respectively, are used.
-
Good 5 is neither produced nor used as an input in this production vector.
Limits to the set of feasible production plans
-
technological constraints
-
legal restrictions
- prior contractual commitments.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Transformation function F()
-
It shows how inputs are transformed into outputs
-
It is a function used to describe the production set Y
-
Hence it has the property that
Y = {y  RL : F(y)  0 } and F(y) = 0 if and only if y is an element of the boundary of Y
-
F(y) < 0 leaves some slack in the production possibilities.
-
F(y) = 0 exhausts all production possibilities.
Transformation frontier:
-
The set of boundary points of Y
{y  RL : F(y) = 0 }
-
A production plan is efficient if there is no way to produce more output with the same inputs
or to produce the same output with less inputs.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Isoquant
• The isoquant gives all input bundles that produce exactly y units of
output.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Commonly aassumed properties of production sets
(1) Y is nonempty (or non-triviality)
The firm has something it can plan to do.
(2) Y is closed
-
The set Y includes its boundary.
-
The limit of a sequence of technologically feasible input-output vectors
is also feasible.
-
The limit production plan y is also in Y.
- It guarantees that points on the boundary of Y are also feasible
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
3) No free lunch
-
It is not possible to produce something from nothing.
4) Possibility of inaction :
-
0  Y.
Complete shutdown is possible.
Variation
-
For a firm that has already made some production decisions, or signed irrevocable
contracts for the delivery of some inputs, inaction is not possible. (sunk costs)
5) Free disposal ( or monotonicity)
-
if y  Y and y '  y (so that y ' produces at most the same amount of outputs using at least
the same amount of inputs), then y'  Y.
-
Extra amount of inputs (or outputs) can be disposed of or eliminated at no cost.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
6) Irreversibility
- Suppose that y  Y and y  0 then irreversibility means that y  Y.
- i.e. it is impossible to reverse a technologically possible production vector to
transform an amount of output into the same amount of input that was used
to generate it unless it is a non action plan.
(7) Additivity (or free entry)
- Suppose that y  Y and y '  Y then y + y '  Y.
- Or Y + Y  Y.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Marginal rate of technical substitution (MRTS)
(single output case.)
-
Holding the level of output fixed, define the marginal rate of technical substitution (MRTS) of
input ℓ for input k at as
MRTS k ( z ) 
f ( z ) / z
.
f ( z ) / zk
The number MRTSℓ k measures the additional amount of input k that must be used to keep
output constant when the amount of input ℓ is reduced marginally.
-
The marginal rate of technical substitution measures the slope of an isoquant.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Returns to Scale
A production function f(z) is said to exhibit :
(1) constant returns to scale
f(t z) = tf (z) for all t ≥ 0
(2) decreasing returns to scale
f(t z) < tf (z) for all t > 1
(3) increasing returns to scale
f(t z) > tf (z) for all t > 1
-
Constant returns to scale (CRS) means that doubling inputs exactly double outputs.
-
Decreasing returns to scale means that doubling inputs less than doubles outputs.
-
Increasing returns to scale means that doubling inputs more than doubles outputs.
-
A technology has constant returns to scale if and only if its production function is
homogeneous of degree 1.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Example: Returns to Scale for a Cobb-Douglas Production Function
f( z1, z2) =
z1 z12
f(2z1, 2z2) = 2   f(z1, z2)
   = 1 gives constant returns to scale
   < 1 gives decreasing returns to scale
   > 1 gives increasing returns to scale
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Elasticity of scale
-
-
A local measure of returns to scale based on elasticity of scale is useful.
Elasticity of scale measures the percent increase in output due to a one percent increase in all inputs
Example for a production function y = f(z)
- Let t be a positive scalar,
- Consider the function y(t) = f(t z)
- t = 1 defines the current scale of operation
- All inputs are scaled up by t if t > 1
- All inputs are scaled down by t if t < 1.
Calculation of the elasticity of scale
-
The elasticity of scale is
e(z) =
-
dy (t ) t
df (tz ) t
=
dt y t 1
dt f (tz ) t 1
Evaluate the expression at t = 1 to calculate the elasticity of scale at the point z.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Profit Maximization
Assumption:
- A firm will always choose the most profitable production plan from the production set.
Definition of profit
(total revenue minus total cost)
p·y =
-
where

L
1
p y .
p > 0 is a given a price vector
y  L is a production vector
Economic profit
-
Defined as the difference between the revenue a firm receives and the costs that it incurs.
-
All explicit and implicit costs are included in the calculation of profit.
-
Both revenues and costs of a firm depend on the actions taken by the firm.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
The firm’s profit maximization problem
Max
y
s.t.
p y
yY
(*)
- where the production set Y is the technological constraint
Constraints to the optimal policy
-
Technological constraints that are specified by production sets and
-
Market constraints that concern the effect of actions of other agents on the
firm.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Profit Function  (p):
-
 (p) = Max { p·y : y  Y }
-
The function  (p) gives the maximum profits as a function of the prices.
-
It is the solution to the profit maximization problem (*)
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Perfectly Competitive markets
- Determining how market prices are determined by the actions of individual agents.
Competitive market assumptions
- Large number of buyers and sellers | price-taking behaviour
- Unrestricted mobility of resources among industries: no artificial barrier or
impediment to entry or to exit from market.
- Homogeneous product. All the firms in an industry produce an identical production.
- All relevant information is common knowledge. Firms and consumers have all the
information necessary to make the correct economic decisions.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Competitive firm assumption
- It takes the market price of output as given (i.e. it cannot control input and output
prices.)
- Price is independent of each individual firm's actions.
- Actions of all firms taken together determine the market price.
- Demand curve for the competitive firm
0
𝐷 𝑝 = 𝑎𝑛𝑦 𝑎𝑚𝑜𝑢𝑛𝑡
∞
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𝑖𝑓 𝑝 > 𝑝
𝑖𝑓 𝑝 = 𝑝
𝑖𝑓 𝑝 < 𝑝
Implications for the firm
- The firm is free to set a price it wants.
- The firm can produce a quantity it is able to produce.
Problem for firm
- No one will purchase its product if it sets a price above the prevailing market price.
- Setting its price below the market price reduces its profits unnecessarily since it can
get as many customers as it wants at the market price.
Caveat
- No market is perfectly competitive.
Note: The models of perfect competition give useful insights about real-world markets.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Profit maximisation problem
-
Choose output y to solve py  c(y).
- First and second order conditions for an interior solution:
p = c (y*) and c (y*)  0.
Inverse supply function p(y)
- It measures the price that makes it profitable for a firm to supply a given amount of
output.
-
First order conditions imply that the inverse supply function is
p(y) = c (y) when c (y) > 0.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Supply function y(p)
- It gives the profit-maximising output at each price.
- It identically satisfies the first order and second order conditions
p = c  (y(p)) and the c (y(p))  0
Supply response to a change in price
-
Normal case: c´(y) > 0, hence y´(p) > 0.
- Implication: The supply curve of a competitive firm has a positive slope.
Msc. Economics and Finance Lecture Slides
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Cost Curves
- Total cost = variable costs + fixed costs
c(y) = cv(y) + F.
- Fixed costs must be paid even if output is zero.
Profitability
- Producing a positive level of output is profitable when the profits from doing so
exceed the profits(losses) from producing nothing.
p y(p)  cv(y(p))  F   F .
- Rearrange to obtain the price inequality
p
cv ( y( p))
y ( p)
-A firm will produce a positive level of output when price is greater than average
variable cost.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Industry Supply Function
- It is the sum of the individual firm supply functions
-yi(p) is the supply function for the ith firm for an industry with m firms.
Industry inverse supply function
- It gives the minimum price at which the industry is willing to supply a given
amount of output.
- Each firm that produces a positive amount of output must have the same marginal
cost.
- Reason: Each firm must choose a level of output where price equals marginal cost.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Example: Different cost functions:
-
Cost functions: c1(y) = y 2 and c2(y)= 2y 2
-
Individual supply functions : y1 = p/2 and y1 = p/4.
-
Industry supply curve is Y(p) = 3p/4.
-
For any level of industry output Y, the marginal cost of production in each firm is 4Y/3.
Example: Identical cost functions:
-
Number of firms: m
-
Identical cost function for firms: c(y) = y 2 + 1.
-
Supply function of the firm: y(p) = p/2
-
Industry supply function: Y(p, m) = mp/2.
-
Inverse industry supply function: p = 2Y/m.
-
Slope of the inverse supply function declines with the number of firms.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Market equilibrium
-
Equilibrium price: Price where the amount demanded equals the amount supplied.
where xi(p) is demand for individual i.
Example for demand curve
X(p) = a  bp
-
Industry supply curve is
Y(p, m) = mp/2
-
Equilibrium implies that
a  bp = mp/2
or that p* =
- Hence, the equilibrium price decreases as the number of firms increases.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Long run
- The number of firms in an industry becomes variable.
- Firms making persistent loses exit the industry.
- Firms making profit continue to produce.
- Entry or exit of firms depend on entry or exit costs, foresight etc.
-
If the number of equilibrium firms is large, the supply curve will be flat and the
equilibrium price will be close to the break-even price where profits are zero at the
optimal supply of output.
- The level of output is set where average cost equals marginal cost.
- The equilibrium number of firms is the largest number of firms that can break even
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Relationship between the equilibrium price and the break-even price
- The equilibrium price can be larger than the break-even price.
- Even though the firms in the industry make positive profits, entry is inhibited since potential
entrants with foresight determine that their entry would result in negative profits.
- Positive profits then become an economic rent or "rent to being first."
- The rent is the (opportunity) cost of remaining in the industry.
Profits in equilibrium
-
Firms earn zero profits in equilibrium.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Example when
c(y) = y 2 + 1
- Breakeven level of output is found by setting average cost equal to marginal cost:
y + 1/y = 2 y ,
- Thus
y = 1.
- When y = 1 marginal cost is 2.
- The breakeven price is also 2.
-
Firms will enter the industry as long as they determine that they will not drive the
equilibrium price below 2.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Welfare
Scenario:
- A representative consumer intending to maximise utility.
- A representative firm intending to maximise profits.
Representative consumer
- Utility function for consumer :
u(x) + y.
- y is the amount of money left over for purchasing other goods after the consumer
makes the optimal expenditure on the good x.
- Market demand: x(p)
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Representative firm
- Cost function for firm: c(x).
- The production of x units of output requires c(x) units of the good y.
- Profit maximizing supply function of the representative firm:
p = c  (x).
Equilibrium level of output
- It is the level of output at which the marginal willingness to pay for good x just
equals its marginal cost of production
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Consumer and producer surplus
- Consumer surplus: CS(x) = u(x)  px.
-
It is the difference between the total benefits from consumption of good x and
expenditure on the good x.
- Producer surplus : PS(x) = px  c(x).
-
It measures the profits earned by the producer.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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The welfare maximization problem
- Entails maximizing both consumer surplus and producer surplus
- The maximization of total surplus CS(x) + PS(x) is the same as the maximization of
u(x)  c(x)
- Reason:
CS(x) + PS(x) = u(x)  px + px  c(x) = u(x)  c(x).
Changes in policy variable (taxes and subsidies )
- The price paid by buyers of a good, Pd, differs from the supply price, Ps, received by
the suppliers of the good by the amount of the tax or subsidy.
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- When quantity tax is a tax levied on the amount of a good consumed then the price paid by
the consumers is greater than the price received by the suppliers by the amount of the tax:
Pd = Ps + t.
- A value tax is a tax levied on the expenditure of a good.
- It is usually expressed as a percentage.
- A value tax at rate, r, gives
Pd = (1 + r) Ps.
- A quantity subsidy, s, means that the seller receives s dollars more per unit than the buyer
pays, i.e.
Pd = Ps  s.
- Equilibrium is when demand equals supply.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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- From the case of a quantity tax equilibrium is calculated from the equations
D(Pd) = S(Ps) where Ps + t.
- Utility to the consumer at x* is
u(x*)  Pd x*
- Profits to the firm are
Ps x*  c(x*)
- Revenue to the government is
t x* = (Pd  Ps) x*
- Net welfare is
W(x*) = u(x*)  c(x*)
It is the area below the demand curve minus the area below the marginal cost curve.
- The difference between the surplus achieved with the tax and the welfare achieved
in the original equilibrium is a deadweight loss.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Price
Supply
pd*
Tax
Revenue
p*s
Deadweight
loss
Demand
Quantity
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Monopoly
- Any situation in which some firm or small group of firms has the exclusive control of a
product in a given market.
- A monopolist has market power.
Meaning
-
The amount of output that a monopolist is able to sell is a function of the price it charges.
Comparison of a competitive firm and a monopolist
- If a competitive firm charges a price higher than the prevailing market price its sales drop
to zero.
- Hence a competitive firm is a price-taker, whereas a monopolist is a price- maker.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Constraints facing the monopolist
- Technological constraints where only certain combinations of inputs and outputs
that are technologically feasible. These are summarized using a cost function.
- Demand constraints based on consumer behaviour.
The monopolist's profit maximization problem
Max py  c(y)
s.t.
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D(p) = y.
Explanations for f.o.c. and s.o.c.
- The first-order condition (f.o.c.) says that at the profit-maximizing choice of output
marginal revenue must equal marginal cost.
- The second-order condition (s.o.c.) requires the derivative of marginal revenue to
be less than the derivative of marginal cost; i.e., the marginal revenue curve crosses the
marginal cost curve from above.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Quality Choice
- Monopolies choose not only output levels, but also other dimensions of the
products they produce such as product quality.
Monopoly Non-Simple Pricing
-
If the monopolist sells all goods at the same price then it sells less than the optimal
amount of the output for the society.
-
If the monopolist could raise the price on some items but not others, it could earn
higher profits and still sell the efficient quantity.
- Non-linear pricing and two-part tariffs can achieve the objective.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Non-Linear Pricing
Assumptions
- The monopolist uses a price scheme where each unit is sold for a different price.
- Consumers are not able to resell items once they buy them.
Solution to the monopolist profit-maximization problem
-
Design a scheme that maximizes the profit earned from any one consumer and
then apply this scheme to all consumers.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Illustration of non-linear pricing
-
Consumer demand curve: P = 100 − Q
-
Monopolist’s marginal cost: 10.
-
At a price of 90 the consumer demands 10 units of output.
-
The consumer does not purchase more output at a price of 90.
-
The consumer purchases more output if the price is lower.
-
The monopolist could sell the first 10 units of output at a price of 90
the second 10 units at a price of 80,
21-30 at a price of 70,
31-40 at a price of 60,
41-50 at a price of 50, etc., and
71-80 are sold at a price of 20.
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100
10
0
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100
Two-Part Tariffs
- A two-part tariff consists of a fixed fee and a price per unit consumed.
Example
-
An amusement park charges an admission fee and a price for each ride,
-
A country club charges a membership fee and a fee for each round of golf the
member plays.
- We have a fixed fee, F, and the user fee, p.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Price Discrimination
- Typically, the monopolist faces the problem that some consumers will have a high
willingness to pay for the product and others will have a low willingness to pay.
- Price discrimination involves selling different units of the same good at different
prices, either to the same or different consumers.
- A monopolist usually desires to sell additional output if it can find a way to do so
without lowering the price on the units it is currently selling.
- In order for price discrimination to be a viable strategy for the firm, it must be able
to discriminate consumers and to prevent resale.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Examples
-
Business travellers will be willing to pay more for airline tickets than leisure travelers
-
Working-age adults may be more willing to pay for a movie ticket than senior citizens.
-
Price discrimination is the monopolist’s attempts to charge different prices to different groups
of people.
Traditional classification of price discrimination [Pigou (1920).]
-
Frequently, three types of price discrimination are identified, although the distinctions are, at
least to some extent, arbitrary.
-
They are called first-degree, second-degree, and third-degree price discrimination.
Msc. Economics and Finance Lecture Slides
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First-degree price discrimination
- Also known as perfect price discrimination
- It involves the seller charging a different price for each unit of the good in such a way that the
price charged for each unit is equal to the maximum willingness-to-pay for that unit.
- The monopolist will choose to produce a Pareto efficient level of output where the marginal
willingness-to-pay equals marginal cost.
- However the producer will also manage to capture all the benefits from this efficient level of
production and obtain the maximum possible profits, while the consumer is indifferent to
consuming the product or not.
- Secondly, the monopolist in this market produces the same level of output as a competitive
industry would.
- A competitive industry will produce where price equals marginal cost and supply equals
demand
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Practicality of first-degree price discrimination
- The monopolist must be able to identify the consumer’s willingness to pay (or
demand curve) and charge a different price to each consumer.
-
It is difficult to identify a consumer’s willingness to pay which they will not say.
-
It is often impractical or illegal to tailor your pricing to each individual.
- First-degree price discrimination is an extreme example of the maximum (but
rarely attainable) profit the monopolist can achieve.
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Second-degree price discrimination
- Also known as nonlinear pricing.
-
The monopolist cannot perfectly identify the consumers.
-
It occurs when prices differ depending on the number of units of the good bought, but not
across consumers.
-
Each consumer faces the same price schedule, but the schedule involves different prices for
different amounts of the good purchased.
- This involves such practices as quantity discounts, where the revenue a firm collects is a
nonlinear function of the amount purchased
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
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Golf course example
-
Some people use a golf course every week and are willing to pay a lot for each
use
-
Others only use the course once or twice a season and place a relatively low
value on a round of golf.
- The owner of the golf course wishes to charge a high price to the high-valued
users, and a low price to the low-valued users
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Problem encountered
-
On any given day, people have an incentive to act like low-valued users and pay the
lower price.
-
The monopolist wishes to charge the high-valued people a high price and the
low-valued people a low price,
Solution to the problem
- The monopolist designs a pricing scheme such that the high-valued people do not
want to pretend to be low-valued people.
- This is also called, including second-degree price discrimination, non-linear pricing,
and screening.
Msc. Economics and Finance Lecture Slides
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Third-Degree Price Discrimination
- Third-degree price discrimination refers to a situation where the monopolist sells
to different buyers at different prices based on some observable characteristic of
the buyers.
Example
-
Senior citizens may be sold movie tickets at one price, while adults pay another
price, and children pay a third price.
-
Self-selection is not a problem since the characteristic defining the groups is
observable and verifiable, at least in principle.
-
e.g. You can always check an identity document to see if someone is eligible for the
senior citizen price or not.
Msc. Economics and Finance Lecture Slides
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Implications of third-degree price discrimination
-
Suppose that there are two types of demand for football tickets.
- Workers have demand pa (qa) = 100 − qa , while students have demand ps (qs) = 20
− 0.1 qs.
-
Suppose the marginal cost of an additional ticket is zero.
Msc. Economics and Finance Lecture Slides
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Number of tickets of each type to be sold
-
Set MR = MC in each market:
20 − 0.2 qs
= 0
100 − 2 qa
= 0,
- Thus, q*a = 50 and q*s = 100 as well.
- Workers tickets are sold at pa = 50.
-
Student tickets are sold at price ps = 10.
- Total profit:
50  50 + 100  10 = 2500 + 1000 = 3500.
-
Suppose that the stadium capacity is 151 seats and that all seats must be sold.
Msc. Economics and Finance Lecture Slides
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Should the remaining seat be sold to workers or students?
-
For each additional seat is sold to workers,
qa = 51, pa = 49.
- Total profit on workers sales is 2499
- Total profit becomes 3499.
-
For each additional seat is sold to students,
qs = 101, ps = 20 − 0.1(101) = 9.9.
- Total profit on student sales is 999.9.
- Total profit is 3499.9.
- It is better to sell the extra ticket to the students even though the price paid by the
workers is higher.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate
Reason
-
Consider the difference between the changes in marginal revenue and marginal
cost.
-
In either case selling an extra unit of output decreases total profit.
-
By selling an additional workers ticket, you have to lower the price more than
when selling another student ticket.
- Hence it is better to sell the ticket to a student, not because more is made on the
additional ticket, but because less is lost due to lowering price on the tickets sold
to all other buyers in that market.
Msc. Economics and Finance Lecture Slides
2016- Advanced Micro Economics
Bernard Banda PhD Candidate