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Competitive Markets
10. The Firm's Profit-Maximizing Output Decision
Assume firms behave in such a way as to maximize their profits ( π ).
π = total revenue (TR) - total costs (TC)
A necessary condition for a level of output to be profit-maximizing is that it solves:
∂π ∂TR ∂TC
=
−
=0
∂y
∂y
∂y
Defining marginal revenue (MR) as the rate of change of total revenue as output changes
and recalling that marginal cost (MC) is the rate of change of costs as output changes we
have
MR = MC
as a requirement for a profit-maximizing level of output.
Note that at this stage no assumptions have been made about the structure of the market.
But we do need that costs of production are the minimum costs of production associated
with any given level of output (which by construction they are). Similarly we need that
total revenue is maximized for any particular level of output.
Some market structures would be such that the firm has no options beyond selling its
entire output at a single price, that price conceivably changing as the level of output sold
changes. In that case:
TR = Py
∂TR
∂P
MR=
= P+
y
∂y
∂y
i.e. The additional revenue generated by an additional unit of output is the price received
for that unit of output plus any (typically negative) impact that additional unit of output
has on the price consumers pay for 'existing' output.
The market structure determines the way that revenue depends on the level of output and
how changes in the level of the firm's output impact the price it receives for its output.
11. (Perfectly) Competitive Market Structure
Assumptions:
- Large Numbers
There are a sufficiently large numbers of both sellers (firms) and buyers (consumers) that
no individual entity is responsible for a significant proportion of sales or purchases. The
key part of this assumption is that no firm or consumer has any individual control over
the prevailing market price and thus behaviour is 'price-taking'.
- Perfect Information
All firms have access to the same production technology. All firms and consumers are
aware of the price in any transaction, which helps justify the notion of a single marketclearing price.
- Product Homogeneity
Each firm is selling the same thing as any other firm. What matters to consumers is how
much of the product they buy, not which particular units of it they buy. (These things also
help justify the single market price.)
- Perfect Mobility of Resources
Inputs can costlessly move from one market to another.
- Independence
This is not something necessary for a competitive market structure, but is an assumption
we make throughout this course.. This assumption requires that any consumer's utility
from consumption depends solely on his own consumption (and not other consumers'
consumption nor the production decisions of firms). It also requires that any firm's level
of output depends entirely on the bundle of inputs it uses (and not other firms' behaviour
nor any consumer's behaviour). When this assumption is not satisfied, there can still be a
competitive equilibrium but it typically won't be a very desirable outcome. We'll make
this assumption throughout the course. Examining what happens when it doesn't hold is
the stuff of courses like environmental economics, public economics, health economics,
etc.
12. The Short-Run Output Decision in Competitive Markets
In a competitive market, an individual firm can't affect the market price so:
MR = P
And thus the profit-maximizing level of output is found where:
SMC = P
Assuming SMC is upward-sloping where it equals the price.
Draw a graph of SMC for both cases and analyse how the sign of the slope of SMC
affects SMC=P being a profit-maximizing level of output.
Also draw a graph with two intersections of price and SMC with the profit function
graphed below.
Also assuming P>=AVC at the level of output where SMC=P.
The individual firm's supply curve gives the profit-maximizing level of output as a
function of the market price. y*(P)
Graphically, it is the SMC curve where it is
i) upward-sloping
ii) above or equal to AVC
(Below or equal to AVC, y*(P)=0.)
e.g. 1) What is the firm's supply function if SMC=y2-4y+5?
2) What is the firm's supply function if VC=(1/2)ay2?
13. Market Supply Curve
In a competitive market each firm's profit-maximizing output decision is independent of
the other firms' levels of output (because of price taking behaviour).
So the market supply curve (a.k.a. industry supply curve), relating price to output of all
firms together, is the horizontal sum of all firms in the market.
e.g. 1) There are 10 identical firms with short-run supply functions y*(p)=p/5. What is
the aggregate supply function?
2) There are two firms in one market. The first firm has a supply function y1*(p)=p-5
if p>5 and y1*(p)=0 if p ≤ 5. The second firm has y2*(p)=p-3 if p>3 and y2*(p)=0 if p ≤ 3.
What is the aggregate supply function?
14. Short Run Competitive Equilibrium and the Gains from Trade
The market demand function, D(p), at any given price, shows how much consumers
wish to purchase at that price. Or alternatively, for any given level of output, shows the
maximum amount consumers will pay for that much output.
Short-Run Competitive Equilibrium occurs where D(p) and Y*(p) intersect.
If D(p)>Y*(p) expect p to rise. If D(p)<Y*(p) expect p to fall.
e.g. Suppose D(p)=100-p and Y*(p)=4+2p. What is the short-run competitive
equilibrium price and quantity?
Consumer's Surplus: The consumer's surplus associated with any unit of output is the
difference between the most the consumer would have been willing to pay for it and what
they actually had to pay. i.e. the vertical distance between demand and price. The
consumers' surplus associated with a market is the total of all individual units of output's
consumer's surpluses. i.e. the area under demand and above price up to the quantity
traded.
Producer's Surplus: The producer's surplus associated with any unit of output is the
difference between what the producer received for their output and the least they would
have been willing to accept. i.e. the vertical distance between price and supply. The
producers' surplus associated with a market is the total of all individual units of output's
producer's surpluses. i.e. the area below price and above supply up to the quantity traded.
Total Surplus: The sum of producers' surplus, consumers' surplus, and anyone else's
(usually government) surplus associated with the operation of a market.
A competitive equilibrium results in maximum total surplus.
Note some of the assumptions implicit in arriving at this result:
Independence of individual demands so that, for example, the good is not 'shared' like,
say, a highway, park, or a police force.
Independence of individual supplies so that, for example, one firm's level of output
doesn't affect another's costs as for example pollution could. (Think of Alberta's dairy
and oil industries.)
Perfect information: consumers and firms know everything relevant to the operation of
the market. For example, they know what prices all firms post, and there is no hidden
information as with, say, used cars or medical care.
15. Long Run Competitive Equilibrium
In the long run all inputs are variable. If we view entry and exit from a market as being
an adjustment of the fixed input from zero or to zero, then in the long run firms can enter
and exit a market, but in the short run they cannot.
Equilibrium involves stability so in a long run equilibrium for some particular market,
firms must not want to adjust the quantity of their fixed input. This implies that in the
long run, conditions must be such that firms do not want to enter or exit the market.
Define long run profits as the maximum profits attainable in a market given the price. i.e.
profits at the profit maximizing level of output when the output is produced using a costminimizing bundle of inputs.
Suppose that a firm wishes to exit the market. That means the firm has better options for
the use of the resources it employs to produce in the market. So the cost of staying in the
market (the value of the inputs in their best alternative use) is greater than the benefit. i.e.
long run profits are negative. Similarly, if long run profits are negative, a firm would
wish to exit the market.
Suppose that a firm wishes to enter the market. Then the market offers higher benefits
than the use of resources used to produce in it would provide in any other market. So the
cost of operating in the market must be less than the benefit. i.e. long run profits are
positive. Similarly, if long run profits are positive, a firm would wish to enter the market.
Thus for a market to be in long-run equilibrium, profits must be zero.
Draw an individual firm's long run average cost curve and long run marginal cost curve.
Assume LRAC is U-shaped.
Label the price corresponding to the minimum level of long run average cost P*.
Consider a price above P*. Use LRMC to find profit maximizing output at this price.
Note that since LRAC is upward-sloping here, LRMC>LRAC, so P>LRAC, so profits
are positive. This makes firms enter the market, increasing supply, and lowering price.
Consider a price below P*. Use LRMC to find the profit maximizing level of output at
this price. Since LRAC is downward-sloping here, LRMC<LRAC, so P<LRAC, so
profits are negative. This makes firms leave the market, decreasing supply, and
increasing price.
Thus in the long run, firms enter or exit a market until price is equal to the minimum
value of LRAC.
The level of output for an individual firm that achieves the minimum of LRAC is called
the efficient scale of production. Note that if you want a market to produce some
particular level of output, there is no less costly way to do it than to have all firms
producing at the efficient scale of production and this is what the competitive market will
achieve in the long run.
Summary:
In long run equilibrium, price is always equal to the minimum of LRAC and all firms
produce at the efficient scale of production.
Example:
Suppose a constant cost competitive market has firms all with long run costs given by:
TC = y3 - 4y2 + 7y and demand given by D(P) = 240 - 4P. How much does each firm
produce? What is the equilibrium price? How many firms are there?
16. Long Run Supply
If long run equilibrium for firms' behaviour can only be established when price is equal to
the minimum value of LRAC, then the long run supply curve is a line at that price.
Now relax the assumption that input costs are constant. We still assume that from the
individual firm's point of view they are constant, but that as the industry grows and total
output increases, input costs change. This gives us two additional cases besides this
constant-cost one.
Increasing-cost case: As market output increases, input cost(s) increase.
Suppose there is a price greater than the minimum of LRAC (say, because demand
increased from a long run equilibrium situation). Existing firms earn profit so other firms
enter. Output increases so input cost(s) increase. This increases LRAC at every level of
output (see the comparative statics lecture) and therefore the minimum of LRAC
increases. Thus when market equilibrium is re-established where price equals the
minimum of LRAC, that price will be higher than before. Thus in this case, long run
supply is upward-sloping.
Decreasing cost case: As market output increases, input cost(s) decrease.
Same as above in reverse. As market output increases, LRAC falls for each firm.
Equilibrium will be re-established at a lower price resulting in a downward-sloping
demand curve.
17. Analysis Using Supply and Demand
'Supply' refers to the supply curve. 'Quantity supplied' refers to the amount of output all
of the firms in an industry actually produce or want to produce.
Supply will shift if the costs of inputs or technology changes (because these things
change the costs of production) but not in response to a change in demand. In response to
a change in demand, price and quantity supplied will change.
'Demand' refers to the demand curve. 'Quantity demanded' refers to the amount of output
all of the consumers in an industry wish to purchase.
Demand will shift if incomes, preferences, prices of related goods, or expectations
change but not in response to a change in supply. That will change price and quantity
demanded.
Quotas
Consider an effective quota--one that restricts quantity to a level less than what would
otherwise be traded in the market. It reduces total surplus (by the triangle shaped area to
the right of the quota line) and this reduction is the dead-weight loss (DWL) associated
with the quota. It also transfers welfare from producers to consumers since the price has
risen.
Note that the quotas create rent for their owners (income earned through the ownership of
factors of production that are in fixed supply). When introduced, the quotas create gains
for their owners. The quotas then have a market value equal to the present value of the
stream of rents they will earn, so that if a firm enters the market by purchasing a quota,
they will still earn zero profits.
Note that the textbook doesn't point out that if quotas are transferable, high-cost
producers will be driven out of the market.
Price Controls
Price controls take two basic forms: ceilings (a maximum price) and floors (a minimum
price). A price control is effective if it actually changes the market price. i.e. a floor
above the market price or a ceiling below the market price.
Consider an effective price ceiling. Quantity supplied is less than quantity demanded and
a dead-weight loss exists. Additionally, welfare is transferred from firms to consumers.
This, however, is not the full story.
Consider that if the reduction in quantity were achieved via a quota, price would rise to
the level of the demand curve and the higher price would ensure that those that value the
good most are the ones that receive it. This is no longer the case with the price ceiling
and increases the welfare loss of the ceiling because consumers' surplus will be smaller
than it would with the equivalent quota.
There will still be pressure for the price to rise. If firms can figure out some way to
increase the effective price they have an incentive to do so. Consider for example rent
control. Key money, damage deposits, finder's fees, sublet fees, maintenance and
furniture rental all provide means for the landlord to effectively raise the price of renting
an apartment when they are forbidden to increase the rent they charge. This will have
two effects: it will reduce the transfer to tenants and reduce the misallocation to people
with lower willingness to pay.
Taxes
Consider first a market equilibrium algebraically without a tax:
QD = QD(P) is demand
QS = QS(P) is supply
QD = QS is our equilibrium condition.
This gives us three equations and three variables to solve for (QD, QS, and P) so finding
or characterising equilibrium is just a matter of solving these three equations together.
We will analyse an excise tax--a tax that involves some fixed amount of dollars paid for
every unit traded.
An excise tax results in the price consumers pay being greater than the price firms receive
by the amount of the tax: PD = PS + t.
Supply with the tax depends solely on the price firms receive: QS=QS(PS)
Demand with the tax depends solely on the price consumers pay: QD=QD(PD)
The equilibrium condition is unchanged: QD = QS
Now we have four equations and four unknowns to solve for. Note that we didn't need to
know who the government collects the tax from in order to fully characterise the
outcome.
The proportion of the tax that shows up as an increase in the price to consumers: (PC-P)/t
(P is the pre-tax price) is the tax burden (or tax incidence) of consumers. The proportion
of the tax that shows up as a decrease in the price to firms: (P-PS)/t is the tax burden of
firms. In this analysis, the legal tax incidence (who, by law, is responsible for giving the
government money) has no bearing on the economic tax incidence.
Use diagrams to analyse how the tax incidence depends on elasticity of demand and
elasticity of supply. The higher the elasticity of supply, the less the firms' tax incidence.
The higher the elasticity of demand, the less the consumers' tax incidence.
18. Applications of Supply and Demand
Natural Resources
Consider the following argument:
Many natural resources are in fixed supply: oil, land, minerals, metals. Population
growth increases geometrically and therefore so does demand for these natural resources.
Therefore at some point in the future the available stock of natural resources will be
unable to meet the requirements of the world's economy. (Which will be very bad.)
Consider the validity of this argument if "available stock" means quantity supplied and
"requirements" means quantity demanded.
This is how, for example, the textbook interprets these sorts of arguments. This form of
the argument can be dismissed by noting that prices will increase until they are equal.
Consider the validity of this argument if "available stock" means supply and
"requirements" means demand.
This would presumably require supply and demand such that they never intersected no
matter how high the price gets. This seems like a not unreasonable characterisation of a
supply curve for some non-renewable resource, but not for a demand curve.
An allegedly influential work that relies on these arguments:
The Limits to Growth: A Report to the Club of Rome
The only mention of prices in the 12 page abstract:
"As resource prices rise and mines are depleted, more and more capital must be used for
obtaining resources, leaving less to be invested for future growth."
Unless of course the capital stock increases enough or technology improves.
Additionally, substitutes for these high-priced resources might be found. Also it's not
clear that more capital must be used to obtain a quantity of resources that is presumably
smaller since its price has risen. In fact, the inclusion of the reference to prices points
towards this means by which the prediction might be flawed while it is arguably worded
as if the rising prices buttress the authors' argument.
The textbook neatly summarises a typical laissez-faire economist's reaction to such
positions:
"Human history provides one example after another of exogenous shocks resulting in
high prices, followed by the development of new substitutes, new methods of production,
and new sources of supply....Our equilibrium model suggests we should be optimistic
about the future of the planet, not doomsayers."
The main problem with this sort of position is that the exogenous shocks can result in
painful transitions to new less-desirable equilibria. Consider, for example, the years of
stagflation in North America brought on by the exogenous shock of higher oil prices
resulting from the formation of OPEC.
Home Heating
The cost of heating is given by TC = pHB(Ti-To)
where:
Ti is the temperature inside
pH is the cost of heating (dollars per unit heat energy)
To is the temperature outside
B is the barrier to heat loss
B reflects the various factors that quality of insulation, size and shape of the home.
(Higher B corresponds to less insulation.)
Consider two homes that are identical except for the outside temperature—one house is
in a colder climate. Graph TC of interior temperature. Graph MC of interior
temperature. Assuming that demand for heat is the same in both homes, what
temperature do they choose?
Now assume that in the house in the colder climate, the insulation is better and thus B is
lower. Add this 3rd house's TC and MC curves to the other diagrams.
The end result should be that the lower marginal cost of heat in the better-insulated house
of the colder climate is expected to have a higher interior temperature.
While this is an interesting analysis of what people might do when making decisions
about home heating, it strikes me as odd that the textbook authors consider this an
example of applying the competitive model.
Crime
The textbook's analysis:
For our supply and demand diagram here we put the number of crimes committed on the
horizontal axis and the marginal costs and benefits on the horizontal axis. Note that
demand here represents the net marginal benefit of crime rather than, say, how much
crime people wish to purchase. The supply of crime represents the marginal cost of
committing crimes which is largely a reflection of the value of alternative uses of the
time involved (i.e. legal employment). We expect it to be upward-sloping to reflect the
fact that some people have low potential non-crime earnings and so are the first to engage
in crime and represent the producers at low quantities. Equilibrium occurs where demand
and supply cross resulting in some quantity of crime.
This suggests two ways of lowering crime: increase the costs of crime (decrease supply)
or decreasing the benefits of crime (decrease demand). The benefits can be decreased by
increasing penalties for people caught committing crimes. The cost can be increased by
increasing outside options, say by improving social services (welfare) or the employment
prospects of low earners.
Why I think the textbook's analysis is flawed:
Why is demand downward-sloping? Presumably because some crimes involve more
benefit than others. e.g. stealing the Mona Lisa versus stealing someone's car versus
stealing a pack of gum. Thus when we draw the demand curve, the crimes are ordered
according their net benefit. But when we draw the supply curve, the crimes are ordered
according by the outside options of the perpetrator. i.e. we can't meaningfully put these
things on the same graph. One way of looking at the problem of applying the supply and
demand framework here is that the assumption of homogeneity has been violated. In its
proper application, the marginal value of a widget doesn't depend on who made it nor on
any attribute of that particular widget itself. It depends on who is consuming it and how
many widgets they already have; give that person a different widget and it has the same
marginal value as the old one did. This is not true of these crimes of different value since
giving a person a different crime might, say, leave them with a pack of gum instead of the
Mona Lisa.
The result of this is that there is no guarantee that only people with costs below the
equilibrium price of crime are the only ones who commit crimes nor that only crimes
with benefits above the price of crime are committed. e.g. If I had the option of working
as a medical doctor (cost above price), it might still pay to steal the Mona Lisa. Similarly
if I had no employment options I might turn to a career of stealing gum (benefit below
price). This may seem somewhat realistic, but it is at odds with the competitive result.
This sort of objection can be addressed by assuming that crimes are transferable. i.e. the
potential medical doctor works as a doctor but hires the potential gum-stealer to instead
steal the Mona Lisa and pays the going competitive price for crime; the doctor works and
the gum remains unstolen. But this effectively leaves out whole categories of crime such
as rape, murder, assault, and traffic violations (because they aren't transferable) and
leaves perhaps only theft and robbery.
This is not to say that it's not possible to affect crime rates by affecting the costs and
benefits associated with criminal activity. Just that the competitive model is probably not
a very good way to think about this.
The Marriage Market
The Textbook's Analysis (and my comments in brackets):
Assume all males are the same and all females are the same. (This is, presumably, so
that husbands-as-a-good and/or wives-as-a-good are a homogenous product.) Draw
demand and supply curves for the marriage market noting that the supply of husbands is
the same as the demand for wives and the supply of wives is the same as the demand for
husbands. (Why these curves have a non-zero slope is a bit of a mystery to me. If the
supply curve for husbands involves some high-cost husbands a) what about them is
different from the low-cost husbands given the assumption that all men are the same and
b) why are they also the source of the high value being placed on a wife when this thing
is viewed as a demand curve for wives?) When the supply of men is reduced, the price of
a husband increases and this price could include pre-marital sex. So when the men go
off to war, this explains the increase in out-of-wedlock births. (This might be valid if
most of the out-of-wedlock births were attributable to the men that didn't go off to war.
The same reasoning would suggest that an influx of men to fight a war would result in a
decrease in out-of-wedlock births.)
As with the crime example, it seems possible one might be able to analyse usefully the
way that social, political, cultural, or economic changes affect the costs and benefits of
marriage to men or women and thus their behaviour in the marriage market. I doubt that
the competitive model would ever be of much use here.