# q 2

```Summary
(almost) everything you need to
in 30 minutes
Production functions
Q=f(K,L)
 Short run: at least one factor fixed
 Long run: anything can change
 Average productivity: APL=q/L
 Marginal productivity: MPL=dq/dL
 Ave prod. falls when MPL&lt;APL
 MPL falls, eventually (the „law” of diminishing
marginal productivity)

Isoquants
 All
combinations of factors that allow
same production
Stolen from: prenhall.com
Substitution
 MRTSKL=-MPK/MPL
 (how
many units of labor are
necessary to replace one unit of
capital)
 MRTS is the inverse of the slope of
the isoquant
Economies of scale
 f(zK,zL)&gt;&lt;=zf(K,L),
z&gt;1
 Shows whether large of small
production scale more efficient
 Example: Cobb-Douglas:
 (zK)α(zL)β=z(α+β)KαLβ
 Thus economies of scale are
constant (increasing, decreasing) if
α+β equal to (greater than, smaller
than) 1.
Costs
Economist’s and accountant’s view
 Opportunity costs
 Sunk costs („bygones are bygones”)
 TC(q)=VC(q)+FC
 ATC(q)=TC(q)/q
 MC(q)=dTC(q)/dq
 MC assumed to go up, eventually
 AVC(q) and ATC(q) minimum when equal
to MC

Cost minimization
 Cost
minimization with fixed
production
 Dual problem to maximizing
production with fixed costs
Perfect competition


Assumptions
– Many (small) firms
– New firms can enter in the long run
– Homegenous product
– Prices known
– No transaction or search costs
– Prices of factors (perceived as) constant
– Market price perceived as constant (firm is a „pricetaker”)
– Profit maximisation
– Decreasing economies of scale
Main feature: perfectly elastic demand for a single firm
Perfect competition-analysis
Magical formula: MC(q)=P
 Defines inverse supply f. for a single firm
 Aggregate supply: S(P)=ΣSi(P)
 In the long run:
– Profit=0
– P=min(AC)
– S=D
 Efficiency:
– Lowest possible production cost
– Production level appropriate given
preference

Monopoly
 Sources
of monopolistic power
Poczta Polska)
networks)
– Patents
– Cartels (the OPEC)
– Economies of scale
 The magic formula: MR(q)=MC(q)
Monopoly-cont’d







By increasing production, monopoly negatively
affects prices
Thus MR lower than AR(=p)
E.g. with P=a+bq:
TR=Pq=(a+bq)q=aq+bq2
MR=a+2bq
Another useful formula: link with demand
elasticity:
MR=P(q)(1+1/ε)
Thus always chooses such q that demand is
elastic
Inefficiency: production lower than in PC, price
Plus, losses due to rent-seeking
Monopoly: price discrimination
 Trying
to make every consumer pay
as much as (s)he agrees to pay
 1st degree (perfect price disc. –
every unit sold at reservation price),
– production as in the case of a
perfectly competitive market
– (thus no inefficiency)
– No consumer surplus either
Price discrimination-cont’d
 2nd
degree: different units at
different prices but everyone pays
the same for same quantity
 Examples: mineral water, telecom.
 3rd degree: different people pay
different prices
– (because different elasticities)
– E.g.: discounts for students
Two-part tarifs
Access fee + per-use price
 Examples: Disneyland, mobile phones,
vacuum cleaners
 Homogenous consumers:
– Fix per-use price at marginal cost
– Capture all the surplus with the access
fee
 Different consumer groups
– Capture all the surplus of the „weaker”
group
– Price&gt;MC
– OR: forget about the „weaker” group

Game theory
Used to model strategic interaction
 Players choose strategies that affect
everybody’s payoffs
 Important notion: (strictly) Dominant
strategy – always better than other
strategy(ies)

Example
left
middl
e
4,1
right
Strategy „left” is
dominated by „right” up
2,2
1,3
 Will not be played
2,5
2,2
 up, down, middle and dow 6,1
right are rationalizable n
 Nash equilibrium: two strategies that are
mutually best-responses (no profitable
unilateral deviation)
 No NE in pure strategies here
 NE in mixed strategies to be found by equating
expected payoffs from strategies

Repeated games
 Same
(„stage”) game played multiple
times
 If only one equilibrium, backward
induction argument for finite
repetition
 What if repeated infinitly with some
discount factor β?
Repeated games-cont’d
„prisoner’s dillema”





Low
price
High
price
Low
price
1,1
High
price
3,0
0,3
2,2
Consider „trigger” stragegy: I play high but if you play
low once, I will always play low.
If you play high, you will get 2+2β+2β2+…
If you play low, you will get 3+β+β2+…
Collusion (high-high) can be sustained if our βs are .5
or higher
(though low-low also an equilibrium in a repeated
game)
Sequential games
A tree (directed graph with no cycles) with
nodes and edges
 Information sets
 Subgame: a game starting at one of the
nodes that does not cut through info sets
 SPNE: truncation to subgames also in
equilibrium
 Backward induction: start „near” the final
nodes
 Example: battle of the sexes

Oligopoly: Cournot
 Low
number of firms
 Firms not assumed to be price-takers
 Restricted entry
 Nash equilibrium
 Cournot: competition in quantities
 Example: duopoly with linear
demand
Cournot duopoly with linear demand
 P=a-bQ=a-b(q1+q2)
 Cost
functions: g(q1), g(q2)
 Π1=q1(a-b(q1+q2))-g(q1)
 Optimization yields q1=(a-bq2MC1)/2b
 (reaction curve of firm 1)
 Cournot eq. where reaction curves
cross
 Useful formula: if symmetric costs:
q1 =q2 =(a-MC)/3b
Oligopoly: Stackelberg
 First
quantity
 Follower react to it
 SPNE found using backward
induction:
Π2=q2(a-b(q1+q2))-TC2
Reaction curve as in Cournot:
q2= (a-bq1-MC2)/2b
 For constant MC we get:
q1 =2q2 =(a-MC)/2b
Comparing Cournot and Stackelberg
 Firm
2 reacts optimally to q1 in either
 But firm 1 only in Cournot
 Firm 1 will produce and earn more in
vS
 Firm 2 will produce and earn less
 Production higher, price lower in
Stackelberg if cost and demand are
linear
Oligopoly: plain vanilla Bertrand
 Both
firms set prices
 Basic assumption: homogenous
goods
 (firm with lower price captures the
whole market)
 Undercutting all the way to P=MC
 If firms not identical, the more
efficient one will produce and sell at
the other’s cost
More realistic: heterog. goods
 Competitor’s
price affects my sales
negatively
 (but not drives them to 0 when just
slightly lower than mine)
 Example:
q1=12-P1+P2
TC1=9q1, TC2=9q2
q1=12-P2+P1
P1=P2=10&gt;MC
Before the exam
 Look
up www.miq.woee.pl