Summary (almost) everything you need to
know about micro theory 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<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)><=zf(K,L), z>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 „price-
taker”)– 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
–Administrative regulations (e.g. Poczta Polska)
–Natural monopoly (railroad 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
higher – deadweight loss 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>MC– OR: forget about the „weaker” group
altogether
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
Strategy „left” is dominated by „right”
Will not be played up, down, middle and
right are rationalizable 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
left middle
right
up 2,2 4,1 1,3
down
6,1 2,5 2,2
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
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)
Low price
High price
Low price
1,1 3,0
High price
0,3 2,2
„prisoner’s dillema”
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-bq2-MC1)/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 player (Leader) decides on 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>MC
Before the exam
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