Esercitazione 1: Teoria della Produzione

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Esercitazione 1 Teoria della Produzione
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Esercizio 1 Cobb-Douglas con input fissi e
variabili
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Solution cannot be at a corner
Put the production function in a more manageable
form
A quick check on the isoquant
Write down the Lagrangean
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maximisation
Differentiate the Lagrangean to get the FOCs
The l is an unknown we need to eliminate it
And rearrange
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Manipulating the FOCs
From the FOC
Use the constraint (the production function)
Rearrange to get l
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Deriving the conditional input demand functions
Substitute this
into this
to get this
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The cost function
multiply these
by wi and then add to get this
Increasing with Q if glt1
Differentiate to get marginal cost
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Short run Formulation
Again put the production function into a
convenient form
Constant in the short run

• So this is the only bit that is variable in the
  short run.
• But this means that the problem has exactly the
  same structure (but with different parameters).
• Therefore the solution has the same structure
  (but with different parameters).

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Short run Results

• So we get

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Short run MC and supply
Differentiating the cost function we get marginal
cost

• This forms the short-run supply curve for the
  firm
• Clearly the elasticity falls if gk falls ep
  ?k/(1- ?k).
• gk falls if k is reduced
• Same applies for conditional input demand ew
  ai/?k-1

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Points to remember

• Get the constraint into a convenient form
• Get a simple view of the problem by deriving ICs
• Use a little cunning to simplify the FOCs
• Re-use your solution for other problems that have
  the same structure

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Esercizio 2Impresa in concorrenza perfetta e
aggregazione di imprese
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• Total cost

• Marginal cost

• Average cost

• MC intersects AC at

For Q below this get IRTS and vice versa

• This is where

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• Price at Q is

Identical this portion of MC

• Price P separates regimes

Exactly two values here
Cheaper to close down here
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If a ß b 1 Q1
AC
Demand high a
Supply (one firm)
?
MC
Demand low a
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• Two such firms extra point half-way between
  green blobs
• Four such firms points at quarter, half,
  three-quarter positions between blobs.
• Infinity of these firms all points between blobs

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• High demand unique equilibrium on upper part of
  supply curve
• Very low demand equilibrium with zero output.
• In between no equilibrium for a single firm?

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• If there is equilibrium SupplyDemand gives us

• Using definition of P

• This is only valid if P gt P

• This yields the condition

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• Equilibrium condition SupplyDemand now gives us

• Outcome depends on which of three regimes
  applies to demand, high, medium or low

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• High demand

P 4 a ß N b

• Medium demand.
• Average output per firm is

Qi (a - 4 a ß)/Nb

• Achieve this by putting a proportion q at Q
  a/ß and 1q at 0 where

• Low demand.
• Output per firm is 0

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Esercizio 3 Monopoly and competition
Part 1 Competition
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Find Firms Supply Curve
Integrate MC in the question to get total costs
Divide by Q to get average costs
Differentiate to find minimum AC at
Where average costs are
Given a price can then find output from supply
curve
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The Firms Supply Curve
marginal cost
abQ
supply curve
P
P
average cost
F/Qa0.5bQ
Q
Q
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Monopoly and competition
Part 2 Unregulated monopoly
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Find monopolists equilibrium
Given the demand curve (AR), total revenue is
So, MR is
FOC for the monopolist (MRMC) is
Solving for Q we get
And from this we have
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Monopolists equilibrium
P
abQ
marginal cost
F/Qa0.5bQ
P
average cost
c
average revenue
A - 0.5bQ
A - bQ
marginal revenue
Q
Q
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Monopoly and competition
Part 3 Regulated monopoly
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Introduce price ceiling
A price ceiling alters the effective demand curve
So AR is now
Multiply by Q and then differentiate to get MR
Note that MR is discontinuous, exactly where AR
is kinked Effect of price ceiling depends on
position of MC relative to this discontinuity
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effect of high price ceiling
P
(Output unchanged)
marginal cost
Pmax
P
c
average revenue
marginal revenue
Q0
Q
Q
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effect of low price ceiling
P
(Output falls)
marginal cost
P
Pmax
c
average revenue
marginal revenue
Q0
Q
Q
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effect of intermediate price ceiling
P
(Output rises to Q0)
marginal cost
P
Pmax
c
average revenue
marginal revenue
Q0
Q
Q
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intermediate price ceiling (2)
P
(Output rises)
marginal cost
P
Pmax
c
average revenue
marginal revenue
Q0
Q
Q
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Points to remember

• Make good use of a helpful diagram to see the
  problem
• Re-use the solution to one part of the problem to
  build the next.
• Dont be fazed by the presence of a discontinuity
  everything is nice and regular either side of
  it.