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Esercitazione 1: Teoria della Produzione

Esercitazione 1: Teoria della Produzione. Esercizio 1: Cobb-Douglas con input fissi e variabili. z. 2. z. 1. 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:. maximisation.

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Esercitazione 1: Teoria della Produzione

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  1. Esercitazione 1: Teoria della Produzione

  2. Esercizio 1: Cobb-Douglas con input fissi e variabili

  3. z 2 z 1 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:

  4. maximisation Differentiate the Lagrangean to get the FOCs: The l is an unknown: we need to eliminate it And rearrange:

  5. Manipulating the FOCs From the FOC: Use the constraint (the production function): Rearrange to get l:

  6. Deriving the conditional input demand functions Substitute this into this to get this:

  7. The cost function multiply these: by wiand then add to get this: Increasing with Q if g<1 Differentiate to get marginal cost:

  8. 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).

  9. Short run Results • So we get:

  10. 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: εp = γk/(1- γk). • gk falls if k is reduced • Same applies for conditional input demand: εw = ai/γk-1

  11. 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

  12. Esercizio 2:Impresa in concorrenza perfetta e aggregazione di imprese

  13. Total cost: • Marginal cost: • Average cost: • MC intersects AC at: For Q below this get IRTS and vice versa • This is where:

  14. Price at Q is: Identical this portion of MC • Price P separates regimes: Exactly two values here Cheaper to close down here

  15. P 10 9 8 7 6 5 4 3 2 1 Qi 0 0 0.5 1 1.5 If α = β = b =1; Q=1 AC Demand: high a Supply (one firm)  MC Demand: low a

  16. 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

  17. 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?

  18. If there is equilibrium Supply=Demand gives us • Using definition of P: • This is only valid if P > P: • This yields the condition:

  19. Equilibrium condition Supply=Demand now gives us • Outcome depends on which of three regimes applies to demand, high, medium or low

  20. High demand P = 4 α β; = N b ; • Medium demand. • Average output per firm is Qi = (a - 4 α β)/Nb • Achieve this by putting a proportion q at Q =α/β and 1–q at 0 where • Low demand. • Output per firm is 0

  21. Esercizio 3: Monopoly and competition Part 1: Competition

  22. Find Firm’s 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

  23. Q*= P - a —— b The Firm’s Supply Curve marginal cost a+bQ supply curve P P average cost F/Q+a+0.5bQ Q Q

  24. Monopoly and competition Part 2: Unregulated monopoly

  25. Find monopolist’s equilibrium Given the demand curve (AR), total revenue is So, MR is FOC for the monopolist (MR=MC) is Solving for Q we get And from this we have

  26. Monopolist’s equilibrium P a+bQ marginal cost F/Q+a+0.5bQ P** average cost c** average revenue A - 0.5bQ A - bQ marginal revenue Q** Q

  27. Monopoly and competition Part 3: Regulated monopoly

  28. 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

  29. effect of high price ceiling P (Output unchanged) marginal cost Pmax P** c** average revenue marginal revenue Q0 Q** Q

  30. effect of low price ceiling P (Output falls) marginal cost P** Pmax c** average revenue marginal revenue Q0 Q** Q

  31. effect of intermediate price ceiling P (Output rises to Q0) marginal cost P** Pmax c** average revenue marginal revenue Q0 Q** Q

  32. intermediate price ceiling (2) P (Output rises) marginal cost P** Pmax c** average revenue marginal revenue Q0 Q** Q

  33. 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. • Don’t be fazed by the presence of a discontinuity – everything is nice and regular either side of it.

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