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Exercise 3.3

Exercise 3.3. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 3.3(1) Question. purpose : to derive competitive supply function method : derive AC, MC. Ex 3.3(1) Costs. Total cost is: F 0 + ½ aq i 2 Marginal cost: aq i Average cost: F 0 / q i + ½ aq i

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Exercise 3.3

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  1. Exercise 3.3 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

  2. Ex 3.3(1) Question • purpose: to derive competitive supply function • method: derive AC, MC

  3. Ex 3.3(1) Costs • Total cost is: F0 + ½ aqi2 • Marginal cost: aqi • Average cost: F0/qi + ½ aqi • Therefore MC intersects AC where: • This is at output level q where: • At this point AC is at a minimum p where: • For q below q there is IRTS and vice versa

  4. Ex 3.3(1) Supply • If p > p the firm supplies an amount of output such that • p = MC • If p < p the firm supplies zero output • otherwise the firm would make a loss • If p = p the firm is indifferent between supplying 0 or q • in either case firm makes zero profits • To summarise the supply curve consists of :

  5. Ex 3.3(1): Supply by a single firm • Average cost p • Marginal cost • Supply of output q qi

  6. Ex 3.3(2) Question • purpose: to demonstrate possible absence of equilibrium • method: examine discontinuity in supply relationship

  7. Ex 3.3(2): Equilibrium? • AC,MC and supply of firm p • Demand, low value of b • Demand, med value of b • Demand, high value of b • Solution for high value of b is where Supply = Demand AC Supply (one firm)  MC qi

  8. Ex 3.3(2) Equilibrium • Outcome for supply by a single price-taking firm • High demand: unique equilibrium on upper part of supply curve • Low demand: equilibrium with zero output • In between: no equilibrium • Given case 1 “Supply = Demand” implies • This implies: • But for case 1 we need p≥p • from the above this implies

  9. Ex 3.3(3) Question • purpose: to demonstrate effect of averaging • method: appeal to a continuity argument

  10. Ex 3.3(3) Average supply, N firms • Define average output • Set of possible values for average output: • Therefore the average supply function is

  11. Ex 3.3(3) Average supply, limit case • As N the set J(q) becomes dense in [0, q] • So, in the limit, if p = p average output can take any value in [0, q] • Therefore the average supply function is

  12. q Ex 3.3(3): Average supply by N firms • Average cost (for each firm) p • Marginal cost (for each firm) • Supply of output for averaged firms q

  13. Ex 3.3(4) Question • purpose: to find equilibrium in large-numbers case • method: re-examine small-numbers case

  14. Ex 3.3(4) Equilibrium • Equilibrium depends on where demand curve is located • characterise in terms of (price, average output) • High demand • equilibrium is at (p, p/a) where p = aA / [a+b] • Medium demand • equilibrium is at (p, [A – p]/b) • equivalent to (p, bq) where b := a[A – p] / [bp] • Achieve this with a proportion b at q and 1–b at 0 • Low demand • equilibrium is at (p, 0)

  15. q Ex 3.3(4): Eqm (medium demand) • AC and MC (for each firm) • Supply of output (averaged) p • Demand • Equilibrium • Equilibrium achieved by mixing firms at 0 and at q 1b here b here q q*

  16. Ex 3.4: Points to remember • Model discontinuity carefully • Averaging may eliminate discontinuity problem in a large economy • depends whether individual agents are small. • Equilibrium in averaged model may involve identical firms doing different things • equilibrium depends on the right mixture

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