Frank Cowell: Microeconomics Exercise 3.3 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

Frank Cowell: Microeconomics Ex 3.3(1) Question purpose: to derive competitive supply function purpose: to derive competitive supply function method: derive AC, MC method: derive AC, MC

Frank Cowell: Microeconomics 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 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

Frank Cowell: Microeconomics 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 :

Frank Cowell: Microeconomics Ex 3.3(1): Supply by a single firm qiqi p   Average cost   Marginal cost   Supply of output q

Frank Cowell: Microeconomics Ex 3.3(2) Question purpose: to demonstrate possible absence of equilibrium purpose: to demonstrate possible absence of equilibrium method: examine discontinuity in supply relationship method: examine discontinuity in supply relationship

Frank Cowell: Microeconomics Ex 3.3(2): Equilibrium? AC MC  qiqi p   AC,MC and supply of firm   Demand, low value of b   Demand, high value of b Supply (one firm)   Solution for high value of b is where Supply = Demand   Demand, med value of b

Frank Cowell: Microeconomics 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

Frank Cowell: Microeconomics Ex 3.3(3) Question purpose: to demonstrate effect of averaging purpose: to demonstrate effect of averaging method: appeal to a continuity argument method: appeal to a continuity argument

Frank Cowell: Microeconomics Ex 3.3(3) Average supply, N firms Define average output Set of possible values for average output: Therefore the average supply function is

Frank Cowell: Microeconomics 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

Frank Cowell: Microeconomics Ex 3.3(3): Average supply by N firms p   Average cost (for each firm)   Marginal cost (for each firm)   Supply of output for averaged firms q q 

Frank Cowell: Microeconomics Ex 3.3(4) Question purpose: to find equilibrium in large-numbers case purpose: to find equilibrium in large-numbers case method: re-examine small-numbers case method: re-examine small-numbers case

Frank Cowell: Microeconomics 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,  q) where  := a[A – p] / [bp]   Achieve this with a proportion  at q and 1–  at 0 Low demand   equilibrium is at (p, 0)

Frank Cowell: Microeconomics Ex 3.3(4): Eqm (medium demand) p   AC and MC (for each firm)   Supply of output (averaged) q   Demand   Equilibrium q*   Equilibrium achieved by mixing firms at 0 and at q  here 1  here q 

Frank Cowell: Microeconomics Ex 3.4: Points to remember Model discontinuity carefully Model discontinuity carefully Averaging may eliminate discontinuity problem in a large economy 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 in averaged model may involve identical firms doing different things  equilibrium depends on the right mixture