Frank Cowell: Microeconomics Exercise 5.2 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

Frank Cowell: Microeconomics Ex 5.2(1): Question purpose: construct a simple model of investment in the context of household supply purpose: construct a simple model of investment in the context of household supply method: combine answer to Ex 5.1 and a straightforward income-maximisation problem method: combine answer to Ex 5.1 and a straightforward income-maximisation problem

Frank Cowell: Microeconomics Ex 5.2(1): Find optimal investment Net income is y = pR 1 + R 2 + bp[1 – e –z ] – z Maximise this where dydy — = bpe –z – 1 = 0 dz Solve this to get z = log (bp), which is positive if bp  1. So optimal investment is z * (p,b) = max (log (bp), 0) Note dependence on price p and on investment productivity b

Frank Cowell: Microeconomics Ex 5.2(2): Question method: Use the result from part 1 Use the result from part 1 Work out the modified expression for income Work out the modified expression for income Combine this with answer to Ex 5.1 Combine this with answer to Ex 5.1

Frank Cowell: Microeconomics Ex 5.2(2): Find supply of rice Maximised income is y * (p,b) = pR 1 + R 2 + bp[1 – exp(–z * (p,b))] – z * (p,b) Using the formula for optimal investment z * we find   if bp >1: y * (p,b) = pR 1 + R 2 + bp[1 – 1/bp] – log (bp)) = pR 1 + R 2 + bp – 1 – log (bp))   otherwise y * (p,b) = pR 1 + R Using answer to Ex 5.1 supply of rice is now y * (p,b) – k S * (p,b)  R 1  –  ————— p   where  and k are parameters of the utility function   and y * (p,b) takes one of the two forms above   depends on whether or not p > 1/b

Frank Cowell: Microeconomics Ex 5.2(3): Question method: Use the result from parts 1 and 2 Use the result from parts 1 and 2 Differentiate with respect to p and b to get responses Differentiate with respect to p and b to get responses

Frank Cowell: Microeconomics Ex 5.2(3): Investment response Optimal investment is z * (p,b) = max (log (bp), 0) If p  1/b then, differentiating:  z * (p,b) 1 ————  — > 0  p b   z * (p,b)  1 ————  — > 0  b p Otherwise z * is constant at 0

Frank Cowell: Microeconomics Ex 5.2(3): Investment response p z*z*

Frank Cowell: Microeconomics Ex 5.2(3): Supply response (i) If p  1/b maximised income is pR 1 + R 2 + bp – 1 – log (bp) So supply of rice is R 2 – k – 1 – log (bp) S * (p,b)  [1–  ]R 1 +  b –  ————————— p Differentiating:  S * (p,b) R 2 – k – log (bp) ————   ———————  p p 2   S * (p,b) 1 – bp ————   ———  b bp If R 2 > k positive for small p maybe negative for large p If R 2 > k positive for small p maybe negative for large p non-positive

Frank Cowell: Microeconomics Ex 5.2(3): Supply response (ii) If p < 1/b maximised income is pR 1 + R 2 So supply of rice is R 2 – k S * (p,b)  R 1 –  ——— p Differentiating:  S * (p,b) R 2 – k ————   ——  p p 2   S * (p,b) ————   b If R 2 > k positive for all p < 1/b

Frank Cowell: Microeconomics Ex 5.2: Points to remember Make good use of answers to previous parts in each stage Make good use of answers to previous parts in each stage When analysing optimal investment take care about the corner solution When analysing optimal investment take care about the corner solution Use the two cases (corner, interior solution) in modelling the agent’s response to price and productivity changes Use the two cases (corner, interior solution) in modelling the agent’s response to price and productivity changes