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

Ex 5.2(1): Question 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

Ex 5.2(1): Find optimal investment Net income is y = pR1 + R2 + bp[1 – e–z] – z Maximise this where dy — = 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

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

Ex 5.2(2): Find supply of rice Maximised income is y*(p,b) = pR1 + R2 + bp[1 – exp(–z*(p,b))] – z*(p,b) Using the formula for optimal investment z* we find if bp >1: y*(p,b) = pR1 + R2 + bp[1 – 1/bp] – log (bp)) = pR1 + R2 + bp – 1 – log (bp)) otherwise y*(p,b) = pR1 + R2 Total output of rice is now R1 + b[1 – exp(–z*(p,b))] So, modifying answer to Ex 5.1, supply of rice is now y*(p,b) – k S*(p,b) = R1 + b[1 – exp(–z*(p,b))] – a ————— p where a 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

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

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

Ex 5.2(3): Investment response z* p

Ex 5.2(3): Supply response (i) If p  1/b maximised income is pR1 + R2 + bp – 1 – log (bp) So supply of rice is R2 – k – 1 – log (bp) S*(p,b) = [1–a ]R1+ ab – a ————————— p Differentiating: S*(p,b) R2 – k – log (bp) ———— = a ——————— p p2 S*(p,b) 1 – bp ———— = a ——— b bp If R2 > k positive for small p maybe negative for large p non-positive

Ex 5.2(3): Supply response (ii) If p < 1/b maximised income is pR1 + R2 So supply of rice is R2 – k S*(p,b) = R1 – a ——— p Differentiating: S*(p,b) R2 – k ———— = a —— p p2 S*(p,b) ———— = 0 b If R2 > k positive for all p < 1/b

Ex 5.2: Points to remember Make good use of answers to previous parts in each stage 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