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

2. 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

3. 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

4. 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

5. 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

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

7. 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

8. Ex 5.2(3): Investment responsez* p

9. 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

10. 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

11. 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