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York University – ECON2350 – V. Bardis Answers to Practice Set 2

1. See notes.

2. Letting b = 2 gives q = f (L) = a(L − t)1/2. Then Jerry’s marginal product is

f

(L)

=

1 a(L

t)−1/2

2

Setting pf (L) = w gives Jerry’s ‘input demand’

L∗ =

ap 2 +t

2w

Substituting L∗ in the production function gives

q∗ = a(L∗ − t)1/2 ⇒ q∗ = a2p 2w

Is the latter equation Jerry’s supply? Almost, but not exactly. A complete answer has to

distinguish between the case where Jerry chooses how much to supply before he travels to the

forest (‘long-run’) and after he has gone there (‘short-run’). The latter is the easier one to deal

with. If Jerry has gone to the forest he has incurred the ‘travel cost’ wt. If he ﬁnds out while

there that the price is p he will supply as many units as the above equation speciﬁes. His proﬁt

will be

π∗

=

pq∗

wL∗

=

a2p p

w

ap

2
+t

= a2p2 − wt

2w 2w

4w

If instead he chooses to supply zero then his proﬁt will be π(0) = −wt 0, i.e., the more productive he is

the

more

he

will

supply

at

any

price

(supply

curve

shifts

down);

∂q ∂p

=

a2 2w

>

0,

i.e.,

higher

price

increases

quantity

supplied;

and

∂q ∂w

=

a2p 2w2

<

0,

i.e.,

an

increase

in

the

wage

will

reduce

supply

(supply curve shifts up). Finally, a change in t aﬀects only ﬁxed costs and so it has no eﬀect on

the

short-run

supply

(t

is

nowhere

to

be

seen

in

the

short-run

supply

function,

so

that

∂q∗ ∂t

=

0).

If Jerry decides before he goes to the forest (a ‘long-run decision’) then he will go only if his

proﬁt is not negative, that is,

π∗ = a2p2 − wt ≥ 0 4w