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Simon Board
Economics 11: Solutions to Practice Problems 5 (Week 6)
November 4, 2008
1. Complements and Substitutes
Suppose
u
(
x
1
,x
2
) =
x
1
x
2
. Are
x
1
and
x
2
gross complements or substitutes? Are they net
complements or substitutes?
Solution
Marshallian demand is given by
x
*
1
=
m/
2
p
1
and
x
*
2
=
m/
2
p
2
. Since demand for good
i
is
independent of
p
j
these are neither gross complements or gross substitutes.
Hicksian demand is given by
h
1
= (
up
2
/p
1
)
1
/
2
and
h
2
= (
up
1
/p
2
)
1
/
2
. Since
h
i
is increasing in
p
j
, these goods are net substitutes.
2. Choice with Partial Subsidies
Suppose an agent chooses how much bread,
x
1
, and wine,
x
2
, to consume. The agent views the
goods as perfect substitutes and has utility function:
u
(
x
1
,x
2
) =
x
1
+
x
2
The price of bread is
p
1
= 4, while the price of wine is
p
2
= 2. The agent has income
m
= 4.
(a) Solve for the agent’s optimal consumption.
(b) In order to encourage the consumption of bread, the Government subsidises the ﬁrst unit
by $1. Hence the ﬁrst unit costs
p
1
= 3, while each unit thereafter costs
p
1
= 4. (i) Draw the
agent’s budget constraint. (ii) Solve for her optimal consumption. (iii) How much does the
subsidy cost the Government?
(c) The Government increases it’s subsidy on the ﬁrst unit to $3. Hence the ﬁrst unit costs
p
1
= 1, while each unit thereafter costs
p
1
= 4. (i) Draw the agent’s budget constraint. (ii)
Solve for her optimal consumption. (iii) How much does the subsidy cost the Government?
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 Fall '08
 cunningham
 Economics

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