301_mid1as10.pdf

ECO 301 – Summer 2010

Instructor: Dr. Michael Malcolm

Instructions: You can use any written materials you would like in completing this

exam, and a calculator.

Statement of academic honesty:

This exam entirely reflects my own work. I have not received assistance from

anyone or given assistance to anyone in completing this exam

Signature: _________________________________

Name:

_____________________________________

a. A price ceiling is a regulated price under the equilibrium price. This means that QS <QD and so supply controls the quantity exchanged:

300 200 100 P P = + ⇒ =

b. A price floor is a regulated price above the equilibrium price. This means that QD <QS and so demand controls the quantity exchanged:

300 1000 4 175 P P = − ⇒ = Problem 2

The first and third observations can be used to estimate a Marshallian price elasticity:

20 40 % 30 1 20 10 % 15 m Q P ε − ⎛ ⎞ ⎜ ⎟

Δ _{⎝ ⎠}

= = = −

−

Δ ⎛ ⎞

⎜ ⎟

⎝ ⎠

The first and second observations can be used to estimate an income elasticity:

80 40 % 60 1 4000 2000 % 3000 Q Y ξ − ⎛ ⎞ ⎜ ⎟

Δ _{⎝ ⎠}

= = =

−

Δ ⎛ ⎞

⎜ ⎟

⎝ ⎠

Furthermore, notice that for all three observations, the budget share for good 1 is 1 1

0.4 P x

Y

θ = = .

We can then find the Hicksian elasticity via the Slutsky equation:

( )( )

1 0.4 1

0.6 m h

h

h

ε ε θζ

ε ε

This utility function is a hybrid of perfect complements and perfect substitutes. A bundle containing both x₁ and x₂ is a perfect substitute with a bundle containing both x₃ and x₄. With the first option, the consumer needs to buy x₁ and x₂ together in a 1-for-1 ratio, costing P₁+P₂. With the second option, the consumer needs to buy x₃ and x₄ together in a 1-for-1 ratio, costing

3 4

P +P. Thus, the demand functions are:

If P₁+P₂ <P₃+P₄: ₁

1 2

Y x

P P

=

+ 2

1 2

Y x

P P

=

+ x3 =0 x4 =0

If P₃+P₄ < +P₁ P₂: x₁ =0 x₂ =0 ₃

3 4

Y x

P P

=

+ 4

3 4

Y x

P P

= +

With anything on the budget line (in the correct ratios) if P₁+P₂ =P₃+P₄

Problem 4

a. No bundle Y on BC2 is RP to bundle X since X is not affordable with BC2. Thus, any bundle Y on BC2 satisfies WARP since it is impossible for X RP Y and Y RP X.

a. In this case, the endpoints are 5 widgets and 15 gadgets. The budget line is the linear segment connecting them. Affordable bundles are those on or under the budget line.

b. Any scaling of income and prices in the same proportion leaves the budget set

unchanged. In the example below, all income and prices are doubled. Any other scaling of all three by the same proportion is also correct.

Income = $300

Price of a widget = $60 Price of a gadget = $20

a. Michael gets no utility from C but increases his utility by consuming more M:

(

)

U C M =M

Any other increasing function of M works just as well.

b. Michael is indifferent over the level of chalk, but gets higher utility as he gets more markers:

c. Michael spends all his income on markers since chalk is useless:

0 boxes of chalk and 10 boxes of markers

d. Michael spends all his income on markers since chalk is useless:

0 boxes of chalk and 2 boxes of markers

e. There is no substitution from markers to chalk, so the entire decline is due to the income effect rather than the substitution effect:

The decline of 8 boxes of markers is due to the income effect.

$50, Michael buys 2 boxes of markers:

g. The entire reduction in demand is due to the income effect, with no substitution effect. Since the Hicksian demand curve represents the substitution effect, the Hicksian demand curve is perfectly inelastic. Another way to think about this is that, if we start at 10 markers, the consumer needs 10 markers to maintain the same level of utility at any price since there is no substitution with chalk possible.

a. Preferences are monotonic in x₁ when utility rises as x₁ rises:

1 1

1

10 2 0 5

U

x x

x

∂ _{= − > ⇒ <} ∂

b. The consumer’s objective is:

2

1 1 2

max 10x −x +x s.t. P x_{1 1}+x₂ =Y (since we set P₂ =1)

The Lagrangian is:

(

)

2

1 1 2 1 1 2

10

L= x −x +x +λ Y−P x −x

First order conditions, with the first two solved for λ:

1

1 1

1 1

10 2

10 2 0 x

L

x P

x λ λ P

− ∂

= − − = ⇒ =

∂

2

1 0 1

L

x λ λ

∂ _{= − = ⇒ =} ∂

1 1 2 0

L

Y P x x

λ

∂

= − − =

∂

Equating the expressions for λ:

1 1 1 1 1 1 1 10 2 1 10 2 10 1 5 2 2 x P x P P x P − = − = − ⇒ = = −

Substituting back to the budget line:

1 1 2

1 1 2

2

2 1 1 1 1

1 5 2 1 1 5 5 2 2

P x x Y

P P x Y

x Y P P Y P P

c. x₁ is a substitute for x₂ when demand for x₂ rises as P₁ rises:

2

1 1

1

5 0 5

m

x

P P

P

∂ _{= − + > ⇒ >} ∂

So x₁ is a substitute for x₂ when P₁>5, but a complement for x₂ when P₁<5.

d. Price elasticity is:

1 1 1 1

1

1 1 1 1

1

2 5 0.5 10

m m

x P P P

P x P P

ε =∂ ⋅ = −⎛_⎜ ⎞_⎟⎛_⎜ ⎞_⎟= −

∂ _{⎝ ⎠⎝} − _⎠ −

Demand is therefore price inelastic when:

1 1 1 1 1 1 1 10 10

2 10 5

P P

P P

P P

− > −

− − > − +

< ⇒ <

Another way to see this is that the demand curve x₁m is linear, and the demand curve is always inelastic to the right of the midpoint of a linear demand curve.

e. Expenditure on good 1 is:

1 1 1

2

1 1 1 1

1 1

5 5

2 2

E P x

P P P P

=

⎛ ⎞

= _⎜ − _⎟= −

⎝ ⎠

Expenditure on x1 rises when P1 rises if:

1

1 1

1

5 0 5

E

P P

P

∂

= − > ⇒ < ∂

1 1 2

min P x +x =Y s.t. 10x1−x12+x2 =U

The Lagrangian is:

(

)

1 1 2 10 1 1 2

L=P x + +x λ U− x +x −x

First order conditions, with the first two solved for λ:

1

1 1

1 1

10 2 0

10 2 P L

P x

x λ λ λ x

∂ _{= − + = ⇒ =}

∂ −

2

1 0 1

L

x λ λ

∂ _{= − = ⇒ =} ∂

2

1 1 2

10 0

L

U x x x

λ

∂ _{= − + − =}

∂

Equating the expressions for λ:

1 1

1 1

1

1 1

1 10 2

10 2

10 1

5

2 2

h

P x

P x

P

x P

= − = −

−

⇒ = = −

The Hicksian and Marshallian demand curves are the same in this case. To see why, notice that demand for x₁ does not depend on income. The difference between Hicksian and Marshallian demand is the income effect. Since there is no income effect in this case, the two are identical.