301_mid1as11.pdf

Midterm Exam 1 – Solutions

ECO 301 – Summer 2011

Instructor: 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:

_____________________________________

Problem 1

a. Here are some bundles giving U=1:



       







c. The constrained maximization problem is: max FG2 s.t. F Y P G_G Lagrangian: L F G2





First order conditions solved for :

1 0 1

L

F  

 _{    }



2

2 G 0

G

L G

G P

G   P

 _{     } 

0 G

L

Y P G F



 _{   }



Equating the expressions for : 2 1 2 2 G G G G P P

G P G



  

Substituting back to the budget constraint:

2 1 2 2 G G G G

F Y P G

P

F Y P F Y P

 

 

  _{ }  

 

d. He earns income of 1 2

2 2

G

G G G

P

P GP _ _ P

  from garbage, as a fraction of total income:

e. Substituting the Marshallian demands back to the utility function:

2

2 2

2 2 2

1 1

2 2

1 1 1

2 4 4

G G

G G G

V F G

Y P P

Y P P Y P

 

   

_  _{ } _

   

    

Indirect utility actually rises as P_G rises in this case. The interpretation is just that he’s selling the rights to dump G rather than buying it, so an increase in P_G is good.

f. Solve the indirect utility function for income:

2

2 2

1 4

1 1

4 4

G

G G

V Y P

Y V P E u P

 

    

Expenditure actually falls as P_G rises in this case. Again, the interpretation is just that he’s selling the rights to dump G, not buying it.

g. Suppose that P_G rises:

Income effect: Consumer’s real income rises so he does not need to accept as much garbage  G falls

Problem 2

a. Bundle A contains 0 boxes of fish and 12 boxes of chips. b. Bundle D contains 3 boxes of fish and 6 boxes of chips.

c. MRS < price ratio since the indifference curve is flatter than the budget line. d. Less fish, since MRS < price ratio.

e. After the law, Lloyd will buy 2 boxes of fish and 8 boxes of chips. Lloyd wants to buy more than 2 boxes, so he will certainly buy up to the most he is allowed to.

f. See the diagram below.

Problem 3

a. Solving the expenditure function for u:





 

 

 

,

E P u u g P

E Y

u V

g P g P

 

  

b. Using Roy’s Identity:

 

 





 

 

 

g P Y

g P g P Y

V p x

V Y g P

g P            

c. From the definition of income elasticity:

 

 

 

 

 

 

 

 

i i m i i m i i

g P g P Y g P

x Y Y

g P Y

Y x g P g P g P Y

Problem 4

a. The price paid by buyers is $3 lower than the exchange price:





35 5 3 10 10 35 5 15 10 10

15 60 4

P P

P P

P P

    

    

  

Substituting back gives Q30. The government pays a subsidy of $3 per bottle over 30 bottles, costing the government $90.

b. Here, the price paid by buyers is $3 higher than the exchange price:





35 5 3 10 10 35 5 15 10 10

30 60 2

P P

P P

P P

    

    

  

Substituting back gives Q10. The government collects a tax of $3 per bottle over 10 bottles, so the government receives $30 of revenue.

Problem 5

This is a hybrid of perfect complements and perfect substitutes. The consumer must equate x₁ with either x₂ or x₃. He picks whichever is cheaper.

Suppose that p₂  p₃, then the consumer will set x₁ x₂, and substituting into the budget line gives:

1 1 2 2

1 1 2 1 1 2

1 2 1 2

p x p x Y

Y Y

p x p x Y x x

p p p p

 

     

 

On the other hand, if p₃ p₂, then the consumer buys x₁ and x₃. Summarizing, the Marshallian demands are:

For p₂  p₃:





1 2 1 2

, , Y , Y , 0 x x x

p p p p

 

  _{ } 

 

For p₃  p₂:





1 3 1 3

, , Y , 0, Y x x x

p p p p

 

  _{ } 

 

Let us derive the Hicksian demands for the p₂  p₃ case first. First find the indirect utility function:

1 2 1 2 1 2

min Y , Y 0 Y

V

p p p p p p

 

 _  _

  

 

Inverting to solve for the expenditure function gives:





Eu PP

You can then use Shepherd’s Lemma to differentiate the expenditure function and find the Hicksian demands. For example, ₁

1 h E x u p    

Using this reasoning, the Hicksian demands are: For p₂  p₃:



 



Problem 6

Start with a table summarizing the expenditures:

Bundle X Bundle Y Prices X 16 12 2 

Prices Y 31 3 5  Here, x y if Bundle Y was affordable when Bundle X was chosen:

12 2 16

2 4 2



 

    

Similarly, y x if Bundle X was affordable when bundle Y was chosen:

31 3 5

28 5 5.6



 

 

  