Solution to HW2 90-907 Econometrics II
Problem 1

Problem 2

Since the supply curves are flat, it is easy to measure the effects of taking into account the environmental costs. Since the average price of coal is 1.54, the 0.5 sales tax is about 32.47% of the current cost. Thus, taking into account the sales tax would raise the price of coal by about 32.47% (which would raise the log price by about 0.3247). We can just plug this into the demand equations to calculate the change in the log quantities:
 ln
 p c
 ln
 q c

2

  ln ln q c p c
/

2

0 .
3247 /

0 .
11
 
2 .
95
 ln
 p g
 ln
 q g

13



 ln ln q g p g
 
14
 ln
 
14
 ln p c p c

/

13


0

0 .
0206 ( 0 .
3247 )

/

0 .
21

0 .
318 e

2 .
95 e
0 .
318

0 .
0523 ,

1 .
375
So, if the coal were imposed .50 sales tax, the consumption of coal will decrease by about 0.0523, and the consumption of gas will increase by 1.375.
Since the supply curve for coal is flat, indicating that the supply for coal is very

elastic. Therefore, it will be the consumer of coal who would pay for the tax. Because coal and gas are substitutes (

14
,

3

0 ), the consumption of gas increases as the result of increased coal price. Producers of gas benefit from the sales tax on coal consumption.
Problem 3 a) If we estimate equation (1) by OLS,
^

1 , OLS
 
1


( X i

( X i

_
X )
 i

_
X )
2
 p

1

Cov ( X ,

)
V ( X ) b) If we estimate equation (2) by OLS,
Y


X

0




1


1
0
X

 
1

1
Y

1

1
 (3)
From the original model Y
 
0
 
1
X
  we know that E (

| Y )

0 , since Y is a function of

, therefore OLS estimates for equation (2) will be inconsistent.
^

1 , OLS

1

1
  p



2
1

1

Cov ( Y ,
 
)



2

1
2
V ( X )
V ( Y )

1

1

Cov (

0
 
1
X
V ( Y )
 
,
 
)

1

1



2
V ( Y )
3) If we estimate equation (1) using Z as our instrument for X,
^

1 , IV


( Z i

X i
( Z i

_
Z ) Y i

_
Z )
 
1


( Z i

_
Z )
 i

X i
( Z i

_
Z ) p

1

Cov ( Z ,

)
Cov ( X , Z )
 
1
Since Cov ( Z ,

)

0 , and Cov ( Z , X )

0

4) If we estimate equation (2) using Z as our instrument for Y, using equation (3)
Cov ( Y , Z )

Cov (



1

1
Cov ( X
, Z )

0
, Z )

Cov (

, Z )
 
1
Cov ( X , Z )

0
So Z is a good instrument for Y. Thus:
^

1 , IV
 p
1

1

Cov ( Y ,
Cov ( Y

, Z


1
)
)

1

1
is consistent.
5) From above analysis, IV estimation gives us consistent estimates regardless of which variable we use as our left hand side variable.