Recall the definition of linear independence. The columns of a matrix [M] are said to be linearly dependent if there exists a vector [vecv!=0] with [Mvecv = 0] . We will say that the columns of [M] are linearly independent if [Mvecv = 0] implies [vecv = 0] . Let [A] be a square matrix. Show that if the columns of [A] are linearly dependent, [A^(-1)] cannot exist. Hint: [vecv] cannot be both zero and not zero at the same time. Solution If the columns of [A] are dependent, then there is some [vecv!=0] such that [Avecv=0.] If [A^(- 1)] existed, then by definition of [A^(-1)] we would have [A^(-1)Avecv=Ivecv=vecv,] where [I] is the appropriately sized identity matrix. But [Avecv=0,] so [A^(-1)Avecv=A^(-1)(0)=0,] because [A^(-1)] is also a linear transformation, so it takes the zero vector to the zero vector. So we would have [vecv=A^(-1)Avecv=0,] but this contradicts what we said above. Therefore, [A^(-1)] can\'t exist..

1. Recall the example of four firms drawing water from an open-access lake (Chapter 4 of the
Text). Suppose that the fifth firm, commissioning its production plant on the lake, would
increase the cost of cleaning up of water for each of the existing four firms to an extent of $10
thousands (i.e., a total of $40 thousands) per every unit of the product produced by the fifth firm.
Suppose that the following are the marginal (private) costs and marginal WTP for the product
produced by the fifth firm.
Quantity supplied (Q)
0
10
20
30
40
50
60
70
Marginal Cost in thou. $
50
60
70
80
90
100
120
130
Marginal WTP in thou. $
130
120
110
100
90
80
70
60
(a) What would be the optimal level of market supply of the fifth firm when the firm has an
open access to the lake. Show your answer with the help of a graph. Compute the total costs and

2. total WTP at this level. Would you consider this level of supply as private-market efficient level
or socially efficient level?
(b) Suppose that the other four firms had the exclusive right on the lake and could stop the fifth
firm from opening its plant on the lake side. However, the fifth firm agrees to pay the increased
cost of cleaning up the water (which is an external cost to the firm) to all the four firms. Based
on this agreement, the fifth firm will start its operation. Would this agreement affect the fifth
firm
Quantity supplied (Q)
0
10
20
30
40
50
60
70
Marginal Cost in thou. $
50
60
70
80
90
100
120
130
Marginal WTP in thou. $
130
120
110
100
90
80
70
60
Solution