problem 2 1 for economists/sm...too. if f increases then fixed costs have increased but marginal...

2.1 (a) ( )1PNB bx x f cx= − − −

(i) FOC 2 0 *2b cb x x xb−

− − = ⇒ =

(ii) SOC 2 0b− < OK (iii)

( )

2 2 2 2 2

2

* 12 2 2

2 2 22

4 2 2 4

4

b c b c b cPNB b f cb b b

b c b c b cf cb b

b c cb c b c cbf fb b b b

b cf

b

− − − = − − −

− + − = − −

− + −= − − + = −

−= −

(iv)

( ) ( ) ( ) ( )

2 2

2 2 2

2 2

2 2 2 2 2

2 2

* 12 2 2

* * 10 02

4 22*4 4 4

2 2 24 4

x b c cb b b bx xf c b

b b c b bc cb c b cPNBb b b bb bc b bc c b c

b b

∂ −= − =

∂∂ ∂

= = − <∂ ∂

− − − +− −∂= − =

∂− − + − −

= =

which is > 0 for internal solutions since if b c≤ then * 0x ≤ .

( )

* 1 0

2* *4 2

PNBf

b cPNB b c xc b b

∂= − <

∂

−∂ −= − = − = −

∂

which is < 0 for internal solutions. The derivatives of PNB* could also be derived using the implicit function theorem:

( )* 1PNB b PNB b x x∂ ∂ = ∂ ∂ = − (when evaluated at x*), * 1PNB f PNB f∂ ∂ = ∂ ∂ = − , and *PNB c PNB c x∂ ∂ = ∂ ∂ = − (when

evaluated at x*). (v) If b increases then marginal benefits are higher, so the agent should

do more of the activity to get MB = MC; with more of the activity and higher benefits for any given level of activity, the maximum value of net benefits will be higher too. If c increases then marginal costs are higher, so the agent should do less of the activity to get MB = MC; with less of the activity and higher costs for any given level of the activity, the maximum value of net benefits will be lower

too. If f increases then fixed costs have increased but marginal benefits and marginal costs haven’t changed. So the agent should do the same amount of the activity but, with higher fixed costs, the maximum value of net benefits will be lower.

(b) From the FOC, ( )* 2 .x b c b= − So if b c≤ then the optimal value of x is zero. b is marginal benefit as x gets marginally greater than zero; likewise, c is marginal cost as x gets marginally greater than zero (ignoring for the moment the fact that the fixed cost f has to be paid too). So if b c< the MB of the very first bit of the activity is less than its marginal cost. In addition, if f is large enough then a border solution x = 0 is best since (0) (0) (0) 0PNB B C= − =

while ( )2 2* 4 0PNB b c b f= − − < if f is large enough. These are costs that don’t have to be paid if the agent does none of the activity, but which don’t change as the amount of activity increases, once it’s positive. So they’re like fixed but not sunk costs: they’re the equivalent of firm fixed costs that can be recovered (for example by selling physical assets) if the firm decides to shut down.

(c) ( )1SNB bx x cx f ex= − − − − so

(i) FOC 2 0b bx c e− − − = , **2

b c exb

− −=

(ii) SOC 2 0b− < OK

(iii) 2 2

b c e b cb b

− − −< since e > 0. Social marginal cost is greater than

private marginal cost so the optimal output level is lower.

(iv) ** **2

b c eE ex eb

− − = =

(v) ( )

2 2

2 2

** 1 02 2 2

** 1 ** 10 02 2

** 02 2 2

** 02

** 22 2 2

b c ex c eb b b bx xc b e bE e b c e c ee eb b b bE ec bE b c e e b c ee b b b

− −∂ += − = >

∂∂ ∂

= − < = − <∂ ∂∂ − − + = − = > ∂ ∂

= − <∂

∂ − − − −= − =

∂

which isn’t signable without knowing the values of the parameters. (vi) If b increases then marginal benefit (social as well as private) is

higher, so social net benefits will increase if there is more of the activity. But this will increase the total amount of external harm as

well. If c increases then marginal cost (social as well as private) is higher, so social net benefits will increase if there is less of the activity, which will also reduce the total amount of external harm. If e increases then private marginal cost hasn’t increased but social marginal cost has. So social net benefits will increase if there is less of the activity. Since the external cost per unit of the activity is higher, though, the total amount of external harm might increase even though there is less of the activity.

(d) The FOC implies that ( )** 2x b c e b= − − but if c e b+ > then the optimal amount of the activity will be zero rather than the value that solves the FOC. So maybe b c> so * 0x > but c e b+ > so the social optimum is at the border solution of no activity. The private marginal benefit of the very first bit of the activity is greater than the private marginal cost, so the private agent will engage in some of the activity. But once external marginal cost is added, the marginal benefit of the first bit of the activity is not high enough to offset its social marginal cost. Also, social net benefits can be forced below zero more easily than can private net benefits by high fixed (but not sunk) costs f : with a bunch of tedious algebra you could show that

( )( )** * 2 ** * *SNB PNB e x x PNB= − + < . So it’s possible that * 0PNB > (so that the private agent will choose to engage in some of the activity) but

** 0SNB < when evaluated at the internal solution, so the socially optimal amount of activity is zero.

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2-1

Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.

2.2

From the text, the solutions to the four cases are:

and

(a)

so when decreases, all solutions except decrease.

(b)

so when decreases, and decrease while the others do not change.

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2-2


2.3

(a) � is exogenous: its value is not determined within the model.

� is the relative weight put on the goal of income maximization as opposed to rentmaximization.

(b)

(c)

When the solution for rent maximization.

When the solution for income maximization.

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2-3


2.4

with K fixed

(a)

From FOC,

(b)

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2-4


2.5

(a)

For

(b)

(i) so

(ii)

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2.6

(a)

which holds when

(b)

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2-6


2.7

Yes, these answers are the same.

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2-7


2.8

(a)

(b)

SOC:

(c)

(d)

(e)

The first-order condition for choosing t to maximize T is

which is cubic in t , with no easy solution.

2.9 a bQ Q cQ f eQ so

(a) FOC 2 0a bQ c e . The SOC 2 0b is satisfied. So if there is an

internal solution, * 2Q a c e b and

* * 2 2P a bQ a a c e a c e .

(b) * *2

a c eE eQ eb

(c) * 1 02

Qe b

. The monopolist will have to pay a higher fine, and the fine

increases with output. The monopolist’s MC has increased, so the monopolist will choose to produce less.

(d) With no fine, optimal output is *2 2a c eQ Qb b

. So if e is large enough, Q

could be positive but * 0Q , which would lead to a border solution. (Also, if there are fixed but not sunk costs, they wouldn’t have to be as large to drive the socially optimal output level to zero if the firm has to pay a fine equal to external costs.) If the socially optimal output level is already zero, an increase in e won’t affect output. If * 1 2Q b then a marginal increase in e will drive the socially optimal output to zero, but this will be less (in absolute value) than the derivative shown in part c.

(e) The per-unit tax should equal the marginal external cost. e works exactly the same way in this model as t did in section 2.4.2. To achieve the social optimum, the external cost has to be internalized into the firm’s decision making. This can be done with a fine based on the amount of external harm or it can be done more indirectly by taxing output, as long as the tax is set at the right level.

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2-8


2.10

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2-9


2.11

(a)

by symmetry

(b)

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2-10


2.12

(a) Solutions from text:

equation (2.70):

equation (2.71):

equation (2.72):

(b) equilibrium n is when

so

Substituting into solutions for

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2-11


Note also that

(c)

The first-order condition for choosing t to maximize T is

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2-12


2.13

(a) From the text,

and

But in this example, c = 0.

(b)

which is the monopoly price.

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2-13


2.14

(equation (2.84))

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2-14


2.15

(a)

g > 0 is sufficient for to be positive and finite.

(b) , which is smaller than the usual multiplier,

(c)

When g is larger, there are two possible effects: government spending increases by more

when equilibrium GDP is lower than the target level, but government spending decreases

by more when equilibrium GDP is higher than the target level. So if the target level is

greater than equilibrium GDP, and a higher value of g will make equilibrium

GDP higher. But if the target level of GDP is lower than equilibrium GDP,

and a higher value of g will make equilibrium GDP lower. Either way, a higher value of

g will move equilibrium GDP closer to the target level.

2.16(a) 2 2L A a B b

(b) FOC 2 2 0A a a B b b SOC 2 22 2 0a b OK

(c) 2 2 2 22 2 2 2 *a b Bb Aa t Bb Aa a b

(d) 22 2 2 2 2 22 2

* 2 *2 0A Bb Aa A aaa a b a b a ba b

(e) 2* B b so * * * *2 B b ba a a

which, when

evaluated at * , equals 2

2 2 2 2

* *2 2Bb Aa a B Aabb B b ba a b a a b

which has sign

(-)(-)(+) > 0.

problem 2 1 for economists/sm...too. if f increases then fixed costs have increased but marginal...

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