Question

Suppose a firm is the sole producer of unique type of surf board wax. The wax can be produced at a constant marginal cost of $4/ounce. The firm faces two types of customers for their product -- west coast surfers and east coast surfers-- and the demand for both groups is given by the following inverse demand functions:

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Answer #1

The firm is the sole producer thus the firm is a monopolist.

Marginal Cost, MC = $ 4 / ounce

P20 0.lq w

First determine the Total Revenue, TR = PQ

(20 0.1q) x qu TR=P

TR 20q-0.1q

Differentiating TR wrt qw

0.1q) d(20qau- dqw dqw

=20 0.24 u MR=

The monopolist can maximize profit by equating MR to MC

20 0.2q= 4

0.2 =20-4

16 0.2

gw =80

Now plug in this in the demand function of west coast we get

Py=20- 0.1q= 20-0.1 x 80 20 8 = $12

Elasticity can be determined using the following formula

$e = \frac{\% \triangle Q}{\% \triangle P} = \frac{dq}{dp}\times \frac{P}{q}$

the demand function of the west coast is,

P20 0.lq w

$\implies 0.1q_w = 20 - P_w$

$\implies q_w = 200 - 10P_w$

Differentiating above equation wrt Pw we get

$\frac{dq_w}{dP_w} = \frac{d(200 - 10P_w) }{dP_w}$

dqw dPu -10

Now determining the elasticity for west coast market

e dpw X

$\implies e = -10\times \frac{12}{80}$

$\implies e = -1.5$

Now determining the Price quantity and elasticity for east coast market

Now equate MR to MC

$\implies 0.1q_e = 12 - 4 = 8$

ge = 80

Plug in this in inverse demand function,

$P_e = 12 - 0.05\times 80 = 12 - 4 = \$ 8$

Now express the demand function in terms of P

Differentiating above equation wrt qe

$\frac{dq_e}{dp_e} = - 20$

Elasticity, $e = \frac{dq_e}{dp_e}\times \frac{P_e}{q_e}$

$\implies e = -20\times \frac{8}{80}$

$\implies e = -2$

	Price	Quantity	Elasticity
West Coast	$ 12	$ 8	1.5
East Coast	80	80	2

Please contact if having any query will be obliged to you for your generous support. Please help me i need your help it mean a lot to me. Thank you.

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Answer 1

The firm is the sole producer thus the firm is a monopolist.

Marginal Cost, MC = $ 4 / ounce

P20 0.lq w

First determine the Total Revenue, TR = PQ

(20 0.1q) x qu TR=P

TR 20q-0.1q

Differentiating TR wrt qw

0.1q) d(20qau- dqw dqw

=20 0.24 u MR=

The monopolist can maximize profit by equating MR to MC

20 0.2q= 4

0.2 =20-4

16 0.2

gw =80

Now plug in this in the demand function of west coast we get

Py=20- 0.1q= 20-0.1 x 80 20 8 = $12

Elasticity can be determined using the following formula

$e = \frac{\% \triangle Q}{\% \triangle P} = \frac{dq}{dp}\times \frac{P}{q}$

the demand function of the west coast is,

P20 0.lq w

$\implies 0.1q_w = 20 - P_w$

$\implies q_w = 200 - 10P_w$

Differentiating above equation wrt Pw we get

$\frac{dq_w}{dP_w} = \frac{d(200 - 10P_w) }{dP_w}$

dqw dPu -10

Now determining the elasticity for west coast market

e dpw X

$\implies e = -10\times \frac{12}{80}$

$\implies e = -1.5$

Now determining the Price quantity and elasticity for east coast market

Now equate MR to MC

$\implies 0.1q_e = 12 - 4 = 8$

ge = 80

Plug in this in inverse demand function,

$P_e = 12 - 0.05\times 80 = 12 - 4 = \$ 8$

Now express the demand function in terms of P

Differentiating above equation wrt qe

$\frac{dq_e}{dp_e} = - 20$

Elasticity, $e = \frac{dq_e}{dp_e}\times \frac{P_e}{q_e}$

$\implies e = -20\times \frac{8}{80}$

$\implies e = -2$

	Price	Quantity	Elasticity
West Coast	$ 12	$ 8	1.5
East Coast	80	80	2

Please contact if having any query will be obliged to you for your generous support. Please help me i need your help it mean a lot to me. Thank you.