EconS425 - Homework #4 (Due on April 3rd, 2012) Ï = Ï
EconS425 - Homework #4 (Due on April 3rd, 2012) Ï = Ï
EconS425 - Homework #4 (Due on April 3rd, 2012) Ï = Ï
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<str<strong>on</strong>g>Ec<strong>on</strong>S425</str<strong>on</strong>g> - <str<strong>on</strong>g>Homework</str<strong>on</strong>g> <str<strong>on</strong>g>#4</str<strong>on</strong>g> (<str<strong>on</strong>g>Due</str<strong>on</strong>g> <strong>on</strong> <strong>April</strong> <strong>3rd</strong>, <strong>2012</strong>)<br />
1. Exercise 2 from Chapter 9 in Zhy (page 249)<br />
Answer<br />
a) The expected profit of a firm engaged in R&D, given that the other two firms are also<br />
engaged in R&D, is the prize times the sum of the probability that the firm discovers<br />
while the other two do not, plus twice the probability that it discovers while <strong>on</strong>e<br />
competing firm discovers and <strong>on</strong>e does not, plus the probability that all the three<br />
discover simultaneously, minus innovati<strong>on</strong> cost. Formally, for each firm ii , 1,2,3<br />
1 1 1 1 1 1 V 1 1 1 V 7V<br />
E i<br />
V ( 2)<br />
I I<br />
2 2 2 2 2 2 2 2 2 2 3 24<br />
Thus, all the three firms find it profitable to engage in R&D if V 24 I / 7 24 / 7 (since<br />
I 1).<br />
b) Now the single firm can operate zero, <strong>on</strong>e, or two labs.<br />
Operating a single lab: In this case, E V /2<br />
I<br />
1 lab<br />
Operating two labs: In this case, the probability of discovery is <strong>on</strong>e minus the<br />
probability that n<strong>on</strong>e of the labs discovers. Thus,<br />
1 1 4V<br />
E (1 ) V 2I 2I<br />
2 labs<br />
2 2 3<br />
E<br />
E<br />
Now 2 labs 1 lab if V 4I<br />
which c<strong>on</strong>stitutes a sufficient c<strong>on</strong>diti<strong>on</strong> for having the<br />
firm choosing to operate two labs.<br />
2. A m<strong>on</strong>opoly with c<strong>on</strong>stant marginal costs m = $10 faces the inverse demand curve p =<br />
50 – Q. The interest rate is r = 10%. An inventor discovers a way to reduce marginal<br />
costs to m = $6 (with no additi<strong>on</strong>al fixed costs) and receives a permanent patent for that<br />
inventi<strong>on</strong>. Up to how much is the m<strong>on</strong>opoly prepared to pay for a permanent license to<br />
use the inventi<strong>on</strong><br />
(i) if it is certain that the inventor will not offer the inventi<strong>on</strong> to any other firm<br />
(ii) if it knows that the inventor may offer the inventi<strong>on</strong> to a potential entrant instead<br />
Answer<br />
(i)<br />
If the m<strong>on</strong>opoly is certain that the inventor will not offer the inventi<strong>on</strong> to any other<br />
firm, it will maximally pay the present value of the difference between its future<br />
m<strong>on</strong>opoly profits with and without the inventi<strong>on</strong>, or<br />
1<br />
Instructor: Ana Espinola
(ii)<br />
If the m<strong>on</strong>opoly knows that the inventor may offer the inventi<strong>on</strong> to a potential<br />
entrant instead, it will maximally pay the present value of the difference between its<br />
future m<strong>on</strong>opoly profits with the inventi<strong>on</strong> and its duopoly profits if a rival firm<br />
obtains the license and enters its market, or<br />
The potential entrant will maximally pay the present value of its future duopoly profits, or<br />
3. A government wants to stimulate research and development and must choose between two<br />
policies to do so. The first <strong>on</strong>e c<strong>on</strong>sists of increasing the length of the patent from 17 to 18<br />
years, while the relevant interest rate is projected to stay at 7%. The sec<strong>on</strong>d policy would leave<br />
patent length at 17 years, but reduce the interest rate from 7% to 6%. Say the inverse annual<br />
demand for the yet-to-be invented new good is p = 100 – 2Q, is c<strong>on</strong>stant over the years, and<br />
marginal cost is c<strong>on</strong>stant at 20. Which policy is more effective Do you need all the informati<strong>on</strong><br />
above to answer that questi<strong>on</strong><br />
Answer<br />
In a simple model, we can normalize the m<strong>on</strong>opoly profits at $1 per year for the length of the<br />
patent. (Other than the assumpti<strong>on</strong> that profits are c<strong>on</strong>stant over the years, the informati<strong>on</strong><br />
about marginal cost and demand is not required to answer this questi<strong>on</strong>.) The value to the<br />
patent holder of a <strong>on</strong>e-year extensi<strong>on</strong> is thus the present value of $1 during the eighteenth year.<br />
The value of that dollar is PV(18th year for 1.00)=1.00×(1/1.07) 18 ≅ 0.296<br />
The value to the patent holder of a reducti<strong>on</strong> in interest rate (making the present value of future<br />
profits larger) is the difference between two streams of payments. We can calculate the present<br />
value of annuities of $1 during 17 years by subtracting the value of a perpetual annuity starting<br />
in 18 years from <strong>on</strong>e starting today:<br />
The difference between the two is 0.769, larger than 0.296, so a reducti<strong>on</strong> in the interest rate is<br />
more valuable to the patent holder (more effective as an incentive to innovati<strong>on</strong>) than a <strong>on</strong>eyear<br />
extensi<strong>on</strong> of the patent.<br />
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Instructor: Ana Espinola
4. An increase in R&D induced by, say, government subsidies could result in a higher rate of<br />
product obsolescence, with new products replacing older <strong>on</strong>es at a faster rate. How does this<br />
“side-effect” of the subsidies affect firms’ incentives to invest in research (Hint: distinguish<br />
between firms with and without market power.)<br />
Answer<br />
A higher level of R&D by potentially rival firms reduces the expected value of existing patents, by<br />
increasing the likelihood that newly-developed products will become obsolete before their<br />
patents run out. Competitive firms therefore have less incentive to invest in R&D with the<br />
purpose of obtaining market power. M<strong>on</strong>opolistic firms whose market power is based solely <strong>on</strong><br />
an existing patent, <strong>on</strong> the other hand, may have more incentive to invest in R&D, so as to<br />
maintain their innovative lead over potential rivals.<br />
5. C<strong>on</strong>siders a m<strong>on</strong>opolist with the following demand curve : P=390-2Q. The m<strong>on</strong>opolist faces<br />
MC M =AC M =30.<br />
a. Solve for the profit-maximizing level of m<strong>on</strong>opoly output, price, and profits.<br />
Q=90; P=210 and profits=16,200<br />
b. Suppose a potential entrant is c<strong>on</strong>sidering entering, but the m<strong>on</strong>opolist has a cost<br />
advantage. The potential entrant faces costs MC PE =AC PE =40. Assuming the m<strong>on</strong>opolist<br />
c<strong>on</strong>tinues to profit-maximize, solve the residual demand curve for the entrant.<br />
P=210-q pe<br />
c. Assume the potential entrant follows the Cournot assumpti<strong>on</strong> about the m<strong>on</strong>opolist’s<br />
output. Solve for the potential entrant’s output, price, and profits in this scenario. What<br />
are the new m<strong>on</strong>opoly profits<br />
q pe =42.5; P=125; profits=3,612.5; profits to m<strong>on</strong>opolist = 8,550<br />
d. Is there a price the m<strong>on</strong>opolist could charge to deter entry Solve for the limit price and<br />
output that will completely deter entry. What is m<strong>on</strong>opoly profit at this point<br />
P=40; q=175 and profits = 1,750<br />
6. C<strong>on</strong>sider the problem described in 10.<br />
a. Make an extensive form of the game based <strong>on</strong> your soluti<strong>on</strong>s in problem 5<br />
b. Solve for the Nash equilibrium of the game. If the m<strong>on</strong>opolist threatens ahead of time<br />
to limit price, is the threat credible Explain<br />
Answer<br />
a. The numbers for the figure are from the soluti<strong>on</strong> to Problem 5<br />
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Instructor: Ana Espinola
. The Nash equilibrium is that in each case, the m<strong>on</strong>opolist would produce q = 90.<br />
Given that the entrant knows this, the entrant will enter and earn 3,612.5 in profits,<br />
rather than stay out and earn 0 in profits. The threat to limit price is not credible.<br />
(Note that the possibility of forgoing current profits for future gain is not built into<br />
this game since it is played <strong>on</strong>ly <strong>on</strong>ce.)<br />
c. If the potential entrant trembled and stayed out, the m<strong>on</strong>opolist would choose to<br />
produce 90 (not limit price) and earn $16,200 in profits. A limit-pricing policy would<br />
not be reas<strong>on</strong>able, because it reduces the m<strong>on</strong>opolist’s profits.<br />
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Instructor: Ana Espinola