R&D (Time Of Discovery)

Question 1

Patent requirements (3)

Answer 1

Novel
Non-trivial
Useful

Question 2

Model for patents:

Investment x reduces unit cost c to c-x.
Assume small/minor innovation

What happens to market price?

Answer 2

Recall we said monopoly price is higher, however homogenous goods so cannot charge above, so decide to instead undercut arbitrarily.

So for simplicity keep P=C

Question 3

Assume P=a-Q.

Identify innovators gain, and DWL pg42

B) answers for M and DWL

Answer 3

See a fall in costs to c-x.

M is innovator now earning monopoly profit since costs fall.

Price remains at c, hence why DWL exists

V) M = (a-c)x
DWL = x²/2

Question 4

So assume they earn these monopoly profit M for t periods. After they earn 0, why?

Answer 4

Patent expires so other people can use the innovation to lower their costs to the same level, so P=c-x . CS takes over M and DL

Question 5

How to model the patent lasting till T then earning 0 profits

By using Lemma, finite series
Final equation for lemma

Answer 5

1- p to the t / 1-p

p is discount factor

M x (1-p to the t/1-p) = present value of monopoly profits

Question 6

Recall monopoly profits is M=(a-c)x
and DWL x²/2

Find optimal R&D level x.

B) final equation we get for x
C) using equation what does our choice of R&D levels increase with (4)

Answer 6

Maximise profit
Maxπ(x, T) = M - TC which is
(1-p to the t/1-p)(a-c)x - x²/2

then FOC with respect to x (investment)and rearrange to find X

B) Final equation is
x = [1-p to the t/ 1-p] (a-c)
i.e lemma x (a-c)

C)
as t increases (duration of patent)
as demand a increases
as cost c falls
as discount factor p increases (value future profits, thus invest more)

Question 7

So that is a firm’s choice of R&D level.

What is the optimal length (t) of patent for society?

First regulator chooses T

So what is total welfare for T years
B) welfare onwards

Answer 7

W = CS+M for T years
(simple: just CS+PS)

B) CS+M+DWL

Fuck this too hard

Question 8

Is optimal T finite? i.e do we want unlimited patent?

e.g rather 1 year patent W(1) or infinite W(∞0) which is higher

Answer 8

W(1) > W(∞) when p<0.5!

i.e low discount factor i.e impatient. Impatient societies prefer short patents to maximise CS faster! (of course comes at the cost of perhaps losing incentives for long run innovation since profits get quashed fast!)

Question 9

What if p>0.5, is optimal T infinite

Answer 9

No
T*<∞

i.e T will be longer, but still not infinite - since need to balance innovation and efficiency (DWL CS)!

Question 10

Why would firm invested in R&D license its tech to rival that hasn’t invested?

Assume innovator offers a ø per unit licensing fee that reduces unit cost of the rival firm 2

What is the ø set to

Answer 10

Ø= (c2-c1) - ε
I.e arbitrarily smaller than the difference in costs between them. So there is incentive for rival to pay ø to produce at a lower cost.

So innovator earns additional ø x q2! (While firm 2 profits and quantity barely change)

Question 11

R&D subsidising - assume 2 countries

Airbus and Boeing
Game theory - look at the diagram pg 53

What is the nash equilibrium

Answer 11

There are 2

Either [producer, don’t produce] or don’t [produce, produce]. At each of these points there is no incentive to deviate for either firm

(Whereas for [produce produce] and [don’t,dont], there is incentive to deviate thus no nash equilirbium)

Question 12

What if EU gives airbus subsidy of £15

Draw new game theory now

What is the new nash equilibrim,

Answer 12

Since Airbus will always want to produce as always profitable now given the subsidy, Boeing won’t produce

Question 13

Effect on social welfare

Answer 13

Unclear, Airbus has monopoly, however funded by taxes.

Question 14

So that was subsidising a product.

Now consider subsdising a process innovation.
P=a-Q
Subsidy xi reduces cost c to c-xi
Gov cost TC = x²i/2

2 stage game. Firms take subsidy and max profits
What is firm i’s profit max problem then and steps to get final profit (in terms of subsidy Xi and Xj)

Answer 14

Max πi = (a-qi-qj-c+xi)qi
(Don’t need to include TC into firm I’s problem, since TC is a government cost, so used for society welfare not this) PTO

FOC, rearrange to find qi, by symmetry qj. Then sub qj into qi to get final qi and qj. Add together to get Q, then sub into find P=a-Q. Then can find πi.

Question 15

Then find welfare of country i; expression

b) Differentiate with respect to to Xi to find optimal subsidy Xi

c) intuition of result: IMPORTANT
d) using equation, what does subsidy R&D level increase with (2)

Answer 15

Profit - TC
so just answer from previous FC - TC X²i/2
Differentiate with respect to to Xi to find optimal subsidy Xi

b) Differentiate with respect to to Xi to find optimal subsidy Xi
xi = 4(a-c) -4xj

C) IMPORTANT:
We can see as Xj (country j’s subsidy) increases, Xi falls (country i’s subsidy)! This means subsidies are substitutes Xi=Xj!
xi=xj= 4(a-c) /5

Thus beneficial for at least one firm to subsidise R&D (return from export sales>costs of R&D)

d) subsidy size, increases with demand, decreases with cost c

Question 16

So its beneficial when one country subsidises, but bad when both do!
Why

Answer 16

As we just showed substitutes!
When only one subsidises, the subsidised firm will always produces, other firm doesn’t. Gain in global welfare since other country doesn’t lose anything (given no subsidy war)

However when both subsidies it is worse (overproduction/duplication of R&D despite losses, wasteful inefficient use of public funds)