R&D (Equilibrium R&D + Social Optimal Diagram)

Question 1

3 types of R&D

Answer 1

Basic (research with no specific application)
Applied (research with a particular application)
Experimental (research directed towards new/existing systems)

Question 2

R&D increases total costs. thus need incentive to invest in R&D

Key assumption for R&D

Answer 2

Pr(Innov=1) = Φ(FirmCharacteristics)

I.e probability of innovation depends on firm characteristics

Question 3

R&D as a % of GDP

Answer 3

Around 3%

Question 4

Why is it risky

Answer 4

Doesn’t always lead to innovation

Question 5

How to address these market distortion (of underinvestment due to risk)

Answer 5

public and private funds towards r&d

why both pay? as innovation (progress) benefits both producers and consumers

Question 6

Start with bertrand competition with homogenous goods pg 10

b) Add small innovation in diagram pg 12, what are the effects?

c) Add large innovation pg 13, what are the effects?

Answer 6

Innovation reduces marginal costs, c0 to c1

If innovation is small, their monopoly price is above price, however as homogenous goods, they instead undercut to steal market
(draw by only reduce c0 to c1 a little)

If large innovation reduced c0 to c1 a lot, now monopoly price is lower, so can set price where MC=MR

Question 7

So what is profit for small innovation vs large innovation (see diagrams if can’t rmb notation pg 11 and 12)

Answer 7

Small: (C0 -C1)Q0

Large: (Pm - C2)Q2

Question 8

Why do firm engage in R&D (2)

Answer 8

Monopoly profits
signal of high quality

Question 9

2 factors important for innovation race

Answer 9

investment amount

Number of rivals

Question 10

Assumption for innovation race

b) what if another firm discovers it too
c) what does it earn if no discovery

Answer 10

A firm investing I has probability α of discovering a technology with profit V if the sole discoverer.

or V/2 if another firm discovers it too

c) if no discovery, earns 0

Question 11

Expected profits of firm 1 Eπ(1) if only firm in market.

b) so when should a firm invest (bottom of pg15)

Answer 11

Eπ(1) = aV - I
(Probability of discovery x Discovery profits - Investment cost)

b) invest i=I if expected profits>0
invest i=0 (not invest) if <0

Question 12

Diagram for firms decision to innovate pg 17

Answer 12

Y axis I (investment cost), X axis a
Line is where expected profits=0 i.e breakeven, where I=aV

LHS of line: no R&D as I>aV i.e expected profits ngeative

RHS of line = invest as I<aV , expected profit positive

Question 13

Now add a 2nd firm. what do we have (2)

Answer 13

Technological uncertainty (whether successful)

Market uncertainty as dont know whether you, or other firm or both will innovate.

Question 14

Expected profit of each firm Eπi(2)

Answer 14

Eπi(2)= α(1−α)V + α²(V/2) − I

invest if Eπi(2) > 0
First term shows if one discovers, 2nd term shows if both discover, then - investment cost.

rearrange to get α(2−α)V =I

Question 15

diagram for 2 firms pg 18

Answer 15

Y axis I, X axis a

Upward sloping concave line.

LHS of line: expected profits (of a firm) with 2 firms in the market is negative, so only 1 firm engages.

RHS of line: expected profits (of a firm) with 2 firms in the market is now positive, so both firms engage in R&D!

Question 16

Why is R&D disincentivised

Answer 16

As firms engaging in R&D provide negative externality to other firms

Question 17

Society’s welfare is measured bywhat

Answer 17

the industry’s expected profit Eπs

Question 18

Society Welfare (Eπs) when only 1 firm in market.

b) Society welfare when 2 firms in market

c) when is it socially optimal to have 2 firms vs 1

d) then add the social welfare to the diagram to show when socially optimal/unoptimal to have 2 firms pg19

Answer 18

Eπs(1)=Eπ(1)=αV −I.
(since based on industry profit)

b) basically sum of Eπ(2) twice since 2 firms.
Eπs(2)=ΣEπi(2)= 2α(1−α)V +α²V −2I

c) if social welfare (industry expected profit) is higher with 2 firms than 1.
Eπs(2) > Eπs(1) ie.
2α(1−α)V +α2V −2I > αV −I

Question 19

So diagram has 4 regions.
1. No R&D
2. Only 1 firm
3. 2 firms but not socially desirable
4. 2 firms socially desirable)

Explain reasoning for each of these 4 regions

Answer 19

No R&D as cost of R&D (I>av) relatively high compared to expected profits

Only 1 firm as externalities high enough to prevent 2nd firm entering

2 firm but not socially desirable: market failure (since firms investing in r&d create neg externalities)

2 firms and socially desirable: cost is low enough to compensate the externaltiies

Question 20

2 firms not socially desirable as externalities from R&D.

This is a market failure

how can we express it

Answer 20

α(1−α) < I < α(2−α)V /2

Question 21

So with market failure, this is where government can intervene to correct.

Policy makers can influence level of IR&D

What should they do then (2)

Answer 21

identify determinants of Ir&d

calculate socially optimal thresholds

Question 22

(despite firms engaging in R&D creating negative externalities for other firms and thus disincentivises R&D….) (note: disincentivise firms or society?)

Why do we need more firms engaging in R&D

Answer 22

shortens time of discovery

Question 23

Next topic: Why do we want more firms engaging in R&D

Final equation

Answer 23

Shorterns time of discovery

final equation was
1 / (1-δ)²

Question 24

Discrete time (infinity periods)
Each period an event can occur with probability a

What is the probability of event occuring period 2?

Answer 24

It won’t occur in period 1, so Pr = (1-a)

occurs in period 2 Pr = a

so answer is Pr(t=2) = (1-a)a

Question 25

If 2 rounds (no longer infinity rounds), what is expected probability of event occuring? Pg26

Answer 25

could occur at period 1, (a) Or period 2, given it didn’t occur in period 1 (1-a)a Add the 2 together Expected prob = a + (1-a)a = a+a-a² =a(2-a)

Question 26

Assuming 2 rounds, what is expected DATE ET(t=2) of event happening?

Answer 26

Basically x each term by the period time, in t=1 probability of occuring is a, hence why x1. Probability of occuring in t=2, given it didn’t occur in 1 is (1-a)a , hence why x2. ET(t=2) = α×1 + (1−α)α×2 =3α−2α² =α(3−2α)

Question 27

Pg28 What is the probability of discovery if 3 periods? B) So what is the expected time of discovery? C) relationship found

Answer 27

Recall: Prob of discovery in p1 = a prob of disc in p2 = (1-a)a So answer: prob of disc in p3 = (1-a)(1-a)a i.e (1-a)²a B) same as last FC, x probabilities by their t! ET(n=1) = 1α + 2(1 − α)α + 3(1 − α)²α + .. Which simplifies to ET(n=1) = 1/a b) Shows as prob of discovery (a) bigger, expected time of discovery falls!

Question 28

Now when n=2 (2 firms) What is the probability of neither firm discovering, in any given period? b) given this, what is the remaining probability, and what does it signify

Answer 28

(1-a)² b) 1 - (1-a)² = a(2-a) which is probability of at least one firm making the discovery (since the first equation is probability of NEITHER discovering, so only one firm needs to discover to make untrue)

Question 29

Now we have established probability of neither discovering in any given period (1-a)², and remaining probability of at least one discovering in any given period is a(2-a) What is probability of at least one discovering in period 3? b) we can find expected time of discovery ET(n=2) c) how does this compare to expected time of discovery with 1 firm ET(1)

Answer 29

(1-a)²(1-a)² = (1-a)⁴ b) ET(2) = 1a(2-a) + 2(1-a)²a(2-a) + 3(1-a)⁴a(2-a) + … simplifies to ET(2) = 1/ a(2-a) c) ET(2) < ET(1) i.e having more firms decreases time of discovery!

Question 30

So ET(1) = 1/a ET(2) = 1/a(2-a) For both equations, the denominator is the probability of discovery (first one is just probability of discovery as only 1 firm, second one is probability of at least one firm discovery) So what do we want for lower expected time of discovery (2)

Answer 30

higher a more firms

Question 31

Should firms work together in R&D? 2 firms, 2 stages Assume market demand p = 100 - q What are 2 stages, which do we solve first

Answer 31

Stage 1: firms choose to cooperate or not in R&D expenditure Stage 2: firms compete cournot competion (quantity) we solve stage 2 first for quantity price and profit, then use for stage 1.

Question 32

R&D is cost reducing ci(xi,xj)≡ 50−xi−βxj 1>β>0 B is externalities unit cost of production is decreasing but R&D is costly TCi(xi) = x²i / 2

Question 33

Start by solving stage 2 (cournot) working pg35 b) what after that

Answer 33

Find quantity price and profit functions b) then use profit function for stage 1 (did some working for it on pg 35, to solve for Xi that maximises π1. From then, need to sub that Xnc into C’s of pg34 profit function to get it in terms of X.

Question 34

So that was if R&D levels are chosen individually. What if R&D contributed jointly

Answer 34

Stage 2 still the same, but stage 1 stage 1 now maximise profit function maxπ1 + π2 since jointly maximise profit now. do FOC to find Xc (Xnc previously) that maximises profit. then sub Xc into C’s of original profit function to get in terms of X, to be able to make comparisons

Question 35

Key findings after derivartions: is cooperation in R&D better than individaul noncooperating towards R&D?

Answer 35

yes. Cooperation in R&D increases firms’ profits

Question 36

What if R&D spillover effect is large (β>1/2) vs <1/2

Answer 36

then cooperative R&D level are higher than noncooperative ones but if β<1/2 then better to privately invest in R&D hence why people try to estimate β to find whether government should contribute or not