Additional Appendix (not intended for publication)
A.1 Supplements for Welfare Analysis
In this appendix, we describe some details about the welfare analysis in Section
7.2, where the relation between IPR protection and social welfare is investigated.
Focusing our analysis on the balanced growth path (hereafter, BGP), we assume
that the degree of IPR protection is kept constant at δ and the economy is on theBGP at time t. In order to clarify the dependence of steady state variables on δ, weindicate their values by “∗” and “(δ)”; e.g., w∗ (δ), µ∗ (δ), ... etc.
Since the economy is on the BGP, the log of output at time τ ≥ t is given byln Yτ∗ (δ) = ln Yt∗ (δ) + γY∗ (δ)(τ − t), where Yt∗ (δ) is the present level of output andγY∗ (δ) is the growth rate of output on the BGP, defined by (24). Substituting thisexpression and ct = Yt /L into (1) yields the utility of household as of time t:Ut∗ (δ) =∞Xτ =t

τβ lnµYτ∗ (δ)
L¶1=1−βµln Yt∗ (δ) +¶β∗γ (δ) − ln L .
1−β Y

(37)

From the discussions in Section 3.1, the output of a monopolistic sector and a competitive sector are given by 1/λw∗ (δ) and 1/w∗ (δ), respectively. Then we obtain thepresent level of output Yt∗ (δ) as follows:
Z 1
Z 1
∗
qit di +ln xit diln Yt = (ln λ)00
Z 1= (ln λ)qit di + µ∗ (δ) ln(λw∗ (δ)) + (1 − µ∗ (δ)) ln(1/w∗ (δ))
0Z 1= (ln λ)qit di − µ∗ (δ)(ln λ) − ln w∗ (δ).

(38)

0In the remainder of the appendix, we assume that b is suﬃciently large that therate of economic growth is maximized at δ max . We focus on the welfare eﬀect ofIPR protection when the level of IPR protection is changed within the neighborhoodof δ = δ max . Note that, since there is no leapfrogging in this case (see Panel 1 ofFigure 5), the equilibrium number of researchers and the wage level is determinedby (15). In the neighborhood of δ = δ max , the number of researchers is positive, andi

therefore condition (15) holds with equality. The condition can be written in termsof steady-state variables:
1 − πµ∗ (δ)= βV (δ)g(n∗ (δ)) = w∗ (δ).
L − (1 − µ∗ (δ))n∗ (δ)(39)

From the first equality, we obtain:
1 − πµ∗ (δ) =βV (δ) [Lg(n∗ (δ)) − (1 − µ∗ (δ))G(n∗ (δ))]=βV (δ) [Lg(n∗ (δ)) − δµ∗ (δ)] ,(40)where identity G(n∗ ) = g(n∗ )n∗ (δ) and stationarity condition (1 − µ∗ (δ))G(n∗ (δ)) =δµ∗ (δ) are used. Solving (40) for n∗ (δ) and substituting the result into (39) gives:¶µ1−β∗−1∗(41)w (δ) = Lπµ (δ) .
1−1 − β + βδ
By substituting (41) into (38), the present level of output can be written in terms ofµ∗ (δ) as follows:
ln Yt∗ (δ) = − (ln λ)µ∗ (δ) − ln(1 − β + βδ − (1 − β)πµ∗ (δ))
Z 1+ ln(1 − β + βδ) + ln L + (ln λ)qit di.

(42)0The following examines whether relaxing IPR protection marginally from thegrowth-maximizing level, δ max , improves or deteriorates the social welfare. Diﬀerentiating (37) with respect to δ gives:dγY∗ (δ)
1 d ln Yt∗ (δ)βdUt∗ (δ)
=+.
dδ
1−βdδ
(1 − β)2 dδ

(43)Note that, since γY∗ (δ) is maximized at δ = δ max , its derivative at δ max is zero.
Therefore, at δ = δ max , (43) can be written as:¯¯
1 d ln Yt∗ (δ) ¯¯dUt∗ (δ) ¯¯=¯ max ,dδ ¯δ=δmax
1−βdδ
δ=δ

where, from (42) and (1 − β)/β ≡ r:

d ln Yt∗ (δ)
dµ∗ (δ)1= −(ln λ)+dδdδ
r + δ − rπµ∗ (δ)
iiµ¶dµ∗ (δ)
1rπ−1 +.
dδr+δ

(44)

(45)

Figure 8: The values of parameters are λ = 1.5, β = 0.95, a = 10−8 .

Note that, since γY∗ (δ)/(ln λ) = (1 − µ∗ (δ))G(n∗ (δ)) = δµ∗ (δ) is maximized at δ =δ max , it must be true that:dµ∗ (δ max )µ∗ (δ max )= − max .
dδδ

(46)

Substituting (45) and (46) into (44) gives:¯·¸
1rπµ∗ (δ max ) + δ maxδ maxdUt∗ (δ) ¯¯∗ max=+(ln λ)µ (δ ) −dδ ¯δ=δmax (1 − β)δ max
r + δ max − rπµ∗ (δ max ) r + δ max(ln λ)rπµ∗ (δ max )=−·(47)(1 − β)δ max (r + δ max − rπµ∗ (δ max ))¶¸µ·r + δ max
1r
∗ max
−2 .
−µ (δ ) −rπln λ r + δ max
A marginal increase in δ from δ max (i.e., a marginal relaxation of IPR protection
from the growth-maximizing level) is welfare improving if and only if expression (47)is positive. From (47), it turns out that this is the case if and only if:¶µr + δ max
1r∗ maxmax−2 .
+µ (δ ) < f (δ ) ≡πrln λ r + δ maxiii(48)

It is straightforward to confirm that the value of f (δ max ) is positive and that it takesa larger value when δ max is larger. Figure 8 depicts a representative shape of f (δ max )in (δ, µ) space. A marginal increase in δ from δ max is beneficial for welfare if andonly if the pair (δ max , µ∗ (δ max )) is below the curve of f (δ max ).
Note that the position of the curve of f (δ max ) only depends on β and λ (recallthat r = (1 − β)/β and π = (λ − 1)/λ), whereas the (δ max , µ∗ (δ max )) pair canchange depending on other parameters. For example, as depicted in Figure 7, thegrowth-maximizing level of IPR protection (δ max ) changes significantly with the sizeof population L. The downward-sloping curve in Figure 8 depicts the locus of (δ max ,µ∗ (δ max )) as L is gradually increased. This numerical example shows that relaxationof IPR protection is beneficial whenever the economy has a reasonably large population such that the growth-maximizing level of IPR protection allows more thanaround 1.5 % of state-of-the-art goods to be imitated within a period. For a reasonable range of parameters, we confirmed that µ∗ (δ max ) < f (δ max ) is likely to holdunless δ max is quite small.

A.2 Endogenous Imitation
In the text, it is assumed that the authority can directly choose the probability ofimitation. In this appendix, we discuss how the behavior of the economy changes if
the authority can choose the diﬃculty of imitation by outsiders, and therefore theprobability of imitation is determined endogenously.
Suppose that, in a monopolized sector, outsiders may try to imitate the incumbent
monopolist’s good at the beginning of each period. A worker who attempts thissucceeds with probability c, and fails with probability 1 − c. If he (or she) fails, hecan work neither as a production worker nor a researcher. If he succeeds, he can hireproduction workers and produce the imitated good at the marginal cost of wt (thesame as the monopolist). We assume that, when some outsider successfully imitates
the state-of-the-art good, only the original monopolist and successful imitator canproduce goods. However, from the next period onwards, all outsiders can learniv

costlessly how to imitate. (In reality, imitated goods are unlikely to be protected bypatents.)
In this setting, the incentives of imitators derive only from the profit that canbe obtained in the period when they imitate successfully. In addition, the amountof this profit varies with the number of successful imitators, since it is possible that
more than one imitator can succeed simultaneously. For simplicity, we allow theoriginal monopolist and successful imitators to collude so that the monopoly priceis maintained and the monopoly profit is divided equally among them. Then, if kimitators succeed simultaneously, each successful imitator’s payoﬀ is π/(1 + k).
The expected profit from imitative activity is calculated as follows. Let l denotethe number of outsiders who are trying to imitate the monopolist’s good. Then, theprobability that k of them succeed is given by the binomial distribution:l!
ck (1 − c)l−k .
k!(l − k)!
Conditional upon this, the probability that one imitator succeeds is k/l. Therefore,
the expected profit from imitative activity is given by:ρ(l, c) ≡lX

π kl!
ck (1 − c)l−k ,1+klk!(l−k)!
k=1which is decreasing in l and increasing in c. Likewise, the expected profit of theincumbent monopolist for one period is given by:πe(l, c) ≡lX

π kl!
ck (1 − c)l−k .
1+kl
k!(l−k)!
k=0

Observe that the monopolist earns the largest profit when k = 0, whereas imitators
can earn profits only when k ≥ 1.
The number of imitators is determined by the free entry condition:
ρ(l, c) ≤ w

with equality if l > 0,

which shows that l is determined by c and w so that we can write l = l(c, w). Fromthe property of ρ(·), l(c, w) is increasing in c and decreasing in w. Those results alsovhold for the endogenous rate of imitation, δ = 1 − (1 − c)l . Therefore, we can write:δ = 1 − (1 − c)l(c,w) = D(c, w),(49)

where function D(·) is increasing in c and decreasing in w.
Suppose that the authority can choose the ease of imitation c. Now consider how
c should be set when the authority wants to achieve a desired rate of imitation δ.
By inverting equation (49), we obtain:c = C(δ, w),(50)where C(·) is increasing in both δ and w. That is, imitation must be easier when theauthority desires a higher rate of imitation, or when the equilibrium wage (i.e., theopportunity cost of imitative activity) is higher. Using (50), the number of imitators
can be written in terms of δ and w:el(δ, w) = l(C(δ, w), w).

el(·) is increasing in δ, but the sign of elw is ambiguous.

The remaining analysis proceeds in essentially the same way as in Section 5.

Recall that the BGP is characterized by (20)-(23). Now profit π in (21) and (22)must be replaced by πe(el(δ, w∗ (z, δ)), C(δ, w∗ (z, δ))). In addition, we must subtractl(δ, w∗ (z, δ)) from the denominator of (22).
the aggregate number of imitators µ∗ (z, δ)eThis modified system is solvable in principle since the number of unknown variables is
the same as in the original system. Although we cannot obtain the explicit solution,
it can be expected that the solution would be qualitatively similar to that obtained inSection 5, given that the significance of above modifications in the aggregate economy
is not large.

vi