Page 1 of 26

Online Appendix

A The Optimal Knowledge Stock

The firm then chooses the optimal Ki that maximizes global net profits,

i ≠ c (Ki), i.e.

max zi

Y

]

[

ÿ

n

S

U

A ›Ki

·jmn B‡≠1

Bin

T

V ≠ ÂKk

i

Z

^

\

The first order condition is

(‡ ≠ 1) K‡≠2

i

ÿ

n

S

U

A ›

·jmn B‡≠1

Bin

T

V = kÂKk≠1

i

Ki = Ÿ

Aÿ

n

· 1≠‡ jmnBinB1/[k≠(‡≠1)]

,

where Ÿ © [›‡≠1 (‡ ≠ 1) / (kÂ)]1/[k≠(‡≠1)].

The second order condition is, inserting the expression for the optimal Ki,

(‡ ≠ 1) (‡ ≠ 2) K‡≠3

i

ÿ

n

S

U

A ›

·jmn B‡≠1

Bin

T

V ≠ k (k ≠ 1) ÂKk≠2

i < 0

Ÿk≠(‡≠1) (‡ ≠ 2) K‡≠3

i

ÿ

n

Ë

· 1≠‡ jmnBinÈ

≠ (k ≠ 1) Kk≠2

i < 0

(‡ ≠ 2) K‡≠3

i Kk≠(‡≠1)

i ≠ (k ≠ 1) Kk≠2

i < 0

(‡ ≠ 1 ≠ k) Kk≠2

i < 0

which holds given that k ≠ (‡ ≠ 1) > 0.

B Fixed Exporting Costs

This section develops an extension of the benchmark model with fixed costs of exporting. In

order to serve a market, a firm i must incur a fixed cost f (n) to market n. Net profits selling

40

Page 2 of 26

to n are then fii (n)=(z/·jm (n))‡≠1 Bi (n) ≠ f (n). For analytical convenience, consider

the case with a unit continuum of countries. Without loss of generality, countries are sorted

according to their profitability, fiim (n), from high to low.

The firm faces two choices: first, how much to innovate and second, where to sell. We

start with the second problem. Because of the presence of market-specific fixed costs, firms

will only export to countries that give them positive net profits, fiim (n) > 0. Label the

destination with zero profits n ̄i, i.e. fiijm ( ̄ni)=0. Global profits are then

i =

⁄ n ̄i

0

S

U

A zi

·jm (n)

B‡≠1

Bi (n) ≠ f (n)

T

V dn.

We now turn to the problem of how much to innovate. Maximizing

i ≠c (Ki) and using

Leibniz’ integral rule yields

Ki = Ÿ

3⁄ n ̄i

0

·jm (n)

1≠‡ Bi (n) dn41/[k≠(‡≠1)]

.

In changes, we obtain

Kˆi =

C⁄ n ̄i

Õ

0

Êij (n) ˆ·jm (n)

1≠‡ Bˆi (n) dnD1/[k≠(‡≠1)]

,

where n ̄Õ is the marginal destination country in the counterfactual equilibrium and Êij (n) is

the gross profit shares in the initial equilibrium,

Êij (n) = ·jm (n)

1≠‡ Bi (n)

s n ̄

0 ·jm (o)

1≠‡ Bi (o) do.

We observe that when n ̄Õ

i = ̄ni, we get a similar expression as equation (2) in the main text.

41

Page 3 of 26

C Approximation of the Knowledge Production Function

The expression Kˆi = 1q

n ÊinBˆin·ˆ1≠‡ jmn21/[k≠(‡≠1)] can be approximated by equation (3) in

the main text,

ln Ki = — q

nœ

i Êin

Tjmn + Ái, where — © (1 ≠ ‡) / [k ≠ (‡ ≠ 1)] and

Ái © [k ≠ (‡ ≠ 1)]≠1 q

n Êin

ln Bin.

Proof. The term

ÿ

n

ÊinBˆin·ˆ1≠‡ jmn = ÿ

n

Êine(1≠‡)

ln ·jmn+

ln Bin

¥ ÿ

n

Êin (1 + (1 ≠ ‡)

ln ·jmn +

ln Bin)

= 1+ ÿ

n

Êin ((1 ≠ ‡)

ln ·jmn +

ln Bin),

where we used the fact that ln (1 + x) ¥ x ≈∆ 1 + x ¥ ex for x close to 0. Hence,

ln Ki = 1

k ≠ (‡ ≠ 1) ln C

1 + ÿ

n

Êin ((1 ≠ ‡)

ln ·jmn +

ln Bin)

D

¥

1

k ≠ (‡ ≠ 1)

ÿ

n

Êin ((1 ≠ ‡)

ln ·jmn +

ln Bin)

= 1

k ≠ (‡ ≠ 1) A

(1 ≠ ‡)

ÿ

n

Êin

Tjmn + ÿ

n

Êin

ln BinB

,

where we used

ln ·jmn =

ln (1 + Tjmn) ¥

Tjmn for Tjmn close to 0.

We evaluate the performance of the approximation using a numerical example. We

evaluate the true and approximated knowledge production functions in equations (2) and

(3) in the main text, respectively. Figure 4 shows the change in log knowledge production,

ln Ki, as a function of the tari

in the new equilibrium, T1, under the assumption that the

initial tari

is 20 percent, T0 = 0.2. Given a slope coe

cient — = ≠1, the approximation

performs well, especially for relatively small changes in tari

s. If there is more curvature,

i.e. — > ≠1 or — < ≠1, the approximation performs worse, especially for large changes in

tari

s.

42