Technology Transfer in Oligopoly in Presence of Fixed-Cost in Production

Abstract

This paper discusses different possibilities of licensing between Cournot duopolists with a technology characterized by constant marginal cost and positive fixed-cost. It is shown that optimal fixed-fee and per-unit royalty can be negative. This implies that the licensor can subsidize the licensee in the equilibrium to license its technology even if they compete in the output market in the post-licensing stage. Moreover superiority of up-front fee as an advance on royalties is also identified. The firm having lower marginal cost and higher fixed cost prefers royalty licensing than licensing by fixed-fee, while the other firm having lower fixed cost and higher marginal cost sometimes prefers fixed-fee licensing. Interestingly, a firm that earns less profit in the no-licensing stage may become the licensor of its technology.

Appendices

The author wishes to thank an anonymous referee for suggesting this extension.

Appendix 1

From Eq. 5 of the main text we have

$${\Pi}_{1}^{r_{2}}(\bar{r_{2}})=\frac{[a-2(c_{2}+\bar{r_{2}})+c_{2}]^{2}}{9}-F_{2}={\Pi}_{1} $$

as firm 2 set the royalty rate as high as possible such that firm 1 is indifferent between licensing and not. From the above equation we get

$$a-c_{2}-2\bar{r_{2}}=\sqrt{(a-2c_{1}+c_{2})^{2}+9(F_{2}-F_{1})} $$

and it must be positive because if $a-c_{2}-2\bar {r_{2}}<0$ then ${\Pi }_{1}^{r_{2}}(\bar {r_{2}})<{\Pi }_{1}$ and licensing will never be possible. Hence, $\bar {r_{2}}=\frac {1}{2}\left [(a-c_{2})-\sqrt {(a-2c_{1}+c_{2})^{2}+9(F_{2}-F_{1})}\right ]$ and substituting in ${\Pi }_{2}^{r_{2}}(\bar {r_{2}})$ (defined in the main text) gives

$${\Pi}_{2}^{r_{2}}(\bar{r_{2}})=\frac{(a-c_{2})^{2}}{4}-\frac{5(a-2c_{1}+c_{2})^{2}}{36}-\frac{5(F_{2}-F_{1})}{4}-F_{2}. $$

This is the net profit of firm 2 if licensing is undertaken. Moreover for licensing to be profitable for firm 2, ${\Pi }_{2}^{r_{2}}(\bar {r_{2}})$ must be greater than Π₂. This is possible if

$$ X^{r}(c_{1})=\frac{9(a-c_{2})^{2}-5(a-2c_{1}+c_{2})^{2}-4(a-2c_{2}+c_{1})^{2}}{45}\geq F_{2}-F_{1} $$

(A.1)

which is relation (6) of the main text, the necessary and sufficient condition for licensing with royalty from firm 2 to firm 1.

Appendix 2

As discussed in the main text firm 1 will charge the royalty rate as high as possible ($\bar {r_{1}}$) such that

$${\Pi}_{2}^{r_{1}}(\bar{r_{1}})=\frac{[a-2(c_{1}+\bar{r_{1}})+c_{1}]^{2}}{9}-F_{1}={\Pi}_{2}. $$

At this royalty rate firm 2 is indifferent between licensing and not. Manipulating the above equation we get

$$a-c_{1}-2\bar{r_{1}}=\sqrt{(a-2c_{2}+c_{1})^{2}+9(F_{1}-F_{2})}. $$

Moreover $a-c_{1}-2\bar {r_{1}}$ must be positive. As if $a-c_{1}-2\bar {r_{1}}<0$ then ${\Pi }_{2}^{r_{1}}(\bar {r_{1}})<{\Pi }_{2}$ and licensing will never be possible as firm 2 will always reject the offer. From the above equation we thereby get $\bar {r_{1}}=\frac {1}{2}[(a-c_{1})-\sqrt {(a-2c_{2}+c_{1})^{2}+9(F_{1}-F_{2})}]$. Further substituting $\bar {r_{1}}$ in ${\Pi }_{1}^{r_{1}}(\bar {r_{1}})$ (defined in the main text) gives

$${\Pi}_{1}^{r_{1}}(\bar{r_{1}})=\frac{(a-c_{1})^{2}}{4}-\frac{5(a-2c_{2}+c_{1})^{2}}{36}+\frac{5(F_{2}-F_{1})}{4}-F_{1}. $$

Moreover ${\Pi }_{1}^{r_{1}}(\bar {r_{1}})\geq {\Pi }_{1}$ if

$$ F_{2}-F_{1}\geq \frac{4(a-2c_{1}+c_{2})^{2}+5(a-2c_{2}+c_{1})^{2}-9(a-c_{1})^{2}}{45}=Y^{r}(c_{1}). $$

(B.1)

This is relation (9) of the main text which is the necessary and sufficient condition for licensing to be possible. As otherwise if the above inequality is not satisfied firm 1 will not be better off from licensing its technology and thereby not offer any royalty rate to firm 2.

Appendix 3 General Demand Structure

The author wishes to thank an anonymous referee for suggesting this extension.

1.1 Appendix 3.1 Basic Set-Up

The inverse market demand is P = f(q), where P is the price, q = q ₁ + q ₂ is the total output and q _i , i = 1,2, is the output produced by the firm i. Assume same cost structure as in the main text.

Assumptions 1

• i) f ^′(q)<0,

• ii) ∃ q ₀ > 0 such that f(q) > 0 for q < q ₀ and f(q)≤0 for q≥q ₀ and

• iii) f ^′(q) + q _i f ^″(q)≤0 for i = 1,2.

For firm i the profit function is Π _i = [f(q)−c _i ]q _i −F _i , where c _i and F _i are the unit cost and fixed cost of firm i respectively. For profit maximization by firm i: F.O.C. is f(q)−c _i + q _i f ^′(q)=0 and S.O.C. is 2f ^′(q) + q _i f ^″(q)<0. The above assumptions are standard and guarantee the uniqueness and stability of Cournot equilibrium (see Vives 2001). These assumptions also ensure the uniqueness and stability of Cournot equilibrium is case of licensing. From the F.O.Cs. and S.O.Cs. it can be said that if the unit cost of any firm increases industry output will fall.

1.1.1 No Licensing

If there is no licensing, the profits of firm 1 and firm 2 are ${\Pi }_{1}=[f(q^{*})-c_{1}]q_{1}^{*}-F_{1}$ and ${\Pi }_{2}=[f(q^{*})-c_{2}]q_{2}^{*}-F_{2}$ respectively; $q^{*}=q_{1}^{*}+q_{2}^{*}$, where $q_{1}^{*}$ and $q_{2}^{*}$ are the optimal output of the firm 1 and firm 2 respectively. As c ₁ > c ₂, therefore $q_{1}^{*}<q_{2}^{*}$. These profits are assumed to be positive, which implies that $F_{2}\in \left (0,\bar {F_{2}}=[f(q^{*})-c_{2}]q_{2}^{*}\right )$, $F_{1}\in \left (0,\bar {F_{1}}=[f(q^{*})-c_{1}]q_{1}^{*}\right )$ and $F_{2}-F_{1}\in (0,\bar {F_{2}})$ as F ₂ > F ₁.

1.2 Appendix 3.2 Fixed-Fee Licensing

Firm i offers a fixed-fee (T _i ) to license its technology to firm j. The profit(net) of the firm i and firm j respectively are ${\Pi }_{i}^{T_{i}}$ and ${\Pi }_{j}^{T_{i}}$ for i,j = 1,2 when the offer of T _i is accepted by firm j and technology is transferred.

1.2.1 Offer by Firm 2

Firm 2 offers T ₂ such that ${\Pi }_{1}^{T_{2}}=[f(2q^{a})-c_{2}]q^{a}-F_{2}-T_{2}={\Pi }_{1}$, where q ^a is the output produced by each firm if firm 2 licenses its technology to firm 1. Therefore if ${\Pi }_{2}^{T_{2}}=[f(2q^{a})-c_{2}]q^{a}-F_{2}+T_{2}\geq {\Pi }_{2}$, such that licensing is profitable for firm 2 or

$$ X=2[f(2q^{a})-c_{2}]q^{a}-[f(q^{*})-c_{1}]q_{1}^{*}-[f(q^{*})-c_{2}]q_{2}^{*}\geq F_{2}-F_{1} $$

(C.1)

technology will be licensed at T ₂ > 0. Moreover after licensing,

$$ {\Pi}_{2}^{T_{2}}=2[f(2q^{a})-c_{2}]q^{a}-[f(q^{*})-c_{1}]q_{1}^{*}-2F_{2}+F_{1}. $$

(C.2)

As the unit cost of firm 1 reduces after transfer the industry output also increases after transfer, or $2q^{a}>q_{1}^{*}+q_{2}^{*}$.

1.2.2 Offer by firm 1

Firm 1 offers T ₁ such that ${\Pi }_{2}^{T_{1}}=[f(2q^{b})-c_{1}]q^{b}-F_{1}-T_{1}={\Pi }_{2}$, where q ^b is the output produced by each firm if firm 1 licenses its technology to firm 2.²⁸ As the unit cost of firm 2 increases after transfer the industry output gets reduced after transfer, or $2q^{b}<q_{1}^{*}+q_{2}^{*}$. Moreover, ${\Pi }_{1}^{T_{1}}=[f(2q^{b})-c_{1}]q^{b}-F_{1}+T_{1}\geq {\Pi }_{1}$ must hold, such that licensing is profitable for firm 1 or

$$ Y=[f(q^{*})-c_{1}]q_{1}^{*}+[f(q^{*})-c_{2}]q_{2}^{*}-2[f(2q^{b})-c_{1}]q^{b}\leq F_{2}-F_{1}. $$

(C.3)

If technology is licensed then

$$ {\Pi}_{1}^{T_{1}}=2[f(2q^{b})-c_{1}]q^{b}-[f(q^{*})-c_{2}]q_{2}^{*}-2F_{1}+F_{2} $$

(C.4)

and T ₁≥0 if $L=[f(q^{*})-c_{2}]q_{2}^{*}-[f(2q^{b})-c_{1}]q^{b}\leq F_{2}-F_{1}$ (this is possible as $[f(2q^{b})-c_{1}]q^{b}>[f(q^{*})-c_{1}]q_{1}^{*}$) and T ₁ < 0 otherwise.

Moreover, assume Π₂ > Π₁ or $J=[f(q^{*})-c_{2}]q_{2}^{*}-[f(q^{*})-c_{1}]q_{1}^{*}>F_{2}-F_{1}$ and Y < J (as $[f(2q^{b})-c_{1}]q^{b}>[f(q^{*})-c_{1}]q_{1}^{*}$). Therefore if Y≤F ₂−F ₁ < J, firm 1 even though earning a lower profit in the no-licensing stage, will license its technology to firm 2.

1.3 Appendix 3.3 Royalty Licensing

Firm i offers a per-unit royalty (r _i ) to license its technology to the other firm. Let the profit(net) of the firm i and firm j are respectively ${\Pi }_{i}^{r_{i}}$ and ${\Pi }_{j}^{r_{i}}$ for i,j = 1,2 when the offer of r _i is accepted by firm j and technology is transferred.

1.3.1 Offer by Firm 2

Firm 2 offers a per-unit royalty (r ₂) to firm 1 to license its technology. Let the profits of firm 1 and firm 2 respectively after the licensing be ${\Pi }_{1}^{r_{2}}=[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}-r_{2}]q_{1}^{r_{2}}-F_{2}$ and ${\Pi }_{2}^{r_{2}}=[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}]q_{2}^{r_{2}}-F_{2}+r_{2}q_{1}^{r_{2}}$, where $q_{i}^{r_{2}}$ is the output of firm i.

The F.O.Cs for profit maximization of firm 1 and firm 2 respectively for given r ₂ are

$$\begin{array}{@{}rcl@{}} f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}-r_{2}+q_{1}^{r_{2}}f^{\prime}(q_{1}^{r_{2}}+q_{2}^{r_{2}})=0\\ f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}+q_{2}^{r_{2}}f^{\prime}(q_{1}^{r_{2}}+q_{2}^{r_{2}})=0 \end{array} $$

(C.5)

and the S.O.Cs are $2f^{\prime }(q_{1}^{r_{2}}+q_{2}^{r_{2}})+q_{i}^{r_{2}}f^{\prime \prime }(q_{1}^{r_{2}}+q_{2}^{r_{2}})<0$ for i = 1,2.

Using the F.O.Cs and S.O.Cs it can be shown that

$$ \frac{\delta {\Pi}_{2}^{r_{2}}}{\delta r_{2}}=q_{1}^{r_{2}}f^{\prime}(q_{1}^{r_{2}}+q_{2}^{r_{2}})\frac{\delta q_{1}^{r_{2}}}{\delta r_{2}}+q_{1}^{r_{2}}>0 $$

(C.6)

as $\frac {\delta q_{1}^{r_{2}}}{\delta r_{2}}<0$. Therefore firm 2 will charge r ₂ as high as possible such that ${\Pi }_{1}^{r_{2}}={\Pi }_{1}$ but c ₂ + r ₂ < c ₁, as otherwise technology will never be licensed due to the increase in fixed-cost for firm 1. Finally technology is licensed to firm 1 if ${\Pi }_{2}^{r_{2}}\geq {\Pi }_{2}$ or

$$[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}]q_{2}^{r_{2}}+r_{2}q_{1}^{r_{2}}\geq[f(q_{1}^{*}+q_{2}^{*})-c_{2}]q_{2}^{*}. $$

As, $q_{1}^{r_{2}}+q_{2}^{r_{2}}>q_{1}^{*}+q_{2}^{*}$ and $q_{2}^{r_{2}}<q_{2}^{*}$, r ₂ must be positive. Therefore technology is licensed if

$$ X^{r}=[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}](q_{1}^{r_{2}}+q_{2}^{r_{2}})-[f(q^{*})-c_{1}]q_{1}^{*}-[f(q^{*})-c_{2}]q_{2}^{*}\geq F_{2}-F_{1} $$

(C.7)

and

$$ {\Pi}_{2}^{r_{2}}=[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}](q_{1}^{r_{2}}+q_{2}^{r_{2}})-[f(q^{*})-c_{1}]q_{1}^{*}-2F_{2}+F_{1}. $$

(C.8)

1.3.2 Offer by Firm 1

Firm 1 offers a per-unit royalty (r ₁) to firm 2 to license its technology. Let the profits of firm 2 and firm 1 respectively after the licensing be ${\Pi }_{2}^{r_{1}}=[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}-r_{1}]q_{2}^{r_{1}}-F_{1}$ and ${\Pi }_{1}^{r_{1}}=[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}]q_{1}^{r_{1}}-F_{1}+r_{1}q_{2}^{r_{1}}$, where $q_{i}^{r_{1}}$ is the output of firm i.

The F.O.Cs for profit maximization of firm 2 and firm 1 respectively given r ₁ are

$$\begin{array}{@{}rcl@{}} f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}-r_{1}+q_{2}^{r_{1}}f^{\prime}(q_{1}^{r_{1}}+q_{2}^{r_{1}})=0\\ f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}+q_{1}^{r_{1}}f^{\prime}(q_{1}^{r_{1}}+q_{2}^{r_{1}})=0 \end{array} $$

(C.9)

and the S.O.Cs are $2f^{\prime }(q_{1}^{r_{1}}+q_{2}^{r_{1}})+q_{i}^{r_{1}}f^{\prime \prime }(q_{1}^{r_{1}}+q_{2}^{r_{1}})<0$ for i = 1,2.

As in the previous case, using the F.O.Cs and S.O.Cs it can be shown that

$$ \frac{\delta {\Pi}_{1}^{r_{1}}}{\delta r_{1}}=q_{2}^{r_{1}}f^{\prime}(q_{1}^{r_{1}}+q_{2}^{r_{1}})\frac{\delta q_{2}^{r_{1}}}{\delta r_{1}}+q_{2}^{r_{1}}>0. $$

(C.10)

Therefore firm 1 will charge r ₁ as high as possible such that ${\Pi }_{2}^{r_{1}}={\Pi }_{2}$. Finally firm 1 will license technology if ${\Pi }_{1}^{r_{1}}\geq {\Pi }_{1}$ or

$$[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}]q_{1}^{r_{1}}+r_{1}q_{2}^{r_{1}}\geq[f(q_{1}^{*}+q_{2}^{*})-c_{2}]q_{2}^{*}. $$

Therefore technology is licensed if

$$ Y^{r}=[f(q^{*})-c_{1}]q_{1}^{*}+[f(q^{*})-c_{2}]q_{2}^{*}-[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}}+ q_{2}^{r_{1}})\leq F_{2}-F_{1} $$

(C.11)

and

$$ {\Pi}_{1}^{r_{1}}=[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}}+q_{2}^{r_{1}})-[f(q^{*})-c_{2}]q_{2}^{*}-2F_{1}+F_{2}. $$

(C.12)

Moreover, r ₁≥0 if $L=[f(q^{*})-c_{2}]q_{2}^{*}-[f(2q^{b})-c_{1}]q^{b}\leq F_{2}-F_{1}$ (this is possible as $[f(2q^{b})-c_{1}]q^{b}>[f(q^{*})-c_{1}]q_{1}^{*}$) and r ₁ < 0 otherwise.

Moreover, assume $J=[f(q^{*})-c_{2}]q_{2}^{*}-[f(q^{*})-c_{1}]q_{1}^{*}>F_{2}-F_{1}$ such that Π₂ > Π₁. Here also Y ^r < J (as $[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}]q_{1}^{r_{1}}>[f(q^{*})-c_{1}]q_{1}^{*}$). Therefore if Y ^r≤F ₂−F ₁ < J, firm 1 even though earning a lower profit in the no-licensing stage, will license its technology to firm 2.

1.4 Appendix 3.4 Royalty vs Fixed-Fee

1.4.1 Firm 2’s Choice

Let us assume that Eqs. C.1 and C.7 are satisfied simultaneously such that firm 2 transfers its technology both in case of fixed-fee as well as royalty licensing. In case of fixed-fee licensing firm 2 gets (as in Eq. C.2)

$${\Pi}_{2}^{T_{2}}=2[f(2q^{a})-c_{2}]q^{a}-[f(q^{*})-c_{1}]q_{1}^{*}-2F_{2}+F_{1} $$

and in case of royalty licensing it gets (as in Eq. C.8)

$${\Pi}_{2}^{r_{2}}=[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}](q_{1}^{r_{2}}+q_{2}^{r_{2}})-[f(q^{*})-c_{1}]q_{1}^{*}-2F_{2}+F_{1}. $$

Let

$$ G_{2}={\Pi}_{2}^{r_{2}}-{\Pi}_{2}^{T_{2}}=[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}](q_{1}^{r_{2}}+q_{2}^{r_{2}})-2[f(2q^{a})-c_{2}]q^{a}, $$

(C.13)

where G ₂ is the excess profit that firm 2 earns in case of royalty licensing than in case of fixed-fee.

If optimal r ₂ = 0, then G ₂ = 0 as ${\Pi }_{2}^{r_{2}}={\Pi }_{2}^{T_{2}}$ because $q^{a}=q_{1}^{r_{2}}=q_{2}^{r_{2}}$. Firm 2 always sets optimal r ₂ such that ${\Pi }_{1}^{r_{2}}={\Pi }_{1}$. When optimal r ₂ = 0 then

$$ {\Pi}_{1}^{r_{2}}=[f(2q^{a})-c_{2}]q^{a}-F_{2}=[f(q^{*})-c_{1}]q_{1}^{*}-F_{1}={\Pi}_{1}. $$

(C.14)

In case of fixed-fee licensing also, firm 2 sets optimal T ₂ such that ${\Pi }_{1}^{T_{2}}={\Pi }_{1}$. The above equation thus implies that if optimal r ₂ = 0 then optimal fixed-fee T ₂ = 0 as ${\Pi }_{1}^{T_{2}}=[f(2q^{a})-c_{2}]q^{a}-F_{2}-T_{2}=[f(q^{*})-c_{1}]q_{1}^{*}-F_{1}={\Pi }_{1}$. So it can be argued that if c ₂ = c ₁ and F ₂ = F ₁ then technology will transferred but r ₂ = 0 and T ₂ = 0.

Hence, if c ₁ > c ₂ and F ₁ < F ₂ (as assumed in the model), then technology if transferred will be at r ₂ > 0 (via royalty) or T ₂ > 0 (via fixed-fee) as discussed earlier in the appendix. However (using Eq. C.5)

$$ \frac{dG_{2}}{dr_{2}}=q_{1}^{r_{2}}f^{\prime}(q_{1}^{r_{2}}+q_{2}^{r_{2}})\left[\frac{dq_{2}^{r_{2}}}{d{r_{2}}}+\frac{dq_{1}^{r_{2}}}{d{r_{2}}}\right]>0 $$

(C.15)

as $q_{1}^{r_{2}}>0,f^{\prime }(.)<0$ and $[\frac {dq_{2}^{r_{2}}}{d{r_{2}}}+\frac {dq_{2}^{r_{2}}}{d{r_{2}}}]<0$. Therefore $G_{2}={\Pi }_{2}^{r_{2}}-{\Pi }_{2}^{T_{2}}>0$ always as optimal r ₂ > 0, if firm 2 licenses its technology by per-unit royalty. Hence, firm 2 will prefer royalty licensing than licensing by fixed-fee.

Alternative proof From the Eq. C.13 $[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}](q_{1}^{r_{2}} +q_{2}^{r_{2}})$ and 2[f(2q ^a)−c ₂]q ^a can be considered as the industry profit, when the unit cost of production is c ₂ (for both the firms) in absence of fixed cost, if the industry output is $q_{1}^{r_{2}}+ q_{2}^{r_{2}}$ and 2q ^a respectively. Moreover, $q^{m}<q_{1}^{r_{2}}+q_{2}^{r_{2}}<2q^{a}$ (where q ^m is the monopoly output produced by a firm using similar technology). As under royalty licensing (r ₂ > 0) the effective unit cost of firm 1 is r ₂ + c ₂ for which the industry output is lower in case of royalty licensing than in case of fixed-fee.²⁹ If the industry output is above q ^m (which maximizes the industry profit), the industry profit falls if output increases. Hence, $[f(q_{1}^{r_{2}}+q_{2}^{r_{2}})-c_{2}](q_{1}^{r_{2}}+q_{2}^{r_{2}})$ must be greater than 2[f(2q ^a)−c ₂]q ^a. Therefore $G_{2}={\Pi }_{2}^{r_{2}}-{\Pi }_{2}^{T_{2}}>0$ always as optimal r ₂ > 0 (and T ₂ > 0). Hence, firm 2 will prefer royalty licensing than licensing by fixed-fee.

1.4.2 Firm 1’s choice

Let us assume that Eqs. C.3 and C.11 are satisfied simultaneously such that firm 1 transfers its technology both in case of fixed-fee as well as royalty licensing. In case of fixed-fee licensing firm 1 gets (as in Eq. C.4)

$${\Pi}_{1}^{T_{1}}=2[f(2q^{b})-c_{1}]q^{b}-[f(q^{*})-c_{2}]q_{2}^{*}-2F_{1}+F_{2} $$

and in case of royalty licensing it gets (as in Eq. C.12)

$${\Pi}_{1}^{r_{1}}=[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}}+q_{2}^{r_{1}})-[f(q^{*})-c_{2}]q_{2}^{*}-2F_{1}+F_{2}. $$

Let

$$ G_{1}={\Pi}_{1}^{r_{1}}-{\Pi}_{1}^{T_{1}}=[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}}+q_{2}^{r_{1}})-2[f(2q^{b})-c_{1}]q^{b}, $$

(C.16)

where G ₁ is the excess profit that firm 1 earns in case of royalty licensing than in case of fixed-fee.

If optimal r ₁ = 0, then G ₁ = 0 as ${\Pi }_{1}^{r_{1}}={\Pi }_{1}^{T_{1}}$ because $q^{b}=q_{1}^{r_{1}}=q_{2}^{r_{1}}$. Firm 1 always sets optimal r ₁ such that ${\Pi }_{2}^{r_{1}}={\Pi }_{2}$. When optimal r ₁ = 0 then

$$ {\Pi}_{2}^{r_{1}}=[f(2q^{b})-c_{1}]q^{b}-F_{1}=[f(q^{*})-c_{2}]q_{2}^{*}-F_{2}={\Pi}_{2}. $$

(C.17)

In case of fixed-fee licensing also, firm 1 sets optimal T ₁ such that ${\Pi }_{2}^{T_{1}}={\Pi }_{2}$. The above equation thus implies that if optimal r ₁ = 0 then optimal fixed-fee T ₁ = 0 as ${\Pi }_{2}^{T_{1}}=[f(2q^{b})-c_{1}]q^{b}-F_{1}-T_{1}=[f(q^{*})-c_{2}]q_{2}^{*}-F_{2}={\Pi }_{2}$.

So it can be argued that given c ₁,F ₂ a n d F ₁ there exists a unique c ₂ say $c_{2}^{*}$ such that technology is transferred but r ₁ = 0 and T ₁ = 0. Hence, if $c_{2}>c_{2}^{*}$, then technology if transferred will be at r ₁ > 0 (via royalty) or T ₁ > 0 (via fixed-fee) as Π₂ falls if c ₂ increases. On the other hand if $c_{2}<c_{2}^{*}$, then technology may be transferred at r ₁ < 0 (via royalty) or T ₁ < 0 (via fixed-fee) as Π₂ rises if c ₂ falls. However (using Eq. C.9)

$$ \frac{dG_{1}}{d{r_{1}}}=q_{2}^{r_{1}}f^{\prime}\left( q_{1}^{r_{1}}+q_{2}^{r_{1}}\right) \left[\frac{dq_{2}^{r_{1}}}{d{r_{1}}}+\frac{dq_{1}^{r_{1}}}{d{r_{1}}}\right]>0 $$

(C.18)

as $q_{2}^{r_{1}}>0,f^{\prime }(.)<0$ and $[\frac {dq_{2}^{r_{1}}}{d{r_{1}}}+\frac {dq_{1}^{r_{1}}}{d{r_{1}}}]<0$. Therefore if firm 1 sets r ₁ > 0, then $G_{1}={\Pi }_{1}^{r_{1}}-{\Pi }_{1}^{T_{1}}>0$ and firm 1 will prefer royalty licensing than licensing by fixed-fee. However, if firm 1 sets r ₁ < 0, then $G_{1}={\Pi }_{1}^{r_{1}}-{\Pi }_{1}^{T_{1}}<0$ and firm 1 will prefer fixed-fee licensing than licensing by royalty. As firm 1 subsidizes (negative fixed-fee/negative royalty) if $L=[f(q^{*})-c_{2}]q_{2}^{*}-[f(2q^{b})-c_{1}]q^{b}> F_{2}-F_{1}$, hence if L > F ₂−F ₁ firm 1 will prefer fixed-fee licensing and royalty licensing otherwise.

Alternative proof From the Eq. C.16 $[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}} +q_{2}^{r_{1}})$ and 2[f(2q ^b)−c ₁]q ^b can be considered as the industry profit, when the unit cost of production is c ₁ (for both the firms) in absence of fixed cost, if the industry output is $q_{1}^{r_{1}}+q_{2}^{r_{1}}$ and 2q ^b respectively. Moreover, $q^{m}<q_{1}^{r_{1}}+q_{2}^{r_{1}}<2q^{b}$ (where q ^m is the monopoly output produced by a firm using similar technology) because under royalty licensing if r ₁ > 0 the effective unit cost of firm 1 is r ₁ + c ₁ for which the industry output is lower in case of royalty licensing than in case of fixed-fee. If the industry output is greater than q ^m (which maximizes the industry profit) the industry profit falls if output increases. Hence, $[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}}+q_{2}^{r_{1}})$ must be greater than 2[f(2q ^b)−c ₁]q ^b. Therefore $G_{1}={\Pi }_{1}^{r_{1}}-{\Pi }_{1}^{T_{1}}>0$ if optimal r ₁ > 0 (and T ₁ > 0). Hence, firm 1 will prefer royalty licensing than licensing by fixed-fee if optimal r ₁ > 0 and T ₁ > 0.

However, $q^{m}<2q^{b}<q_{1}^{r_{1}}+q_{2}^{r_{1}}$ if r ₁ < 0 as the effective unit cost of firm 1 is r ₁ + c ₁ (< c ₁) for which the industry output is greater in case of royalty licensing than in case of fixed-fee. If the industry output is above q ^m (which maximizes the industry profit) the industry profit falls if output increases. Hence, $[f(q_{1}^{r_{1}}+q_{2}^{r_{1}})-c_{1}](q_{1}^{r_{1}}+q_{2}^{r_{1}})$ must be lower than 2[f(2q ^b)−c ₁]q ^b. Therefore $G_{1}={\Pi }_{1}^{r_{1}}-{\Pi }_{1}^{T_{1}}<0$ if optimal r ₁ < 0 (and T ₁ < 0). Hence, firm 1 will prefer fixed-fee licensing than licensing by royalty if optimal r ₁ < 0 and T ₁ < 0.