Risk Aversion and Technology Portfolios

Abstract

This paper analyzes the choice of a technology portfolio by risk-averse firms. Two technologies with random marginal costs are available to produce a homogeneous good. If the risks that are associated with the technologies are correlated, then the firms might invest in a technology with a negative expected return or, conversely, might not invest in a technology with a positive expected return. If the technology with the lower expected cost is riskier than the other technology, then this “low-cost” technology will be eliminated from the firm’s portfolio if the risks are highly correlated. With imperfect competition, the portfolios of firms are different, and the difference in risk tolerance can explain the full specialization of the industry: The less risk-averse firms use the low-cost technology, and the more risk-averse firms use the less risky, higher-cost technology.

Appendices

Appendix 1: Supply Curve

1.1 Preliminary Results

The two first-order conditions (4) can be written:

$$\begin{aligned} \lambda \left[ \begin{array}{c@{\quad }c} \sigma _1^2 &{} \sigma _{12}\\ \sigma _{12} &{} \sigma _{2}^2 \end{array} \right] \left[ \begin{array}{c} q_1 \\ q_2 \end{array} \right] =\left[ \begin{array}{c} p-c_1 \\ p-c_2 \end{array} \right] . \end{aligned}$$

(11)

Let us denote $x_1$ and $x_2$ the solutions of these Eqs. (11)

$$\begin{aligned} \left[ \begin{array}{c} x_1 \\ x_2 \end{array}\right] = \frac{1}{\lambda \Delta } \left[ \begin{array}{c} \left( \sigma _2^2 - \sigma _{12}\right) \left( p-c_1\right) + \sigma _{12} \left( c_2 - c_1 \right) \\ \left( \sigma _1^2 - \sigma _{12} \right) \left( p-c_1\right) - \sigma _{1}^2 \left( c_2 - c_1 \right) \end{array} \right] . \end{aligned}$$

(12)

where,

$$\begin{aligned} \Delta = \sigma _1^2 \sigma _2^2 - \sigma _{12}^2= \sigma _1^2 \sigma _2^2 \left( 1 - \rho ^2 \right) . \end{aligned}$$

(13)

The threshold prices $p_1$ and $p_2$ in (7) are the solutions of the equations $x_1=0$ and $x_2=0$ [defined by Eqs. (12)] respectively. Two useful expressions of these prices are:

$$\begin{aligned} p_1=c_1-\frac{\rho \sigma _1}{\sigma _2-\rho \sigma _1}(c_2-c_1) \text { and } p_2=c_1+\frac{\sigma _1}{\sigma _1-\rho \sigma _2}(c_2-c_1). \end{aligned}$$

(14)

Lemma 3

The quantities $q_1(p,\lambda )$ and $q_2(p,\lambda )$ that maximize the firm’s objective (3) are:

• if $x_2\le 0$, then $q_2=0$ and $q_1=(p-c_1)/(\lambda \sigma _1^2)$;

• if $x_2>0$ and $x_1\le 0$, then $q_1=0$ and $q_2=(p-c_2)/(\lambda \sigma _2^2)$;

• if $x_1>0$ and $x_2>0$, then $q_1=x_1$ and $q_2=x_2$.

where $x_1$ and $x_2$ are defined by Eq. (12).

Proof

There is a unique solution to the firm maximization problem because its objective function is strictly concave (because $\lambda >0$).

If $(q_1,q_2)$ maximizes the firm’s profit subject to the constraints $q_t\ge 0$ for $t=1,2$; then the Kuhn-Tucker conditions are:

$$\begin{aligned}&p-c_1-\lambda \left( \sigma _1^2q_1+\sigma _{12}q_2\right) +\nu _1=0 \end{aligned}$$

(15)

$$\begin{aligned}&p-c_2-\lambda \left( \sigma _1^2 q_2+\sigma _{12}q_1\right) +\nu _2=0 \end{aligned}$$

(16)

$$\begin{aligned}&\nu _t q_t=0 \text { and } \nu _t\ge 0,\; q_t \ge 0 \end{aligned}$$

(17)

in which, $\nu _t$ is the Lagrange multiplier that is associated with the inequality constraint $q_t\ge 0$. There is a unique couple $(q_1,q_2)$ solution of these equations.

• a. $q_1=0, q_2=0$ if and only if $p<c_1$;

• b. $q_1>0,q_2>0 \Leftrightarrow q_1=x_1,q_2=x_2\Leftrightarrow x_1>0,\; x_2>0;$

• c. $q_1>0 \text { and } q_2=0 \Leftrightarrow q_1=(p-c_1)/\lambda \sigma _1^2,\; (p-c_2)-\lambda \sigma _{12}q_1 \le 0 \Leftrightarrow p>c_1\text { and } x_2\le 0;$

• d. $q_1=0 \text { and } q_2>0 \Leftrightarrow q_2=(p-c_2)/\lambda \sigma _2^2,\; x_1 \le 0 \Leftrightarrow p>c_2\text { and } x_1\le 0.$

Therefore, for $p>c_1$, the signs of $x_1$ and $x_2$ indicates what is the solution. If $x_2\le 0$, then $q_2=0$ and $q_1=(p-c_1)/\lambda \sigma _1^2$ (by c). Otherwise, $x_2>0$, and $x_1$ is either positive or negative; if it is positive, then $x_1$ and $x_2$ are the solutions of the Kuhn-Tucker solutions (point b) and if $x_1$ is negative, then the only possibility left is $q_1=0$ and $q_2=(p-c_2)/\lambda \sigma _2^2$ (point d).

Note that, for $p>c_1$ the situation $x_1\le 0$ and $x_2>0$ is possible only if $\sigma _{12}>0$ and $(p-c_2)>0$. Furthermore, it is possible that $p>c_1$ and both $x_1$ and $x_2$ are negative; in that case $q_2=0$ and $q_1>0$.

1.2 Proof of Lemma 1

Let us show that $q_1(p,\lambda )/\left( q_1(p,\lambda )+ q_2(p,\lambda )\right)$ is increasing with respect to $p$. This function if continuous and differentiable by parts because $q_1$ and $q_2$ are differentiable. For $p>c_1$ three situations can arise: Either both quantities are strictly positive, or only one of them is positive. In the latter case the share of technology 1 is constant w.r.t. $p$ (it is either 1 or 0). In the former case the two first-order conditions (4) are satisfied, and both quantities are equal to $x_1$ and $x_2$ given by Eqs. (12). Taking the derivative of the Eqs. (12) gives

$$\begin{aligned} \left[ \begin{array}{c} \partial q_1/\partial p \\ \partial q_2/\partial p \end{array} \right] = \frac{1}{\lambda \Delta } \left[ \begin{array}{c} \sigma _2^2 - \sigma _{12}\\ \sigma _1^2 - \sigma _{12} \end{array} \right] . \end{aligned}$$

(18)

And the sign of the derivative of the share of technology 1 in the firm’s portfolio is the sign of (the numerator of its derivative):

$$\begin{aligned} \frac{\partial q_1}{\partial p}q_2-\frac{\partial q_2}{\partial p}q_1 \end{aligned}$$

(19)

which is equal to (from (12) and (18) and the expression of $\Delta$ from (13)):

$$\begin{aligned} \frac{1}{(\lambda \Delta )^2} \left[ \left( \sigma _2^2-\sigma _{12} \right) (-\sigma _1^2)-\left( \sigma _1^2- \sigma _{12}\right) \sigma _{12}\right] = \frac{-1}{\lambda ^2 \Delta }<0. \end{aligned}$$

(20)

1.3 Proof of Proposition 1

If $\rho < \min \left\{ \sigma _1/\sigma _2,\sigma _2,\sigma _1\right\}$, then $p_1<p_2$:

• if $\rho \ge 0$ then from (14) $p_1<c_1 < p_2$;

• if $\rho <0$ then, $(-\rho )(\sigma _1 - \rho \sigma _2) < \sigma _2 - \rho \sigma _1$ because $\rho ^2<1$, and from (14), using $\rho <0$, $p_1<p_2$.

Also $x_t$, for $t=1,2$, is increasing with respect to $p$ because $\sigma _t^2 > \sigma _{12}$.

Then, for $p\in (c_1,p_2]$, $x_2\le 0$ so $q_2$=0 and $q_1>0$ by Lemma 3; for $p\ge p_2$, then $p\ge p_1$ so $x_1>0$ and $q_t=x_t$ for $t=1,2$, and both quantities increase as the price increases. Furthermore, from (7) $p_2=c_2+\rho (c_2-c_1) \sigma _2/(\sigma _1 - \rho \sigma _2)$, so $p_2<c_2$ if and only if $\rho <0$.

1.4 Proof of Proposition 2

If $\rho >\sigma _2/\sigma _1$, then $\sigma _2<\sigma _1$ because $\rho <1$. And, from (7)

• $p_2<p1$: $\rho (\sigma _1-\rho \sigma _2) > \rho \sigma _1- \sigma _2)$ because $0<\rho <1$; then, $p_2<p_1$ from the expressions (14) of $p_1$ and $p_2$ and because $\sigma _1-\rho \sigma _2>0$ and $\rho \sigma _1 - \sigma _2>0$.

• the proposition follows from the application of Lemma 3.

1.5 Proof of Proposition 3

If $\rho >\sigma _1/\sigma _2$, then $\sigma _1<\sigma _2$ and $x_2<0$ for any $p$ larger than $c_1$ from (12) and by Lemma 3 the firm only invests in technology 1.

Appendix 2: Market Equilibrium

Let us introduce a new notation to ease the exposition: $\varPhi (p)$ is the total production of a firm with risk aversion 1 facing a price $p$:

$$\begin{aligned} \varPhi (p)=q_1(p,1)+q_2(p,1). \end{aligned}$$

(21)

This function is increasing with respect to $p$: $\varPhi ^\prime =\frac{1}{\Delta } \left[ \sigma _1^2+\sigma _2^2-2\sigma _{12}\right] >0.$

1.1 Proof of Lemma 2

The quantities that are produced with each technology by a price-taking firm facing price $p$, with risk aversion $\lambda$ are $q_1(p,\lambda )$ and $q_2(p,\lambda )$. From Lemma 3 these functions satisfy:

$$\begin{aligned} q_t(p,\lambda )=\frac{1}{\lambda } q_t(p,1)\text { for }t=1,2. \end{aligned}$$

(22)

At the industry level, for any price $p$ the supply of the industry with technology $t=1,2$ is:

$$\begin{aligned} Q_t=\sum _{i\in I} q_t(p,\lambda ^i)=q_t(p,1)\sum _{i \in I} \frac{1}{\lambda ^i}=\frac{1}{\varLambda } q_t(p,1)=q_t(p,\varLambda ). \end{aligned}$$

(23)

At the market equilibrium, the price $p^*$ satisfies:

$$\begin{aligned} p^*=P( q_1(p^*,\varLambda )+q_2(p^*,\varLambda )). \end{aligned}$$

(24)

The equilibrium production of firm $i$ is

$$\begin{aligned} q^i_t=q_t(p^*,\lambda ^i)=\frac{1}{\lambda ^i} q_t^i(p^*,1)=\frac{\varLambda }{\lambda ^i} q_t^i(p^*,\varLambda ). \end{aligned}$$

1.2 Proof of Corollary 1

At the market equilibrium, the price $p^*$ satisfies:

$$\begin{aligned} p^*=P( q_1(p^*,\varLambda )+q_2(p^*,\varLambda ))=P(\varPhi (p^*)/\varLambda ) \end{aligned}$$

(25)

taking the derivative w.r.t. $\varLambda$:

$$\begin{aligned} \frac{\partial p^*}{\partial \varLambda }&=-\frac{P^\prime }{\varLambda ^2}\varPhi +\frac{P^\prime }{\varLambda } \varPhi ^\prime \frac{\partial p^*}{\partial \varLambda }. \end{aligned}$$

(26)

Then, the sum $\varPhi$ is increasing with respect to $p$ (from (18)), and $P^\prime$ is negative, so $p^*$ is increasing with respect to $\varLambda$.

A reduction of the risk aversion of one firm or the entry of a new firm are both equivalent to a reduction of $\varLambda$ and induce a reduction of the equilibrium price. From Lemma 1, the share of technology 1 increases in each firm’s portfolio and in the industry portfolio.

1.3 Existence and Uniqueness of a Cournot Equilibrium

Let us first show that there is a unique Cournot equilibrium. To do so we write the reaction function of firms as functions of the total production and look for a fixed point.

Let us denote $R(Q,\lambda )$ as the solution of the equation:

$$\begin{aligned} R=\frac{1}{\lambda } \varPhi (P(Q)+P^\prime (Q)R). \end{aligned}$$

(27)

The solution to this equation is unique because the right-hand side is continuous and decreasing w.r.t. $R$ (because $P^\prime <0$ and $\varPhi$ is increasing). $R(Q,\lambda )$ is the production of a firm with risk aversion $\lambda$ when total production is $Q$. This function is differentiable by parts and

$$\begin{aligned} \frac{\partial R}{\partial Q}=-\frac{\varPhi ^\prime }{\lambda }\left( P^\prime +P^{\prime \prime }R\right) +\frac{\varPhi ^\prime }{\lambda } P^\prime \frac{\partial R}{\partial Q}. \end{aligned}$$

(28)

Therefore, $R(Q,\lambda )$ is decreasing w.r.t to $Q$ because $P^\prime +P^{\prime \prime }R<0$ by assumption A.

At a Cournot equilibrium the total production $Q$ and each firm production $q^i$ satisfy $q^i=R(Q,\lambda ^i)$; therefore, the total production is a solution of the equation:

$$\begin{aligned} Q=\sum _{i\in I} R(Q,\lambda ^i). \end{aligned}$$

(29)

This equation as a solution which is unique (the R.H.S. is decreasing w.r.t. $Q$ because so is $R(Q,\lambda )$). Therefore, there is a unique Cournot equilibrium. The total production is the solution of (29) and firms’ production quantities are the unique solutions of

$$\begin{aligned} q^i&=R(Q,\lambda ^i) \end{aligned}$$

(30)

$$\begin{aligned} q^i_t&=q_t(P+P^\prime q^i,\lambda ^i)=\frac{1}{\lambda ^i}q_t(P+P^\prime q^i,1). \end{aligned}$$

(31)

1.4 Proof of Proposition 4

At the Cournot equilibrium $q^i=R(Q,\lambda ^i)$ and the function $R(Q,\lambda )$ is striclty decreasing w.r.t. $\lambda$ when positive (by differentiation of (27)). So,

$$\begin{aligned} \lambda ^i \leqslant \lambda ^j \Leftrightarrow q^i \geqslant q^j. \end{aligned}$$

(32)

Then, let us show that

$$\begin{aligned} \lambda ^i \leqslant \lambda ^j \Rightarrow \frac{q^i_1}{q^i} \geqslant \frac{q^j_1}{q_j}. \end{aligned}$$

(33)

We proceed by contradiction and assume that there are two firms $i,j\in I$, $i\ne j$ such that $\lambda ^i < \lambda ^j$ and $q_1^i/q^i < q_1^j/q^j$. By Lemma 1 $q_1(p,1)/\varPhi (p)$ is increasing w.r.t. to $p$, so (by (31)) $P+P^\prime q^i > P+P^\prime q^j$ and $q^i<q^j$, which is a contradiction.

The Proposition follows from (32) and (33).

1.5 Proof of Corollary 2

We consider a decrease of the risk aversion of firm $i \in I$. The cournot equilibrium production is the solution of Eq. (29). A decrease of $\lambda ^i$ increases the right-hand-side for all $Q$, so the total production increases. We prove the three first points of the Corollary:

• For firm $j\ne i$, its production $R(Q,\lambda ^j)$ decreases because $R$ decreases w.r.t. $Q$ (point (ii)).

• Concerning firm $i$, its production increases because the total production increases and all other firms’ production decreases (point(i)).

• The marginal revenue of all firms decreases:

  • for firm $i$, $P(Q)+P^\prime (Q) q^i$ decreases because $Q$ and $q^i$ increases;

  • for firm $j\ne i$, $P(Q)+P^\prime (Q)q^j$ decreases because $q^j$ decreases and $q^j=\varPhi (P+P^\prime q^j)/\lambda ^j$, and $\lambda ^j$ remains unchanged and $\varPhi$ is decreasing.

  So, from Lemma 1, the share of technology 1 in all firms portfolio increases (point(iii)).

Appendix 3: Generalisation

1.1 Expected Utility

Technology 1 is not used at equilibrium if there is $q_2$ such that

$$\begin{aligned} \mathbb {E}[ U'(\pi )(p-\theta _2-c_2)]=0\text { and } \mathbb {E}[U'(\pi )(p-\theta _1-c_1)]\le 0. \end{aligned}$$

(34)

The left equality should be rewritten: $0=\mathbb {E}[\pi ](p-c_2) -\text {cov}(U'(\pi ), \theta _2)$ and the inequality:

$$\begin{aligned} 0&\le \mathbb {E}[U'(\pi )(p-\theta _1-c_1)]=\mathbb {E}[\pi ](p-c_1) -\text {cov}(U'(\pi ), \theta _2) \nonumber \\&\le \text {cov}(U'(\pi ), \theta _2)\frac{p-c_1}{p-c_2} -\text {cov}(U'(\pi ), \theta _2). \nonumber \end{aligned}$$

1.2 Coherent Risk Measure

Consider the function $\tilde{\rho }(q_1,q_2)=\rho (\theta q_1+\theta _2 q_2)$. We assume that it is homogeneous of degree one; consequently its derivatives are homogeneous of degree zero:

$$\begin{aligned} \frac{\partial \tilde{\rho }}{\partial q_t} (rq_1,rq_2)=\frac{\partial \tilde{\rho }}{\partial q_t} (q_1,q_2). \end{aligned}$$

Let us also assume that $\tilde{\rho }$ is strictly convex. For each firm there is a unique couple of maximizing quantities; if firm $i$ invests in both technologies, the first-order conditions are

$$\begin{aligned} p-c_t=\lambda ^i\tilde{\rho }(q_1^i,q_2^i)\frac{\partial \tilde{\rho }}{\partial q_t}(q_1^i,q_2^i) = \tilde{\rho }(\lambda ^i q_1^i,\lambda ^i q_2^i) \frac{\partial \tilde{\rho }}{\partial q_t}(\lambda ^i q_1^i,\lambda ^i q_2^i). \end{aligned}$$

The quantities $\lambda ^i q^i_1$ are the solutions of a couple of equations that are independent of $\lambda _i$ so that they are independent of $\lambda ^i$. This ensures that the functions $q_t(p,\lambda )$ are homogeneous of degree -1 with respect to $\lambda$.