Portfolio choice of a risk lover

Take the standard portfolio choice problem as presented in MWG (p.188-189), but with a risk loving decision maker:

• with initial wealth $w$
• invests an amount $\alpha$ in a risky asset with a random gross rate of return $z$, where $z$ is distributed according to cdf $F$, and $\int zdF(z)>1$

The decision maker solves the following problem $$\max_{\alpha\in[0,w]} \int u(w-\alpha +\alpha z) dF(z),$$ where $u'>0$, $u''>0$.

Is it true that the optimal solution is $\alpha^*=w$?

Note that since $u$ is convex, the usual first order condition is not sufficient for a maximum.

Intuitively the answer should be yes. But what's the reason, exactly?

Since we are suspecting a corner-solution, it is better to write the problem explicitly with its constraint. Even better, use the Fritz John (FJ) conditions rather than the Karush-Kuhn-Tucker (KKT) ones. We will mention the differences as we go along.

$$\max_{\alpha} \int u[w+\alpha(z-1)] dF(z),\;\; \text{s.t.}\;\; w-\alpha \geq 0$$

The lagrangean under the Firtz John formulation is

$$L_{FJ} = \lambda_0\int u[w+\alpha(z-1)] dF(z)\; + \; \lambda_1(w-\alpha)$$

The new element is the multiplier on the objective function, $\lambda_0$. Without loss of generality we can specify

$$\lambda_0 \in \{0,1\},\;\; \lambda_0 + \lambda_1 \neq 0$$

What is that we gain here, compared to the much more widely known and used KKT-conditions? If a solution necessitates that $\lambda_0 =1$, we obtain the KKT conditions with the constraint qualification satisfied. If a solution necessitates that $\lambda_0 =0$, it reflects, among other special cases, the case where the constraint qualification fails to hold.

(A standard example is the case where the feasible set for $\alpha$ has been reduced to a single point due to the constraints imposed. Then we will find that the only solution dictates that $\lambda_0=0$, which has an intuitive explanation: if $\alpha$ can take one and only one value due to the constraints, then the objective function "plays no role" in the determination of $\alpha$ and so it gets a zero multiplier).

Back to our problem. The first order condition is

$$\frac {\partial L_{FJ}}{\partial \alpha} = \lambda_0\int u'[w+\alpha(z-1)]\cdot (z-1) dF(z) - \lambda_1 \leq 0$$

(note the "lower or equal to zero" which is the case when optimizing under inequality constraints, rather than just "equal").

First, we establish that $\alpha^* >0$. Due to $u'>0$ and the (strict) convexity of the utility function,$u''>0$, and the assumption that $E(z) >1$ we have (using Jensen's Inequality)

$$E[u(w+\alpha(z-1))] > u(w+\alpha(E(z)-1))] > u(w + 0\cdot(E(z)-1))=u(w)$$

We turn now to examine cases. Since the two multipliers cannot be both zero, and $\lambda_0$ takes only two values, there are three possible combinations.

Examine the case $\lambda_0 =1$.
Then $\lambda_1$ can in principle be zero or positive. Examine the case where $\lambda_1 =0$, i.e. the constraint is not binding which implies that $\alpha^* < w$. With this candidate pair of multipliers, $\{\lambda_0=1,\lambda_1=0\}$, the first order condition would become

$$\int u'[w+\alpha(z-1)]\cdot (z-1) dF(z) \leq 0 \Rightarrow E(zu') - E(u') \leq 0$$

Since $u'>0 \Rightarrow E(u') > 0$. Also, since $E(z)>1$ we have that

$$E(u')< E(u')E(z) \Rightarrow E(zu') < E(u')E(z) \Rightarrow \text{Cov}(z, u')<0$$

But this cannot hold, because, since $\alpha^*>0$ and $u''>0$, we have that $u'$ will be strictly increasing in $z$. So the covariance of $z$ and $u'$ cannot be negative. But then the pair of multiplier values $\{\lambda_0=1,\lambda_1=0\}$ cannot be a solution, and this happens due to the assumption $u''>0$.

We are left with the cases $\{\lambda_0=1,\lambda_1>0\}$, or $\{\lambda_0=0,\lambda_1>0\}$. In both these cases, $\lambda_1 >0$ i.e. the constraint is binding , i.e. we will have $\alpha^* = w$. QED.

References

The Fritz John conditions have been stated in "F. JOHN. Extremum problems with inequalities as side conditions. In “Studies and Essays, Courant Anniversary Volume” (K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker, eds.), pp. 187-204. Wiley (Interscience), New York, 1948"

and have been generalized in

"Mangasarian, O. L., & Fromovitz, S. (1967). The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. Journal of Mathematical Analysis and Applications, 17(1), 37-47.