Quasiconvexity of the indirect utility function for Cobb-Douglas utility

Question

I've recently started Mas-Colell's, Green's and Whinston's Microeconomic Theory. In section 3.D, the authors define the indirect utility for a price vector $p$ and wealth $w$ as the utility derived from a utility-maximizing consumption bundle $x^\star$, i.e. $v(p, w) = u(x^\star)$ for (any) $x^\star \in x(p, w)$, where $x(p, w)$ is the Marshallian demand.

Now, example 3.D.2 considers the Cobb-Douglas utility function (in logs) $u(x_1, x_2) = \alpha \ln x_1 + (1 - \alpha) \ln x_2$ for the two-good case, and shows that the indirect utility function is given by $v(p, w) = c + \ln w - \alpha \ln p_1 - (1 - \alpha) \ln p_2$ (where $c$ is an uninteresting constant). Exercise 3.D.2 asks the reader to verify the properties of proposition 3.D.3 for this indirect utility function.

The third of these properties is quasiconvexity of $v$, i.e. convexity of the set $\{ (p, w) : v(p, w) \le \bar v \}$ for any $\bar v$. I'm having trouble with this, and would like to ask for help.

What I've done so far is the following: let $\bar v$ be fixed, and suppose that $p$, $p'$, $w$ and $w'$ are such that $v(p, w) \le \bar v$ and $v(p', w') \le \bar v$. Let $0 \le a \le 1$, and define $p'' = ap + (1-a)p'$, $w'' = aw + (1 - a)w'$. Then I want to show that $v(p'', w'') \le \bar v$; to this end I started calculating,

$$v(p'', w'') = ac + (1-a)c + \ln (aw + (1 - a) w') - \alpha \ln (ap_1 + (1-a)p_1') - (1 - \alpha) \ln (ap_2 - (1-a)p_2')$$

Now $\ln$ is a concave function, so that $\ln (ap_1 + (1 - a)p_1') \ge a \ln p_1 + (1 - a)\ln p_1'$ etc. As such, I can take replace the last two summands to get the inequality

$$v(p'', w'') \le ac + (1-a)c + \ln (aw + (1 - a) w') - a\alpha\ln p_1 - (1-a)\alpha\ln p_1' - a(1-\alpha)\ln p_2 - (1-a)(1-\alpha)\ln p_2'$$

which is almost but not quite $av(p, w) + (1-a)v(p', w') \le a\bar v + (1-a)\bar v = \bar v$. The problem here is the first logarithm, $\ln (aw + (1 - a) w')$. I only know that $\ln (aw + (1 - a) w') \ge a\ln w + (1-a) \ln w'$, but this doesn't help me since I'd need this inequality to hold in the opposite direction.

I've not thought about this for very long, but for the time being I'm stuck. I know that by proposition 3.D.3, quasiconvexity of $v$ must in fact hold - so it ought be possible to verify this by direct calculation. (Of course, as we all know, "is" and "ought" are two very different things in economics.)

I'd appreciate any pointers.

Note: this is not homework. I am taking a microeconomics course using this textbook but I'm solving these exercises in the textbook on my own in order to learn, not because the course requires it.

Answer 1

I've procured the solutions manual for the textbook (Hara, Segal, Tadelis, Solutions Manual for Microeconomic Theory), and want to reproduce the answer given there since it neatly sidesteps the issue I encountered.

In my own words, the answer is as follows. Since by (i) of proposition 3.D.3, the indirect utility function is homogeneous of degree zero (i.e. $v(\lambda p, \lambda w) = v(p, w)$ for all $\lambda > 0$), it in fact suffices to prove that the set $\{ p : v(p, \bar w) \le \bar v \}$ is convex for any $\bar v$ and any fixed (!) $\bar w$; and that this is so can be checked in the same manner I did above.

I'd still be interested whether my original approach can be made to work or not.