Cobb–Douglas utility maximized by spending a "fixed fraction of income on each good"?

Question

Consider a Cobb–Douglas utility function having the form $$u(x) = \prod_{j=1}^n x_j^{a_j} $$ where $x$ is an allocation vector and $a_j$ are utility parameters with $\sum a_j = 1$. My question has to do with the demand of a buyer with a Cobb–Douglas utility function and fixed prices budget. That is, $$\arg \max_x \lbrace u(x) : \pi^T x \leq w \rbrace$$ where $\pi$ is a fixed price vector and $w$ is the budget.

This is easy to solve, but I am confused by the following claim, which I read in an optimization textbook [1]:

It is not difficult to show that a trader with a Cobb–Douglas utility function spends a fixed fraction of her income on each good.

This is clearly true when $a_j = 1/n, \forall j$ and the prices are the same.

But suppose $n = 2$, $\pi = (1, 1)$, $w = 1$, and $a = 0.8, 0.2$. Then "spending a fixed fraction of the budget on each good" would mean spending $w / n = 0.5$ on each good, or buying $0.5$ units of each good, implying $x^* = (0.5, 0.5)$. This is clearly suboptimal, as the following table of values shows:

 x_1  x_2  u(x)
 0.0  1.0  0.0
 0.1  0.9  0.155185
 0.2  0.8  0.263902
 0.3  0.7  0.355399
 0.4  0.6  0.433789
 0.5  0.5  0.5
 0.6  0.4  0.553265
 0.7  0.3  0.590885
 0.8  0.2  0.606287   (*)
 0.9  0.1  0.579955
 1.0  0.0  0.0

Instead, the maximum is at $x = (0.8, 0.2)$, which I would describe verbally as

A trader with a Cobb–Douglas utility function allocates each good in proportion to its utility per unit cost.

In this example, resource 1 yields $a_1 / \pi_1 = 0.8$ utils per dollar, resource 2 yields $0.2$ utils per dollar, and the optimal allocation is a scalar times $(0.8, 0.2)$.

Is my interpretation correct? Is the textbook's interpretation quoted above correct? If both are correct, what explains the discrepancy in the example above?

Please note that I am not asking how to compute the demand of a trader with a Cobb–Douglas utility function. I am asking about the specific claim that the trader spends a "fixed fraction of her income" on each product.

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[1] Nisan, Noam, Tim Roughgarden, Éva Tardos, and Vijay V. Vazirani, eds. 2007. Algorithmic Game Theory. Cambridge University Press.

Answer 1

A "fixed fraction" doesn't mean an "equal fraction", or at least that's not the intended meaning.

It can be easily verified that the solution to \begin{equation} \max_{x_1,x_2}\;x_1^{a_1}x_2^{a_2}\qquad\text{s.t.}\; \pi_1x_1+\pi_2x_2\le w \end{equation} is \begin{equation} x_1^*=\frac{a_1}{a_1+a_2}\frac{w}{\pi_1}\quad\text{and}\quad x_2^*=\frac{a_2}{a_1+a_2}\frac{w}{\pi_2}. \end{equation} Plugging in the values you used: $a_1=0.8,a_2=0.2,\pi_1=\pi_2=1$, the solution is consistent with your numerical simulation.

Generalizing to the $n$-goods case (with $\sum_ia_i=1$), the demand for $x_i$ is \begin{equation} x_i^*=\frac{a_i}{\pi_i}w, \end{equation} where the "fixed" fraction refers to $a_i/\pi_i$.

Answer 2

The "a fixed fraction of her income" means a fixed share of the total expenditure, not the share of total goods purchased. This is a property of a Cobb–Douglas utility function but not a property of a general CES utility function.

This is because that with a CD form, you have the elasticity of substitution to be 1. You can easily find this in a two factor case, where $$\frac{x_1^* \pi_1}{x_2^* \pi_2} = \frac{a_1}{a_2}$$. In other words, the substitution effect exactly compensates the price effect and the factor expenditure shares are entirely determined by technology.

This is not the case in a general CES function where the elasticity of substitution can be any arbitrary number and the substitution effect will not exactly offset the price effect if the elasticity of substitution is not 1.