Cobb-Douglas 效用是通過“在每種商品上花費固定部分的收入”來最大化的嗎？
考慮具有以下形式的 Cobb-Douglas 效用函式 $$ u(x) = \prod_{j=1}^n x_j^{a_j} $$ 在哪裡 $ x $ 是一個分配向量並且 $ a_j $ 是實用程序參數 $ \sum a_j = 1 $ . 我的問題與具有 Cobb-Douglas 效用函式和固定價格預算的買家的*需求有關。*那是，$$ \arg \max_x \lbrace u(x) : \pi^T x \leq w \rbrace $$ 在哪裡 $ \pi $ 是一個固定價格向量並且 $ w $ 是預算。
這很容易解決，但我對以下聲明感到困惑，這是我在優化教科書中讀到的
$$ 1 $$:
不難證明，具有 Cobb-Douglas 效用函式的交易者將其收入的固定比例花在每種商品上。
這顯然是正確的 $ a_j = 1/n, \forall j $ 而且價格是一樣的。
但是假設 $ n = 2 $ , $ \pi = (1, 1) $ , $ w = 1 $ ， 和 $ a = 0.8, 0.2 $ . 那麼“在每件商品上花費固定比例的預算”將意味著花費 $ w / n = 0.5 $ 在每件商品上，或購買 $ 0.5 $ 每種商品的單位，意味著 $ x^* = (0.5, 0.5) $ . 這顯然不是最理想的，如下表所示：
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相反，最大值為 $ x = (0.8, 0.2) $ ，我會口頭描述為
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.
$$ 1 $$ Nisan, Noam, Tim Roughgarden, Éva Tardos, and Vijay V. Vazirani, eds. 2007. Algorithmic Game Theory. Cambridge University Press.
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 $ .