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For a high school multivariable calc class I am writing an essay on an economic utility function and its maximisation using the Lagrange multipliers.

The formula is $output = \prod_i^n x_i^{\sigma_i} $, where n is the number of goods, x is a specific good and $\sigma$ is a multiplier. The constraint needed for the lagrange multiplier method is C > $ \sum_i^n x_i*p_i $ where p is the price and C is the maximum cost of a company.

So essentially you have: $$f(x_i) = \prod_i^n x_i^{\sigma_i} - {\lambda}(C-\sum_i^n x_i*p_i) $$

Now my question is how to go on. Which partial derivative should I take, and also is that possible with the pi and sigma notation?

Thanks in advanced.

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  • $\begingroup$ If you have trouble seeing beyond the sigma and pi notation, write the expression without them and see what you can do. If the arbitrary number of variables is daunting, start with 2 or 3 and see what happens. $\endgroup$ – Matthew Leingang Nov 21 '17 at 14:29
  • $\begingroup$ I have done it already with the standard Cobb-Douglas and that all worked out. Now the challenge is doing it with a general form! $\endgroup$ – Sven van Holten Nov 21 '17 at 14:39
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When differentiating a monomial with respect to one of the variables, the other variables are to be held constant. Therefore, $$ \frac{\partial}{\partial x_j} \left(x_1^{\sigma_1} x_2^{\sigma_2} \dots x_n^{\sigma_n}\right) = x_1^{\sigma_1} x_2^{\sigma_2} \cdots x_{j-1}^{\sigma_{j-1}}\sigma_j x_j^{\sigma_j-1} x_{j+1}^{\sigma_{j+1}}\dots x_n^{\sigma_n} $$ As long as $x_j \neq 0$ you could write it as $$ \frac{\partial}{\partial x_j}\prod_{i=1}^n x_i^{\sigma_i} = \frac{\sigma_j}{x_j} \prod_{i=1}^n x_i^{\sigma_i} $$

When differentiating a linear combination with respect to one of the variables, again the other variables are to be held constant. Therefore, $$ \frac{\partial}{\partial x_j}\left(p_1 x_1 + p_2 x_2 + \dots + p_n x_n\right) = \frac{\partial}{\partial x_j} \sum_{i=1}^n p_i x_i = p_j $$

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