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So I have a consumer with a utility function of the Cobb-Douglas form $v(x_1,x_2)=x^{\frac{1}{2}}_1x^{\frac{1}{2}}_2$.

From that I constructed the demand function for good 1 and good 2:
$x_1=\frac{1}{2} \cdot \frac{m}{p_1}$

$x_2=\frac{1}{2} \cdot \frac{m}{p_2}$

From here, I am to determine whether good 1 and good 2 are substitutes or complements. I was told by my teacher to take the derivative with respect to $m, p_1$ and $p_2$, however I can't get a sensible result. As far as I've come to understand from my book, I need to find $\frac{dx_1}{dp_2}$, the change in the demand for good 1 as the price of good 2 changes. Substituting m back to $p_1x_1+p_2x_2$ becomes messy as well. I understand the problem theoretically, but I simply can't figure out how to properly argue using math, rather than logic.

I hope someone is willing to help

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    $\begingroup$ Would be useful to tell us what you get for each step before it gets "messy." $\endgroup$
    – Art
    Commented Dec 11, 2019 at 4:29

3 Answers 3

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The Marshallian demands are functions $x_i(p_1,p_2,m)$.

You treat $m$ as an independent variable rather than $p_1 x_1 + p_2 x_2$.

Since the demands don’t depend on the cross prices (as they don’t appear in the formulas), i.e.,

$\frac{\partial x_1}{\partial p_2} = 0$,

$\frac{\partial x_2}{\partial p_1} = 0$,

they are not complements nor substitute goods, they’re unrelated goods.

Intuitively, if you were to spend all the budget on one good, you would buy

$x_i = \frac{m}{p_i}$ units of that good.

The demands for both goods $x_1, x_2$ look like that multiplied by $\frac{1}{2}$.

This means you’re spending half the budget on each good independently of their relative prices.

For example, let’s say I’m spending half my budget on pizzas and the other half on hamburgers.

If the price of pizzas suddenly increased, I’d still spend that $\frac{m}{2}$ on pizzas, being able to buy less pizzas with it of course.

But I’d still leave the other spending of $\frac{m}{2}$ on hamburgers intact. Since I’m not changing my spending on hamburgers and their price didn’t change, I’d buy the same amount of hamburgers independently of the price change on pizzas.

Note: A beautiful property of the Cobb-Douglas demand function is that when the exponents add up to $1$, each exponent corresponds to the ratio of the budget you’d spend on that specific good.

If the exponents don’t add up to 1, just divide each exponent by the sum of the exponents to get the ratio.

The marshallian demands for a general Cobb-Douglas $u(x_1,x_2) = C x_{1}^{\alpha} x_{2}^{\beta}$ are

$x_1 = \frac{\alpha}{\alpha + \beta} \frac{m}{p_1}$,

$x_2 = \frac{\beta}{\alpha + \beta} \frac{m}{p_2}$.

Therefore, for every Cobb-Douglas utility function, the goods are unrelated (not complements nor substitutes).

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Substitutes and complements are determined by elasticity of substitution, which is a characteristics of utility function. You can compute it in the following way:

$$\sigma_{ij} = \frac{\frac{\partial (x_j/x_i)}{x_j/x_i}}{\frac{\partial MRS_{ij}}{MRS_{ij}}}$$

See that this elasticity of substitution characterized per each pair of goods (any $x_i$ and $x_j$) in utility function.

Why is this important? Because you know one important thing:

  • If elasticity of substitution is high ($\sigma_{i,j} > 1$), then it is (relatively) easy to substitute your pair of goods among each, therefore you work with substitutes.
  • If elasticity of substitution is low ($\sigma_{i,j} < 1$), then it is (relatively) hard to substitute your pair of goods among each, therefore you work with complements.

Why do we need to know? Because it can be shown that for

  • $\sigma_{i,j}>1$ the cross-price elasticity in case of linear prices will be always positive!
  • $\sigma_{i,j}>1$ the cross-price elasticity in case of linear prices will be always negative!

Cross price elasticity can be characterized as:

$$e_{P_j}^{x_i} = \frac{\frac{\partial x_i}{x_i}}{\frac{\partial P_j}{P_j}}$$

These are percentage changes which can be rewritten in terms of logarithms:

$$e_{P_j}^{x_i} = \frac{ln(x_i)}{ln(P_j)}$$

Therefore:

  • Positive cross-price elasticity means that increase of price of other good by 1 % implies an increase in demand for our good. Therefore, we have substitutes!
  • Negative cross-price elasticity means that increase of price of other good by 1 % implies a decrease in demand for our good. Therefore, we have complements!

So what you have to do is take your Marshallian demand and compute the cross-price elasticity in the following way:

$$e_{P_j}^{x_i} = \frac{\partial x_i}{\partial P_j} * \frac{P_j}{x_i}$$

Where $x_i$ is the demand for good $i$.

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U should check this video, it helped me anyway https://www.youtube.com/watch?v=0BJ4GUpvKHQ

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    $\begingroup$ Please make your answer more explicit! $\endgroup$
    – emeryville
    Commented Jan 27, 2023 at 9:30

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