<p>Cobb-Douglas preferences are just a special case: The <span class="math-container">$p_1$</span> price offer curve is flat and the cross price elasticities of demand are zero.</p>

CC BY-SA 4.0
2021-12-06T09:55:28.037Z
2021-12-06T09:55:28.037Z
VARulle
<p>For a utility function of the form <span class="math-container">$U(x_1,x_2) = x_1^\alpha x_2^\beta$</span> and the standard budget constraint, the utility maximisation problem gives us a demand system characterised by:</p>
<p><span class="math-container">$x_1(\alpha, \beta, y, p_1) = \frac{\alpha}{\alpha + \beta}\frac{y}{p_1}$</span></p>
<p><span class="math-container">$x_2(\alpha, \beta, y, p_2) = \frac{\beta}{\alpha + \beta}\frac{y}{p_2}$</span>.</p>
<p>Demand systems take as arguments the price vector, but in this particular case we can clearly see that we can do without taking all the prices in each demand function. How can this demand system then capture a substitution effect, as in if <span class="math-container">$p_2$</span> increases, shouldn't we see an increase in <span class="math-container">$x_1$</span>?</p>
<p>Thanks.</p>

CC BY-SA 4.0
2021-12-05T15:36:21.920Z
When price change there is both a substitution effect and an income effect. Marshall demand captures both. Maybe Cobb Douglas preferences are a special case in the sense that the income effect exactly nullifies the substitution effect?
Jesper Hybel
<consumer-theory><optimization>
Arguments of the Marshallian demand system of a Cobb-Douglas utility function
Panhaboth K
# Arguments of the Marshallian demand system of a Cobb-Douglas utility function

(Asked by: username_0 on 2021-12-05)

For a utility function of the form $U(x\_1,x\_2) = x\_1^\alpha x\_2^\beta$ and the standard budget constraint, the utility maximisation problem gives us a demand system characterised by:

$x\_1(\alpha, \beta, y, p\_1) = \frac{\alpha}{\alpha + \beta}\frac{y}{p\_1}$

$x\_2(\alpha, \beta, y, p\_2) = \frac{\beta}{\alpha + \beta}\frac{y}{p\_2}$.

Demand systems take as arguments the price vector, but in this particular case we can clearly see that we can do without taking all the prices in each demand function. How can this demand system then capture a substitution effect, as in if $p\_2$ increases, shouldn't we see an increase in $x\_1$?

Thanks.

**Tags:** consumer-theory, optimization

> **Comments:**
> **username_1**: When price change there is both a substitution effect and an income effect. Marshall demand captures both. Maybe Cobb Douglas preferences are a special case in the sense that the income effect exactly nullifies the substitution effect?  

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## 1 answers

### Answer by username_2 on 2021-12-06. Score: 1 | Accepted answer

Cobb-Douglas preferences are just a special case: The $p\_1$ price offer curve is flat and the cross price elasticities of demand are zero.


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