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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.

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    $\begingroup$ 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? $\endgroup$
    – Jesper Hybel
    Dec 5, 2021 at 15:36

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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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