Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p_{x}x+p_{y}y=I$$ where $\alpha+\beta=1$.

It can be shown that the Marshallian demand for $x$ and $y$ is $x^*=\frac{\alpha I}{p_{x}}$ and $y^*=\frac{\beta I}{p_{y}}$ respectively.

Does a change in $p_{y}$ affect the quantity of $x$ demanded? By looking at $x$'s Marshallian Demand, it seems as though such a change will not affect the quantity of x demanded but $p_{x}$ and $p_{y}$ are related by the budget constraint which makes me not quite sure.

  • $\begingroup$ $p_x$ and $p_y$ are not related by the budget constraint, they are not variables but parameters therein. $\endgroup$
    – Giskard
    Oct 15 '16 at 15:53
  • $\begingroup$ @denesp, thank you! So a change in $p_[y}$ does not affect quantity demanded of x is the answer, right? $\endgroup$
    – Omrane
    Oct 15 '16 at 16:04

You correctly derived the Marshallian demand function for Cobb-Douglas utility, you notice that the optimal level of consumption of $x$ or $y$ is a function only of the individual's income and the price of said good. This is an interesting feature of CD-utility, that when the price of good $y$ changes the demand for $x$ doesn't change. This means that $x$ and $y$ are neither substitutes nor complements to one another. Econport has a nice figure of this:

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The first figure shows that as the price of good $x$ changes the quantity of $x$ changes but the quantity of $y$ does not change. The second figure shows that as price increases the amount demanded decreases and vice-versa when price decreases.

As a side note, CD-utility further implies strict quasi-concave utility so that the indifference set is strictly convex, hence $\vec{x}^* (\vec{p},I)$ is single-valued. With constant prices for $y$ and prices $p_a, p_b, p_c$ for $x,$ we choose the optimal consumption level of each commodity at the points $a = (x_a, y_a) ,b = (x_b, y_b) ,c = (x_c, y_c)$, such that the slope of the budget hyperplane and the tangent slope of the indifference curve are equal.


To briefly add to Sunhwa's answer, the intuitive reasoning for why the price of $y$ does not affect consumption of good $x$ is because of the unique properties of the Cobb-Douglas utility function. Changing the price of a good changes the price ratio, which is part of what determines the ratio of goods consumed. So even if the price of $y$ were to go up and cause you to buy less $y$, that would also change the ratio of goods you are buying, even if you don't change $x$. In this case, the optimization condition ends up being so that you don't have to change $x$ at all (for all other things equal).

$$\frac{p_y}{p_x} \cdot \frac{\alpha}{\beta} = \frac{x}{y}$$

Peruse this condition as you see fit.


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