Indirect changes in Marshallian Demand

Question

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.

Answer 1

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.

Answer 2

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.