Finding Marshallian demands for Leontief production function with different powers

Question

Attempting to solve a practice problem, not sure if I'm heading in the right direction since my solution seems pretty messy. Given the following utility function,

$u(x,y)=min\{x^{1/2},2y\}$, find the Marshallian demands.

My answer:

Since Leontief is perfect complements, must be the case that $x^{1/2}=2y$, substituting this into a budget constraint yields the following:

$p_x \times x + p_y \times y = w$, where w is total income. Taking $x^{1/2}=2y$ and squaring this yields $x=4y^2$. Subbing this into constraint would give:

$p_x \times 4y^2 + p_y \times y = w$, at this point I applied the quadratic formula and got a demand function for y as follows,

$$y = \frac{-p_y \pm \sqrt{p_y^2 + 16 p_x w}}{8p_x}$$

This seems messy to me, I figure I can rule out the minus side of the quadratic seems that would imply y is negative. Even if some confirmation that this is the right approach would be appreciated.

Thanks!

Answer 1

As Amit said, your answer is correct. As Toby said, you can rule out negative output.

There is no much insight to add as an answer really. However, I was also fairly puzzled that such a simple problem gives a complex answer. As it turns out, you get a linear demand (on income) only if the exponents on both components is the same.

Consider the more general case:

$$ U(x,y) = \text{min}\{ax^\alpha,by^\beta\} $$

Optimality requires (* notation for optimal values omitted)

$$ ax^\alpha = by^\beta\ $$

Replacing one of these in the budget constraint leads to:

$$ p_x x + p_y \left(\frac{a}{b}\right)^{\frac{1}{\beta}} x^{\frac{\alpha}{\beta}} = w $$

Here we see that only when $\alpha=\beta$, demand will be linear (on income). In this case:

$$ x = \frac{w}{p_x + p_y\left(\frac{a}{b}\right)^{\frac{1}{\alpha}}} $$

It seems an algebraic solution for the general (unrestricted) case cannot be derived, as the polynomial you get might have irrational powers.

Answer 2

An interesting exploration here is whether there are conditions under which such a demand function reflects a negative relation between price and quantity demanded. Normalizing $p_x =1$ and keeping income constant we have

$$y = \frac{-p_y}{8} + {\sqrt{(p_y/8)^2 + \frac14 w}}$$

$$\implies \frac {\partial y}{\partial p_y} = -\frac {1}{8}+ \frac 12\cdot\left((p_y/8)^2 + \frac14 w\right)^{-1/2}\cdot \frac {2p_y}{8^2}$$

$$= -\frac 18 \cdot \left[ 1- \left((p_y/8)^2 + \frac14 w\right)^{-1/2}\cdot \frac {p_y}{8}\right]$$

For this derivative to be negative, we want to expression inside the brackets to be positive. So we require

$$1- \left((p_y/8)^2 + \frac14 w\right)^{-1/2}\cdot \frac {p_y}{8} > 0$$

$$\implies 1> \left((p_y/8)^2 + \frac14 w\right)^{-1/2}\cdot \frac {p_y}{8} $$

$$ \implies \left((p_y/8)^2 + \frac14 w\right)^{1/2} > \frac {p_y}{8}$$

$$\implies (p_y/8)^2 + \frac14 w > (p_y/8)^2$$

which holds always. So we see that even this "strange looking" utiity function with its "messy" demand functions, reflects always the negative effect of price on quantity demanded.