Mixed Partial Derivatives in Profit Function

Question

$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w_xx - w_zz$

Question 1:

Using the first order conditions, we get:

$x = \frac{pa}{w_x}$ $z = \frac{pb}{w_z}$

What do we call these Input demand functions as a function of price i.e. are they just un-compensated demand functions as in consumer theory? . I.e. When it's in terms of output, we call them contingent

Question 2

I'm asked "What is the relationship between $x$ and $z$, are they complements or substitutes". The answer is

These inputs $x$ and $z$ are not related because : $\pi_{w_z,w_x} = - \frac{\partial z}{\partial w_x} = 0$

Could someone kindly explain what is going on here? I believe the $\pi_{w_z,w_x}$ tells us how the marginal profit from a change in the input cost of $z$ changes when the cost of input $x$ changes. But I'm not sure i understand the partial derivative on the right hand side. It looks a bit like the chain rule for implicit differentiation.

Answer 1

1. In my classes, they’re called Marshallian or uncompensated demands, as in consumer theory.

2. The partial derivative on the right hand side means to directly differentiate the demand for $z$ (the optimal $z$ level) with respect $w_x$ (the price of the other factor, $x$).

In consumer theory for goods $x,y$ with prices $p_x,p_y$, complement and substitute goods are defined as:

• Complements

$\frac{\partial x}{\partial p_y} < 0$

Here an increase in the price of good $y$ makes a lower amount of $y$ to be demanded as it is more expensive, and hence a lower amount of $x$ is demanded as $x$ and $y$ are usually consumed together.

• Substitutes

$\frac{\partial x}{\partial p_y} > 0$

Here an increase in the price of good $y$ makes a lower amount of $y$ to be demanded as it is more expensive, and a higher amount of $x$ is demanded as the consumer substitutes good $y$ for the relatively cheaper alternative good $x$.

• Unrelated goods

$\frac{\partial x}{\partial p_y} = 0$

A change in price in good $y$, and hence its quantity demanded, has no effect over the demand for good $x$.

Here you’re working with the analogous notion of complements/substitutes/unrelated goods (here production factors) in firm theory.