# Mixed Partial Derivatives in Profit Function

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

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.

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.