Solving for profit function $\pi (w,p)$ given the output of production function $f(z) = \sqrt{2z_1 + 3z_2}$

Question

Solving for profit function $\pi (w,p)$ given the output production function $f(z) = \sqrt{2z_1 + 3z_2}$.

I approached this problem by trying to solve the $p\nabla f(z) = w$. This is derived from setting up the Lagrangian for the Profit Maximization Problem, \begin{align*} \text{maximize } &pf(z)-w^Tz\\\ \Rightarrow \mathcal{L}(z) &= pf(z) -w^Tz \end{align*} Then taking the partial of the Lagrange to zero, \begin{align*} \frac{\partial \mathcal{L}}{\partial z} = 0 = p\nabla f(z) - w\\ \Rightarrow p\nabla f(z) = w. \end{align*} The issue is, I thought that I could solve for an optimal $z^*$, but that does not seem possible, but I know that a solution exists.

To showing this issue simply, let $q=f(z)$, then the gradient is: \begin{align} \nabla f(z) = \begin{bmatrix} 1/q\\ 3/2q \end{bmatrix} \end{align} So solving our equation $p\nabla f(z) = w$, should let us solve for $z_1,z_2$, but as you can see, \begin{align*} \begin{bmatrix} 1/q\\ 3/2q \end{bmatrix} = \begin{bmatrix} w_1/p\\ w_2/p \end{bmatrix}\\ \Rightarrow \begin{bmatrix} q\\ q \end{bmatrix} = \begin{bmatrix} p/w_1\\ 3p/2w_2 \end{bmatrix}\\ \Rightarrow \begin{bmatrix} \sqrt{2z_1 + 3z_2}\\ \sqrt{2z_1 + 3z_2} \end{bmatrix} = \begin{bmatrix} p/w_1\\ 3p/2w_2 \end{bmatrix} \end{align*} This shows I cannot isolate $z_1$ or $z_2$. Without an optimal $z^*=<z_1^*,z_2^*>$, I cannot find my profit function $\pi(w,p) = pf(z^*) - w^Tz^*$.

EDIT My guess is that actually, for some output $q$, my profit function is what I already solved for, $\pi(w,p)=\max{\{pw_1, 3p/2w_2\}}$

Can anyone confirm this?

Answer 1

The production function has a particular feature: the inputs are perfectly substitutable. One unit of input 1 can be substituted by 2/3 of input 2 to produce the same quantity of output.

Intuitively, a producer would optimally use only one output to produce. Suppose the production plan is $(z_1,z_2)$. By choosing $(z_1-1,z_2+2/3)$, the firm produces the same quantity but profits are increased by $w_1-2/3w_2$. We can conclude: the producer would buy only input 1 ($z_2=0$) if $w_1<2w_2/3$, and they would buy only input 2 ($z_1=0$) if $w_1>2w_2/3$. From that point, you can solve the maximisation problem by distinguishing these two cases.

The question can also be solved as a constrained optimisation problem if you add the constraints $z_1\geq 0$ and $z_2\geq 0$ to the Lagrangian. If $\lambda_1$ and $\lambda_2$ denote the Lagrangian parameters of these two constraints, you obtain the two cases aforementioned when i) $\lambda_1=0$ and ii) $\lambda_2=0$.

Answer 2

Given the production function $\sqrt{2z_1+3z_2}$, cost function can be obtained by minimizing cost: \begin{eqnarray*} \min_{z_1, z_2} \ \ w_1z_1+w_2z_2 \\ \text{s.t.} \sqrt{2z_1+3z_2} \geq q\end{eqnarray*} Solving it we get conditional input demand as follows: \begin{eqnarray*} (z_1, z_2) = \begin{cases} \left(\frac{q^2}{2}, 0\right) \ \text{if } \frac{w_1}{w_2} \leq \frac{2}{3} \\ \left(0,\frac{q^2}{3}\right) \ \text{if } \frac{w_1}{w_2} \geq \frac{2}{3} \end{cases} \end{eqnarray*} Cost function is therefore, $C(w_1,w_2, q)=\left[\min\left(\frac{w_1}{2},\frac{w_2}{3}\right)\right]q^2$.

Now to obtain the profit function we solve the following problem \begin{eqnarray*} \max_{q} \ \ pq- \left[\min\left(\frac{w_1}{2},\frac{w_2}{3}\right)\right]q^2\end{eqnarray*} and we get the supply function as: $q(p, w_1, w_2)= \frac{p}{2\left[\min\left(\frac{w_1}{2},\frac{w_2}{3}\right)\right]}$ and the optimal profit is $\pi (p, w_1, w_2)= \frac{p^2}{4\left[\min\left(\frac{w_1}{2},\frac{w_2}{3}\right)\right]}$