Are There Giffen Inputs? I am studying for my candidacy exams and I came across this question on a previous exam. The question is in the TFD (True, False, Debatable) section of the exam. The claim is:


  There are no Giffen inputs in production.


I think this question is a very fascinating one, and should spark some interesting discussion. My intuition tells me that this is false because if there are Giffen goods on the consumer side then surely there are Giffen goods on the producer side. However, I cannot think of a concrete counterexample to the claim. In consumer theory, they claim that Giffen goods occur when the good is so important to the consumer that when the price increases, they decide to just buy that good and not buy any other goods. For example, economists believe that one of the only real life Giffen good situations is potatoes in the Irish potato famine. They claimed that potatoes were such a staple in the Irish diet that when the prices rose, the Irish people decided not to buy other foods (such as meat) and dedicated all of their food budget to potatoes. 

Are there any situations where we might see a firm/industry act in a similar way? What do you guys think? Are there any Giffen inputs in production?
I believe the answer is true.

Giffen goods are goods where the income effect overpowers the substitution effect.

$$\begin{align}
\max_{\vec x} \ \ \ &amp; U(\vec x) \\
&amp; \text{s.t.} \ \ \ \vec p \cdot \vec x \leq I
\end{align}$$

To start, if you think about the consumer's problem (for example utility maximization, here), a change in a good's price affects both relative substitutability of goods through the marginal rate of substitution AND it affects purchasing power through the budget constraint.



Let us consider a profit maximizing firm with a constraint on how much they can spend. For simplicity let us use a single output technology, with differentiable production function $f(\vec z)$. Let $\vec z$ be a vector of inputs (expressed as negative values), $\vec w$ a vector of input prices, and $p$ the output price.

$$\begin{align}
\max_{\vec z} \ \ \ &amp; pf(\vec z) + \vec w \cdot \vec z \\
\text{s.t.} &amp; \ \ \ \vec w \cdot \vec z \leq B \\
&amp; \ \ \  z_i \leq 0 \\
\end{align}$$

Normally we would have a constraint on production, but instead we have a "budget" constraint. What happens if we form the Lagrangian here?

$$\mathcal{L} = pf(\vec z) - \vec w \cdot \vec z - \lambda(\vec w \cdot \vec z - B) + \vec\mu \cdot \vec z$$

Take first order conditions:

$$\frac{\partial \mathcal{L}}{\partial z_i} = pf_{z_i}(\vec z) - w_i - \lambda w_i + \mu_i = 0 \tag{1}$$

$$\frac{\partial \mathcal{L}}{\partial f(\vec z)} = p = 0 \tag{2}$$

$$\frac{\partial \mathcal{L}}{\partial \lambda} = \vec w \cdot \vec z - B = 0 \tag{3}$$

At an interior solution where the budget constraint binds, we should have the optimum $\vec z^*$ to solve the FOCs

$$p \frac{\partial f(\vec z^*)}{\partial z_i} = w_i$$

but instead you solve out (1):

$$p \frac{\partial f(\vec z^*)}{\partial z_i} = - \frac{\mu_i}{1 + \lambda}w_i$$

and (3) does not provide any help to solve the Lagrangian multipliers. (2) is nonsense.

A better constraint would be something like $y - f(\vec z) \leq 0$, where $y$ represents the scalar of output.

Without an "income effect", there isn't much to study Giffen behavior. Producer theory doesn't use a budget constraint to solve these sorts of problems. Increasing input price will always decrease use of that input except with corner solutions, where there might be no change. So there can't be a Giffen input.