# 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} \ \ \ & U(\vec x) \\ & \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} \ \ \ & pf(\vec z) + \vec w \cdot \vec z \\ \text{s.t.} & \ \ \ \vec w \cdot \vec z \leq B \\ & \ \ \ 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.

• Isn't there an analog of the CMP for consumers though? Doesn't the expenditure minimization problem for consumers mimic the cost minimization problem for producers? If so, wouldn't the same argument rule out Giffen goods for consumers? – DornerA Jun 2 '16 at 18:22
• @DornerA My intuition is that although the UMP and EMP are dual problems for the consumer, EMP assumes utility is exogenous, which doesn't make sense for a consumer (for a social planner, sure). Also note that the PMP and CMP for the producers both do not have input price in the constraints. – Kitsune Cavalry Jun 2 '16 at 18:47
• I agree that the UMP makes more sense from a consumer's standpoint, but again, I think that the same argument applies to producers. The cost minimization problem assumes that you already know what output would maximize profits, which is also weird to think about. – DornerA Jun 2 '16 at 19:06
• We cannot examine the OP's question using the frameworks of cost minimization and profit maximization. In either, the firm can vary its total expenditure, i.e. its budget. But Giffen behavior is examined under the assumption that the budget of the consumer remains constant. The existence of a "budget constraint" is the main difference between Consumer Theory and (standard) Firm Theory: in Firm Theory no "budget constraint" exists. (for some discussion and a reference for the theory of firm under a budget constraint see economics.stackexchange.com/a/5273/61 – Alecos Papadopoulos Jun 2 '16 at 22:28
• @Dugo It is not with me that you disagree. It is with what is considered fundamental microeconomic theory of the firm by a very wide number of scientists and textbooks. – Alecos Papadopoulos Jun 7 '16 at 17:45

There are no Giffen inputs. Suppose there are $l$-goods, including all inputs and outputs. A price system is then a vector $p=(p_1,\ldots,p_l)\in\mathbb{R}^l$. One can give a firms production decision by a production plan $y=(y_1,\ldots,y_l)\in\mathbb{R}^l$. The idea is that $y_j$ denotes the net output produced of good $j$. If it is an input, this entry is negative. This way of writing production plans has the wonderful effect that $$p\cdot y=\sum_{j=1}^lp_jy_j$$ equals revenue minus cost and therefore profit when the firm can actually sell $y$ at the price system $p$. The revenue comes from the positive entries, the output times price, the cost from the negative entries. Now let $p$ and $p'$ be two price systems and $y$ and $y'$ be two production plans such that $y$ is profit-maximizing given the price system $p$ and $y'$ is profit-maximizing given the price system $p'$. Then we must have (we'll see later why) that $$(p-p')\cdot(y-y')=\sum_{j=1}^l(p_j-p_j')(y_j-y_j')\geq 0.$$ If $p$ and $p'$ differ only in the price of good $j$, this gives us $(p_j-p_j')(y_j-y_j')\geq 0$ which shows that an increase of the price of good $j$ can never reduce the amount of net output of good $j$ being produced. If this is an input, so that the entry is negative, there can be never more use of the input.

So let's prove that $(p-p')\cdot(y-y')\geq 0$. Since $y$ is proft maximizing at $p$, $y'$ cannot give a higher profit at $p$. So $p\cdot y-p\cdot y'=p\cdot (y-y')\geq 0$. Similarly, $p'\cdot y'-p'\cdot y=p'\cdot (y'-y)\geq 0$. Therefore, $$(p-p')\cdot(y-y')=p\cdot (y-y')+(-p')\cdot(y-y')=p\cdot (y-y')+p'\cdot(y'-y)\geq 0.$$

# Consumer's problem

We assume monotone concave utility function, that is, diminishing marginal utilities and binding budget constraint.

The first order condition is: $$\frac{P_A}{P_B} = \frac{\text{MU}_B}{\text{MU}_A}$$ where $\text{MU}_i$ is the marginal utility for good $i$.

Now suppose $P_A$ increases, the first order condition should still hold, therefore the right-hand side should also increase. If A is Giffen good, then consumer buys more A, and less B under binding budget. So $\text{MU}_B$ increases, and $\text{MU}_A$ decreases, thus the ratio increases.

# Producer's problem

Without loss of generality, I use two traditional inputs labor $L$ and capital $K$. I also assume diminishing marginal product for both inputs. For interior solutions, \begin{align*} P\cdot\text{MP}_{L} & =w\\ P\cdot\text{MP}_{K} & =r \end{align*} One of the differences between consumer's problem and the firm's problem is that a consumer spends all of the budget, as long as the utility function is strictly monotone. But a firm may choose to leave some or all of the money on the table, if producing more means losing more. But when examine the Giffen behavior, we need to keep the budget constant. So the question should be asked under the assumption that the firm exhausts constant budget both before and after the input price change. Let's assume that's true, due to high enough product price, high marginal products, or low input prices.

Now suppose the wage increases. Labor would be a Giffen input only if the firm use more labor. From the first equation about the labor, we know the marginal product of labor has to increase. Under diminishing marginal products, either of the following could be true:

1. the firm uses less labor, therefore higher $\text{MP}_L$.
2. the firm uses more labor, but still achieve higher $\text{MP}_L$ if capital also increases, due to certain degree of complementarity between inputs.

But the binding budget rules out the second possibility: higher labor cost and more labor implies less capital. Therefore I don't think Giffen input exists for "well-behaved" production functions, at least not for interior choices. But I haven't examined production functions that have pathological properties such as when higher capital stock decreases the marginal product of labor (negative cross partial derivatives).

It is possible to have "Giffen Inputs," but we rarely see them in practice.

We can decompose an output effect and a substitution effect in producer theory. In consumer theory, we used Slutsky decomposition to find income and substitution effects. This is done by setting compensated (Hicksian) demand equal to uncompensated (Marshallian) demand and taking the derivative with respect to the price of the good in question. Similarly, we can find a compensated and uncompensated factor input demand through the derivative of the profit function and the cost function, respectively, with respect to the price of the input we wish to analyze. We then set these equal to one another, and take the derivative again with respect to the input price.

With an increase in input price, we find that the substitution effect will always be negative. If we fix our level of output, the output effect will be zero, and there will never be an inferior or giffen input. However, when we allow output to vary - we can get all three results: normal input, inferior input, and giffen input.

We might imagine a firm using an environmentally unfriendly resource, and facing political pressure from using it. In this case, it might be reasonable for the firm to increase the use of another more environmentally friendly input even though its price is increasing from the outside political pressure (firms are increasing demand for it to save their public image) and decrease the use of this input when its price subsides after the spotlight is gone. This is not a perfect example, but again, giffen things are tricky to find in practice. The theory behind it, however, exists.