How accurate is duality?

Question

In economic theory we know that with the use of some calculus, Hotellings Lemma and Sheppards lemma we can derive a given firms supply function and in term its Profit function.

With data of a given firms costs, can we in fact get an accurate estimate of its profit function and supply function?

Answer 1

Let me focus on the use of duality theory in demand analysis (as this is what I'm most familiar with).

Direct approach

Usually, in order to obtain a demand system (which you can then estimate) you need to take the following steps.

1. specify a utility function $u(x_1, \ldots, x_n)$.
2. maximize this with respect to a budget constraint $\sum_i p_i x_i \le m$. .
3. Obtain a closed form solution for $x_i$ as a function of all prices $(p_1,\ldots, p_n)$ and income $m$.
4. Estimate these functions.

Step 3 is the trickiest part as usually no closed form solutions are possible. To see this, notice that the first order conditions give the following system: $$ \frac{\partial u(x_1, \ldots, x_n)}{\partial x_n} = p_i\,\, \lambda,\\ \sum_i p_i x_i = m. $$ Solving for $\lambda$ we have: $$ \dfrac{\dfrac{\partial u}{\partial x_i}}{\sum_j \dfrac{\partial u}{\partial p_j}} = \frac{p_i}{m} $$ This gives the (normalized) prices on the right hand side as a function of the quantities (so the inverse demand functions). To obtain a closed form solution of the quantities in terms of the functions, this system needs to be inverted. Except in some particular cases (e.g. Cobb-Douglas or CES) we do not know how to do this.

Dual approach

The dual approach starts from a specification of the expenditure function and uses known duality results to obtain a closed form expression for the demand functions. It takes the following steps:

1. Specify an expenditure function $e(p_1, \ldots, p_n,u)$.
2. Use Shephard's lemma to obtain the Hicksian demand function $h_i(p_1,\ldots, p_n, u)$.
3. Invert $e(p_1, \ldots, p_n, u)$ with respect to $u$ to get the indirect utility function $v(p_1, \ldots, p_n, u)$.
4. substitute $v(p_1, \ldots, p_n, u)$ in $h(p_1, \ldots, p_n, u)$ to get the Marshallian demands $x_i(p_1, \ldots, p_n, m)$.

As an exemple, consider the widely used Almost Ideal Demand System specification of Deaton & Muellbauer

Start from the following (flexible) specification for the expenditure function. $$ \ln e(p,u) = \alpha_0 + \sum_i \alpha_i \ln(p_i) + \sum_{i,j}\frac{1}{2}\gamma_{i,j} \ln(p_i) \ln(p_j) + u \prod_i p_i^{\beta_i} $$ We impose $\gamma_{i,j} = \gamma_{j,i}$, and by homogeneity of degree 1 we have that $\sum_i \alpha_i = 1$ and $\sum_{i = 1}^n \gamma_{i,j} = 1$ and $\sum_{i} \beta_i = 0$.

First we can take the derivative $\frac{\partial \ln e(p,u)}{\partial \ln p_i}$ to get the demand share equations. Shephard's lemma gives: $$ \frac{\partial \ln e(p,u)}{\partial \ln p_i} = \frac{p_i h_i(p,u)}{e(p,u)} = \alpha_i + \sum_j \frac{1}{2}\gamma_{i,j} \ln(p_j) + u \beta_0 \beta_i \prod_i p_i^{\beta_i} $$ Here $h_i(p,u)$ is the Hicksian demand for good $i$.

Writing by $v(p,m)$ the indirect utility function, we know by duality between utility maximisation and expenditure minimisation that: $$ m = e(p,v(p,m)) \text{ or } u = v(p,e(p,u)). $$ Using this, we can invert the expenditure function to obtain the indirect utility function. $$ v(p,m) = \frac{\ln m - \alpha_0 - \sum_j \alpha_j \ln(p_j) - \sum_{i,j}\frac{1}{2} \gamma_{i,j} \ln(p_i) \ln(p_j)}{\beta_0 \prod_i p_i^{\beta_i}} $$

Also by duality, if $x(p,m)$ is the Marshallian demand function then: $$ x_i(p,m) = h_i(p,v(p,m)). $$ So we can substitute $v(p,m)$ into $h_i(p,u)$ to get the shares as a function of prices and income.

$$ \begin{align*} s_i(p,m) &= \frac{p_i x(p,m)}{m} = \alpha_i + \sum_j \frac{1}{2}\gamma_{i,j} \ln(p_j) + \beta_i \left(\ln m - \alpha_0 - \sum_j \alpha_j \ln(p_j) - \frac{1}{2}\sum_{i,j} \gamma_{i,j} \ln(p_i) \ln(p_j)\right),\\ &= \alpha_i + \sum_j \frac{1}{2}\gamma_{i,j} \ln(p_j) + \beta_i \ln(m/P). \end{align*} $$ where: $$ P = \alpha_0 + \sum_j \alpha_j \ln(p_j) + \sum_{i,j} \frac{1}{2}\gamma_{i,j} \ln(p_i)\ln(p_j). $$ This is the Almost Ideal Demand System. Conditional on $P$, shares are linear in the log of income and log of prices. The symmetry and homogeneity restrictions immediately translate to conditions on the coefficients of this system.

Answer 2

Here's a paper link where they picked up a $R^2$ of over .99 I don't think this is particularly unusual or strange of a result, either. In estimating costs for firms in public utilities, I recall doing numerous exercises with real data from various utilities. I typically got $R^2$'s in the 0.95's or higher, so I feel they are fairly accurate within sample.

This means economists are fairly good at estimating what a firm's per-unit costs are if we have a good knowledge of their total capital stock and expenditures, and we stay within the bounds of our sample.

However, if you are asking about very far out-of-sample predictions, or systemic shifts in the market, then you may find less success.