Estimating Labor Demand

REFERENCE: Trends in U.S. Wage Inequality: Revising the Revisionists»

This section discusses the approach taken by the empirical literature on labor demand. Applications which we will discuss in class include Card (2009) on the impact of international migration, Boustan (2009) on domestic migration and racial inequality, and Autor et al. (2008) on the impact of technical change.

This literature aims to explore the impact of the relative labor supply of different groups on relative wages. Papers in this literature estimate regressions of the form

We will start with a very general model, and get increasingly specific, until we end up with a model rationalizing such regressions. This approach allows us to discuss the assumptions invoked along the way, as well as their implications. We begin with a **general demand function**, mapping labor supply of various groups, in conjunction with unobserved factors, into the wages of these groups. We then consider the ** neoclassical theory of wage determination**, which assumes that wages correspond to marginal productivities with respect to some aggregate production function. We finally consider a set of specific parametric production functions, including the "**CES-production function**" and some of its variants.

Suppose there are \(j = 1, \ldots, J\) types of labor. As for "types," think in particular of the level of education; however, types might also depend on age, gender and country of origin. For each type \(j\), denote by \(N_j\) the number of people of that type which are employed in a given labor market. We might think of a labor market as a city, as a state, or as a nation. Denote by $$ N=(N_1, \ldots, N_J) $$ the labor supply of each type in this labor market, and by $$ w=(w_1, \ldots, w_J) $$ the (average) wage of each type.

As \(N\) changes, whether through immigration, demographic shifts, or education, this has consequences for wages. We can denote the **counterfactual wages** that would prevail if labor supply were equal to \(N\) (holding all else equal) by

Here "all else" is captured by \(\epsilon\), denoting all other factors influencing wages besides labor supply.

We would like to learn about the function \(w\), so that we can tell to what extent historical changes in \(N\) are responsible for changes in wage inequality. If changes in **labor supply** are **random**, that is independent of \(\epsilon\), this is in principle possible by regressing \(w\) on \(N\). The problem is that we usually have only a few observations, but potentially many variables \((N_1, \ldots, N_J)\) on which to regress, which makes it hard to get precise estimates. The literature therefore imposes restrictions on the function \(w(N, \epsilon)\), which are justified by theoretical models.

If changes in \(N\) are not random, which is likely the case if the labor markets considered are cities, then we additionally need to find valid **instruments** for \(N\). We will get back to this point below.

How are wages determined? Neoclassical theory assumes that there is an **aggregate production function**

which determines the amount of output \(y\) (in US$ terms) that can be achieved for a given level of inputs of different types of labor. Implicit in this formulation is that the supply of capital and demand for products have already been "concentrated out." Concentrating out a variable means to plug in its maximizing value given the other variables. For instance, concentrating out \(K\) from \(F(N,K)\) gives the function \(f(N) := \max_K F(N,K)\).

Neoclassical theory additionally assumes that wages are determined by the **marginal productivity** of different types of labor, that is by the amount output would be increased by increasing inputs by one unit:

This theory is justified by assuming that employers are profit maximizing, and that labor markets clear.

There are many **reasons to be skeptical** about this theory:

- What if effort or the qualification of applicants depend on offered wages? Then employers would be ill-advised to pay just the marginal productivity.
- Who even knows what the marginal productivity of a given type of labor is?
- What about social norms for remuneration, and what about collective bargaining?
- What if employers face upward sloping labor supply, maybe because of search frictions? Then they would depress wages below marginal productivity, acting as a "monopsony."
- What if labor markets don't clear, for whatever reason?

That said, as far as the function \(w(N, \epsilon)\) is concerned, the assumption that wages are determined by the marginal productivity of types of labor with respect to some aggregate production function does not impose much of a restriction. Its only implication for the behavior of the demand function is that it implies the **symmetry** condition

This symmetry is sufficient and necessary for the existence of a function \(f\) such that equation (4) holds.

Usually it is also assumed that the aggregate production function exhibits **constant returns to scale**, so that

for all \(\alpha >0\). This says that if all inputs are increased by a factor \(\alpha\), then so is aggregate output. Note that constant returns to scale in terms of labor inputs implicitly requires an infinitely elastic supply of capital and other factors, as well as an infinitely elastic demand for final products.

Constant returns to scale implies that wages only depend on relative supplies of labor. To see this, differentiate both sides of equation (6) with respect to \(N_j\), which yields

$$ \alpha \cdot \partial_j f(\alpha N_1, \ldots, \alpha N_J) = \alpha \cdot \partial_j f(N_1, \ldots, N_J), $$where I use \(\partial_j f\) to denote the partial derivative of \(f\) with respect to its \(j\)th argument. This in turn implies

Empirical work often assumes a specific functional form for the aggregate production function. The most common form is the "constant elasticity of substitution" production function. This production function takes the form

where the \(\alpha_j\) and \(\rho\) are unknown parameters which we might try to estimate from data. Assuming the marginal productivity theory of wages, equation (4), we get

This implies that the relative wage between groups \(j\) and \(j'\) is given by
$$
\frac{w_j}{w_{j'}} = \frac{\alpha_j}{\alpha_{j'}} \cdot \left (\frac{N_j}{N_{j'}}\right )^{\rho - 1}
$$
Denote
$$
\sigma = \frac{1}{\rho}-1,
$$

so that \(\rho-1 = - 1/\sigma\). \(\sigma\) is called the **elasticity of substitution**. It describes the slope of how relative wages depend on relative supply. This slope is constant for the given production function, lending it the name "constant elasticity of substitution."

In log terms, the relative wage can be written as

Substituting controls (including trends), as well as an unobserved residual \(\epsilon\), for the term \(\log \left (\frac{\alpha_j}{\alpha_{j'}}\right )\), we get our initial regression specification.

The basic CES production function is fairly restrictive. The literature uses various generalizations, where each type of labor is considered to be an aggregate of sub-types.

**Generalization 1: Aggregate types**

Assume that the \(J\) types of labor can be grouped into \(K\) aggregate types, where type \(j\) belongs to aggregate type \(k_j\). Many papers assume that the production function \(f\) is CES with respect to the supply \(L_k\) of these aggregate types. Within them, different types are perfectly substitutable, but might have different marginal productivities (different \(\theta_j\)). Formally,

We get wages, relative wages, and their logarithm to equal

$$ \begin{align*} w_j &= \frac{\partial f(N_1, \ldots, N_J)}{\partial N_j} = \left ( \sum_{k=1}^K \alpha_k L_k^\rho \right )^{1/\rho - 1} \cdot \alpha_{k_j} \cdot L_{k_j}^{\rho - 1} \cdot \theta_j\\ &= y^{1 - \rho} \cdot \alpha_{k_j} \cdot L_{k_j}^{\rho - 1} \cdot \theta_j\\ \frac{w_j}{w_{j'}} &=\frac{\theta_j}{\theta_{j'}} \cdot \frac{\alpha_{k_j}}{\alpha_{k_{j'}}} \cdot \left (\frac{L_{k_j}}{L_{k_{j'}}}\right )^{\rho - 1} \\ \log \left (\frac{w_j}{w_{j'}} \right ) &= \log \left (\frac{\alpha_{k_j}}{\alpha_{k_{j'}}} \cdot \frac{\theta_j}{\theta_{j'}}\right ) - \frac{1}{\sigma}\log\left (\frac{L_{k_j}}{L_{k_{j'}}}\right ) \end{align*} $$

The relative wages of types \(j\) and \(j'\) thus depend only on the relative supply of aggregate types \(k_j\) and \(k_{j'}\).

**Generalization 2: Nested CES**

As in generalization 1, assume that there are \(K\) aggregate types which are substitutable with elasticity \(\sigma_1\). Within these aggregate types, however, the sub-types are not perfectly substitutable, but instead have an elasticity of substitution of \(\sigma_2\). Formally,

$$ \begin{align*} y &= f(N_1, \ldots, N_J) = \left ( \sum_{k=1}^K \alpha_k L_k^{\rho_1} \right )^{1/{\rho_1}}\\ L_k &= \left ( \sum_{j:k_j=k} \theta_j N_j^{\rho_2} \right )^{1/{\rho_2}} \end{align*} $$We get wages, relative wages, and their logarithm to equal

$$ \begin{align*} w_j &= \frac{\partial f(N_1, \ldots, N_J)}{\partial N_j} = y^{1 -\rho_1} \cdot \alpha_{k_j} \cdot L_{k_j}^{\rho_1 - \rho_2} \cdot \theta_j \cdot N^{\rho_2-1}\\ \frac{w_j}{w_{j'}} &=\frac{\theta_j}{\theta_{j'}} \cdot \frac{\alpha_{k_j}}{\alpha_{k_{j'}}} \cdot \left (\frac{L_{k_j}}{L_{k_{j'}}}\right )^{\rho_1 - \rho_2} \cdot \left (\frac{N_j}{N_{j'}}\right )^{\rho_2 - 1} \\ \log \left (\frac{w_j}{w_{j'}} \right ) &= \log \left (\frac{\alpha_{k_j}}{\alpha_{k_{j'}}} \cdot \frac{\theta_j}{\theta_{j'}}\right ) + \left (\frac{1}{\sigma_2} - \frac{1}{\sigma_1} \right )\log\left (\frac{L_{k_j}}{L_{k_{j'}}}\right ) - \frac{1}{\sigma_2}\log\left (\frac{N_j}{N_{j'}}\right ) \end{align*} $$The relative wages of \(j\) and \(j'\) thus depend on both \({L_{k_j}}/ {L_{k_{j'}}}\) and \({N_j}/{N_{j'}}\).

Contrary to what we have assumed so far, variation in labor supply might not be random relative to other factors determining wages. We should especially worry about this when comparing wages across cities. It might for instance be the case that workers with a college degree migrate to cities where the returns to a college degree are highest.

To take care of this endogeneity issue, Card (2009) proposes to instrument changes of labor supply making clever use of prior migration patterns. If you need to review instrumental variables, have a look at chapter 4 of of Angrist and Pischke (2010) or my LECTURE SLIDES. The idea is that new migrants tend to settle in the same cities as previous migrants from the same source countries, while the amount of new migrants at the national level is arguably not affected by city-specific economic conditions. Suppose for instance that prior migrants from country \(m\) happened to settle mostly in Chicago and Los Angeles. If then some political or economic crisis in country \(m\) compels a new set of people from \(m\) to leave their home country, Chicago and Los Angeles will likely experience an increase in their labor force which is unrelated to local economic conditions in these two cities.

Let us make this more formal, cf. (Card, 2009, p8). Let \(m\) index different source countries for migrants, and let \(M_m\) denote the number of new migrants from country \(m\) arriving in a given time period. Let \(\lambda_{m}\) denote the share of prior migrants from county \(m\) living in a given city \(i\), and \(\delta_{m,j}\) the share of migrants from country \(m\) that are of type \(j\). If new migrants from \(m\) have the exact same settlement patterns as prior migrants, then we should see

$$ \sum_m \lambda_m M_m $$new migrants arriving in a given city. Given their distribution of types, this would imply a growth of the population of workers of type \(j\) in the given city by a factor

This is the instrument for changes in \(\log N_j\) that Card proposes. For the relative change of \(N_j\) and \(N_{j'}\) we can use the instrument

$$ Z_{j,j'} = \frac{Z_j}{Z_{j'}}. $$This is a valid instrument if it satisfies the condition

$$ E[Z_{j,j'}\cdot \epsilon_{j,j'}]=0. $$Under this condition, we can estimate \(\beta\) in equation (1) by