6.3Productivity and the Density of Economic Activity

This section provides a description of a study by Ciccone and Hall (1996) [CH hereafter], who evaluated the role of county density of economic activity in explaining the variation of average labor productivity across the U.S. states. Their most important finding is a positive elasticity of productivity with respect to density. Their estimates indicate that doubling average employment density at the county level can increase by 6% the average labor productivity at the state level. This goes again because of diminishing marginal product. Here CH found that workers are more pro st the neoclassical assumption that the marginal product of labor would be lower in denser areas ductive when moved to a denser area.

To explain the mechanism by which density affects productivity, they use two models. The first model is based on geographically localized externalities, comparing agglomeration and congestion effects of density. However, it does not reveal the source of agglomeration effects. The second model gives density an explicit role. It builds on earlier work by Abdel-Rahman (1988) and Rivera-Batiz (1988), and is based on the fact that denser areas exhibit a greater variety of intermediate products, which increases productivity. The last part of CH’s analysis deals with capital and total factor productivity, in an attempt to determine the ways in which density affects productivity. First, assuming constant returns in technology and increasing transportation costs, output will rise with density because firms will avoid transportation costs by concentrating in space. Second, there might also be externalities emerging from the physical proximity of production. Finally, density might affect productivity through a higher degree of specialization.

CH made an important contribution to the literature with their empirical work on productivity based on actual measures of density. Previous studies had assumed agglomeration benefits implicitly only. Theoretically, the economics of agglomeration state that a greater variety of intermediate inputs will increase productivity. The purpose of CH is to consider density explicitly, at the county level, and measure its effect on average labor productivity at the state level.

CH’s first model explains how density affects productivity and how to aggregate across productive units. Their model, which is based on externalities, considers labor and land only as factors of production. They first made the assumption that the externality depends multiplicatively on output per acre, which is the measure of density. The elasticity of output with respect to density is a constant, (λ - 1) / λ, and the elasticity of output with respect to employment is also a constant, α. Elasticity α measures the effect of congestion whereas λ measures the effect of agglomeration. If λ is less than one, the elasticity is negative and there are agglomeration diseconomies. A CES production function gives the output q produced in an acre of space by employing n workers:

f (n,q,a) = nα (q/a)( λ -1)/ λ(6.10)

The county-wide production density function is then given by

qc / ac = ( nc / ac ) γ(6.11)

where γ is the product of the production elasticity, α, and the elasticity of the externality, λ. If α <1 and λ >1 then γ >1 and agglomeration effects exceed congestion. Empirical results show that the net effect favors agglomeration. The technique used to aggregate county production density to at the state level is to calculate average labor productivity in the state by summing the county production densities weighted by each county’s area and dividing by total state employment, Ns

Qs / Ns = [ Σ nc γ ac - ( γ -1) ] / Ns (6.12)

This magnitude is also defined as the factor density index Ds(γ). CH then decomposed this density index into three components. The density effect becomes the product of a national effect, a state effect and a county effect. The neoclassical model assumes γ <1 and leads to the hypothesis that productivity and density are negatively related if congestion effects are greater than agglomeration effects.

In their second model, CH hypothesized increasing returns in production of local intermediate goods, as suggested by Abdel-Rahman (1988) and Rivera-Batiz (1988). This model treats density endogenously, and shows its relationship to productivity. The production function now depends on the amount of labor m, used directly in the making of the final good, and i, the amount of intermediate service input, which cannot be transported outside the acre. The production function for the final good is:

f ( m , i ) = [ m β i (1 - β ) ] α(6.13)

where α and λ still represent respectively congestion and agglomeration effects.

The level of output of i at the zero-profit level is given by the following CES production function

where x(t) denotes the individual differentiated services, indexed by type t, z is the number of different types of individual services produced, and μ is the markup of price (defined as a ratio of the marginal cost that the producer will set in order to maximize profit). If labor is paid at w, the profit function for an intermediate product maker will be π = (μ -1) wx – wv. Because of the monopolistic competitive market situation, competitors can enter freely and eventually profit will be driven to zero, where the level of output will then be x = v / (μ - 1). Substituting into the intermediate production function gives i = zμ x. The productivity of the i-making process will then be z μ -1 and because μ >1, productivity will rise with the available variety of intermediate goods. In a denser area there is more variety of intermediate goods and therefore there is a positive relationship between density and productivity. CH then elaborated an equation that is similar to the county production function found in the first model and concluded “ both models provide a theoretical foundation for the same estimation procedure in state data.” The production function describing output produced in an acre of space is then define as:

As [(e c n c)β k c 1- β ] α (q c / a c) ( λ -1) / λ(6.15)

where As is Hicks-neutral technology multiplier for state s, n c is employment in county c and k c is capital in county c. It is assumed that labor and capital employed in a county are distributed equally among the acres in the county. Finally, e c is a measure of the efficiency of labor, which depends on the average years of education h c, and is defined by e c = h c η (where η is the elasticity of education).

CH dealt with capital by first assuming a uniform rental price of capital r. Then, they substituted factor price for factor quantity in the factor demand function. Further, they defined the elasticity of labor, θ, by the ratio γβ / [1 - γ (1 - β)].

Assuming a log-normal distribution of state productivity around a nationwide level and allowing for mismeasurements in an error term, the final model of production is:

ln ( Q s / N s ) = ln φ + ln Ds ( θ, η) + us(6.16)

where φ is a constant that depends on the interest rate, and us is the measurement error, assuming that errors of different states are uncorrelated.

According to neoclassical assumptions, in equilibrium density should be equal everywhere and nobody would have the incentive to move. However, if θ >1, a worker would be more productive if moved to a denser area. The only equilibrium would then be for all workers to concentrate in one single county. In reality, states and counties have different densities, so how can they be in equilibrium? The answer given by CH is the same one used by urban theorists: some workers simply prefer to live in less dense areas, with lower wages, to avoid the disamenities from agglomeration (pollution and traffic for example).

Many studies have estimated the effects of agglomeration economies for manufacturing industries. The industries studied have been usually 2-digit SIC industries at the regional level. However, as we have seen, Ciccone and Hall built an ingenious model estimating net external economies of employment density for counties. Their output measure is almost the same as the one used by Carlino and Voith (1992): Gross State Product (GSP) minus agricultural and mining output. Their analysis goes beyond the manufacturing sector. Data on employment by county come from the U.S. Bureau of Economic Analysis for the year 1991. Data on education are gathered at the state and county level by the U.S. Bureau of Census. At the state level, it is the number of years of education times the number of hours worked. At the county level it is simply the average years of education.

Using nonlinear least squares, Ciccone and Hall first estimated the returns to scale parameter, θ, and the elasticity of average product with respect to education, η. To prevent their results from being biased by “reverse causation,” they used nonlinear instrumental variables for the density index. The characteristics used as instruments refer to the historical and geographical situations of the states. The instrumental variables estimate for θ is 1.06, which means that doubling the employment density in a county increases labor productivity by 6%. They also found that the estimate of γ was close to 1.04, meaning that doubling employment density in a county would result in an increase of 4% in total factor productivity.

According to the density index Ds (γ), the District of Columbia is the densest area, followed closely by New York state, which also contains the densest county in the U.S., New York City county (with a factor density index of 1.94). Workers in New York city are 22% more productive than in New York state. Plotting productivity by state against education, they showed a significant role of education in determining productivity (positive slope of the regression line).

Finally, CH showed that the differences in productivity were not due to factors such as public capital, and the differences persisted even when controlling for education. The positive relationship between density and productivity was also not due to the size of the market.