Urbanization and the Viability of Local Agricultural Economies

II. The Theoretical Model

Consider a local agricultural economy that includes farmers, agricultural input suppliers, output processors, and consumers. The farmers purchase inputs from the input suppliers and sell their products to the output processors. All farmers are price takers in both the input and output markets, and produce an agricultural commodity using a constant return to scale production technology. The unit of output is normalized such that the production function takes the form y = α + βx − 0.5x², where y is the yield per acre, x is the input application, and α and β are positive parameters. Assume that each farmer chooses input application to maximize profit. Then the optimal input application is x* = β − (w/p), where w and p are the input and output prices, respectively. The optimal yield is y* = γ − 0.5(w/p)², where γ = α + 0.5β²; the per-acre net return is π^a = γp − βw + 0.5(w²/p).²

Let A be the total acres of cropland in the region, then the total demand for the input and the total supply of the output in the region are

[1]

[2]

Note that A is a function of the net return to agriculture π^a (therefore, p, w) as well as the net return to developed land π^d. Based on previous studies (e.g., Alig and Healy 1987; Hardie and Parks 1997), π^d is assumed to be a function of the existing level of urbanization, measured by population density (D) and total developed area (U) and location characteristics of the area (A), such as the distance to the nearest metropolitan center and the level of natural amenities in the area. Thus, A = A(p,w,U,D,X).³

Consider the supply of the agricultural input. Suppose there are multiple input suppliers initially, and the suppliers engage in oligopolistic competition. The exit and entry decisions of the suppliers are conceptualized as a two-stage process following Mas-Colell, Whinston, and Green (1995, 405-7). In the first stage, all suppliers decide “in” or “out.” If a supplier decides “in,” it incurs a setup cost K > 0. Once the setup cost is sunk, all suppliers that have decided “in” play an oligopolistic Cournot game in the second stage and supply the input at a constant marginal cost m. This two-stage model defines a dynamic game. In a subgame perfect Nash equilibrium, no supplier wants to change its exit decision given the decisions of the other suppliers. There is an equilibrium with suppliers choosing to stay in the input market if and only if and , where π^s(N) denotes the profit of a supplier in Stage 2 equilibrium when there are N input suppliers in the market.⁴

Given the input demand function [1], the second-stage output, x^s(N), and profit, π^s(N), for each input supplier can be derived as

[3]

[4]

Note that given N and p, both x^s(N) and π^s(N) increase with A. Thus, if urbanization reduces the total amount of farmland, it will reduce the output and profit of an input supplier, at least in the short run. Solving for N at which π^s(N) = K gives

[5]

The equilibrium number of input suppliers equals the largest integer that is less than or equal to , but for simplicity, we simply treat as from now on. Note that given p, increases with A. Thus, with urbanization, there will be fewer input suppliers in the region if the output price is not affected. However, if urbanization also increases the output price p, it could potentially increase the number of input suppliers in the region. Solving for A at which gives the threshold of farmland acreage below which no input supplier will operate in the region. Specifically, when A <4pK/(βp − m)², the demand for the inputs is so low that even if a supplier can charge a monopolist’s price, it would still not be able cover its total cost.

The total supply of the agricultural input equals the output per supplier multiplied by the number of suppliers in the market, that is, . Setting the total supply equal to the total demand, the equilibrium price of the agricultural input, w^*, can be derived as

[6]

Equation [6] clearly shows that if the output price is not affected by urbanization, farmers will face higher input prices as more farmland is developed. Since A = A(p,w,U,D,X), equation [6] implicitly defines w* as a function of p, m, U, D, and X, which is denoted by w* = w(p,m,U,D,X).

Farmers’ output prices may also be affected by urbanization. Specifically, by substituting A = A(p,w,U,D,X) and w* = w(p,m,U,D,X) into [2], the total supply of the agricultural output in the region can be derived as a function of (p,m,U,D,X). If the output market is assumed to be dominated by a few processors and each of them engages in Cournot competition, the equilibrium output price for farmers can be derived as p* = p(U,D,X,M), where M is a vector of constant marginal costs for the input suppliers and the output processors. As in the input market, it can be shown that if urbanization does not increase the output price of the processors, there will be fewer processors in the region, and the equilibrium price of the agricultural output will be lower.⁵ However, with urbanization, the processors will face increasing demand for their output because of a larger customer base. To obtain more input, the processors may also be willing to pay a higher price for agricultural output supplied by the farmers. In sum, ∂p*/∂U ≤ 0 if developed area has little effect on the demand for agricultural output, and ∂p*/∂U > 0 if development significantly increases the demand for agricultural output (likewise ∂p*/∂D > 0 if an increase in population density significantly increases demand for the agricultural output).

Given the equilibrium prices of the input and output for farmers, the equilibrium number of input suppliers and the farmers’ production cost and net return can be derived as

[7]

[8]

[9]

where , , and are all functions of (U,D,M,X). The effect of urbanization on the number of input suppliers can be derived by differentiating [7] with respect to U:

[10]

If urbanization reduces the total amount of farmland (i.e., ∂A*/∂U < 0) but has no effect on the price of agricultural output (i.e., ∂p*/∂U = 0), [10] is negative, implying that urbanization reduces the number of input suppliers in the region. However, if urbanization also increases the demand for agricultural output and thus the output price p* (i.e., ∂p*/∂U > 0), its impact on the number of input suppliers cannot be determined analytically because input use intensification may occur due to output price increases. Higher agricultural supply and input application per acre could lead to higher demand for agricultural inputs by farmers even if the total acres of farmland has been reduced.

Urbanization can also have a positive or a negative effect on farm net return, depending on its effect on the price of agricultural output. By Hotelling’s rule,

[11]

If urbanization reduces the total amount of farmland but has little effect on the price of agricultural output, then dw*/dU > 0 by[3]. In this case, , implying that urbanization reduces farm net return. However, if urbanization increases the price of agricultural output, can be positive or negative, depending on the relative magnitude of (dw^*/dU) and (dp^*/dU). Likewise, the effect of urbanization on farmers’ production costs can be positive or negative, depending on the elasticity of the demand for agricultural inputs and the effect of urbanization on the input price.

So far, we have assumed that the local agricultural economy produces a single commodity. But the emergence of a new customer base with urbanization may provide opportunities for farmers to grow high-value crops such as flowers and vegetables. To examine such structural changes associated with urbanization, we expand the model to include two commodities; one is a traditional crop such as corn or wheat, and the other is a high-value specialty crop such as a flower or a vegetable. The high-value specialty crop is more input intensive and faces increasing demand with urbanization. The total demand for agricultural input is

[12]

where x_i and x₂ are the per-acre input use for the specialty and traditional crops, respectively; and s₁ and s₂ are the shares of land allocated to the two crops. Assume that each parcel is allocated to the crop that generates the highest profit; in equilibrium the land in the region will be allocated between the two crops such that their marginal profits are equalized:

[13]

where is the marginal profit of increasing land allocation to crop i, and . If for s₂ = 1, then only the traditional crop is planted in the region.

Suppose the demand for the specialty crop increases with urbanization and, as a result, the profit from producing the specialty crop increases faster than the profit fromproducing the traditional crop. With equation [13], it is easy to show that more land will be allocated to the specialty crop. Since the specialty crop is more input intensive than the traditional crop (i.e., x₁ > x₂), the total demand for agricultural input increases even if the total amount of farmland A decreases, because the average input use per acre, (s₁x1 + s₂x₂), increases. Furthermore, if the two crops use different types of inputs, urbanization can also change the mix of agricultural inputs used in the region, by inducing farmers to convert some portion of their land to production of specialty crops.

In sum, urbanization can have a positive or a negative effect on agricultural infrastructure, farmers’ production costs, and net farm returns. If urbanization has little effect on farmers’ output prices, there will be fewer input suppliers in business with urbanization. The input markets will become less competitive and the prices of inputs may increase. However, with urbanization, the total demand for agricultural output may instead increase, which may lead to higher prices for agricultural outputs. In addition, farmers may switch to high-value, input-intensive specialty crops such as flowers and vegetables. In this case, the effects of urbanization on the number of input suppliers, farmers’ production costs, and net farm returns cannot be determined analytically. In the next section, we conduct an empirical analysis to examine how urbanization affects agricultural support sectors, farm production costs, and net farm returns.

III. The Empirical Model

The empirical model is specified based on the theoretical analysis. The model is the empirical counterpart of equations [7] through [9], which specify the number of input suppliers, the number of output processors, the per-acre production cost, and per-acre net return as functions of all exogenous variables included in each equation, including U, D, M, and X:

[14]

[15]

[16]

[17]

where i, n, and t index county, “neighborhood,” and time, respectively. Following Holmes (1999), we define neighborhood as a given county and all counties having centers within 50 miles (as the crow flies) of that county’s center.⁶ The dependent variables of equations [14] through [17] are the number of agricultural input suppliers, the number of agricultural output processors, the average per-acre farm production cost, and the average per-acre net farm income, respectively. Based on equations [7] through [9], these variables depend on the level of urbanization, associated with the amount of developed acres U and population density D. We lag the developed acres and the population density explanatory variables in part because of a concern with potential endogeneity. A substantive reason for lagging these variables is that decisions to start up or shut down a business take time. For instance, as the amount of cropland in a given county declines and farm input suppliers lose their customer base, they may deliberate on the decision of whether to stay in business or exit. Similarly, external effects on farm production costs and net farm returns associated with urbanization and farmland development may occur with a lag. For example, one posited benefit to farmers of urbanization is the possibility of higher farm income from producing and marketing specialty crops to urban consumers. Moving out of more traditional crops and into such specialty crops may take time.

It should be noted that developed acres U and population density D are measured at the neighborhood level, while all other model variables are measured at the county level. We expect that the number of input suppliers, output processors, farm production costs, and net returns to farming in a given county are influenced by urbanization in that county as well as in neighboring counties. Thus, an important advantage of measuring developed acres U and population density D at the neighborhood level is to take into account potential spatial autocorrelation directly.

Equations [7] through [9] suggest that dependent variables are affected by the input suppliers’ and output processors’ constant marginal costs M, including labor costs. Therefore, we include the lagged wage in wholesale trade and food manufacturing in equations [14] through [17]. Vector X is a set of factors hypothesized to affect the net returns to developed land, such as median household income, the level of natural amenities, road density, land quality, land area, and the distance to the nearest metropolitan center. Some of these variables may also attract or deter input suppliers and output processors from locating in a county (Goetz 1997).

An important question for this study is if there is a “critical mass” of farmland necessary to sustain the necessary agricultural infrastructure and local agricultural economies. To provide insights into this issue, equations [14] through [17] are specified as a quadratic function of U and D to identify any potential threshold effects and nonlinear relationships in the equations. In addition, a variable is added to capture the interaction between the effects of urban development and population density.

IV. Data and Estimation

Data for this study come from several sources and concern years 1987, 1992, 1997, and 2002. Information on the number of farm input suppliers and farm output processors comes from the U.S. Census Bureau’s County Business Patterns CD-ROM for years 1992, 1997, and 2002. The farm input suppliers variable is the number of establishments engaged in the wholesale distribution of agricultural machinery and equipment (Standard Industrial Classification code [SIC] 5083 or North American Industry Classification System code [NAICS] 421820) plus the number of establishments involved in wholesale distribution of animal feeds, fertilizers, agricultural chemicals, seeds, and other farm supplies (SIC 5191 or NAICS 422910). The farm output processors variable is the number of food manufacturing establishments (SIC 2000 or NAICS 311).⁷

The Census of Agriculture is the source of data on production cost per farmland acre, net farm income per farmland acre, and harvested cropland acres. Net cash farm income per farmland acre is used rather than net cash farm returns because the latter is unavailable for 2002. Net cash farm income is net cash farm returns plus government payments plus other farm income; net cash farm income does not include income from off-farm employment. To account for inflation we deflate farm production cost and net cash farm income using the Consumer Price Index for All Consumers (CPI-U)(1982-1984 = 1).

From the Census of Population and Housing we obtain data on population density by year. We linearly interpolate values for years 1987, 1992, and 1997 using data for years 1980, 1990, and 2000. As mentioned earlier, we measure population density at the neighborhood level, that is, a given county and all counties having centers within 50 miles (as the crow flies) of that county’s center. The distance weights matrix feature in Geoda software (Anselin, Syabri, and Kho 2006) is used to determine each county’s neighbors. County boundary files for this procedure come from the U.S. Census Bureau. The county map comes unprojected and is projected in ArcGIS with the U.S. Contiguous Albers Equal Area projection as recommended by the U.S. Census Bureau. Total acres of developed (built- up) land at the neighborhood level were estimated using data from the National Resources Inventories.

The data on returns to agricultural land come from the Bureau of Economic Analysis. Returns per acre are measured as by Langpap, Hascic, and Wu (2008) using income from crops to represent revenue and production expenditures to measure costs. Some expenditures are allocated to crop production (e.g., purchase of seeds, fertilizer, and lime), but others (e.g., petroleum products, labor, and other expenditures) are not. To allocate these latter expenditures we calculate the share of total revenues that crop revenue represents and we allocate the same share of these expenditures to agricultural production costs. We adjust the resulting measure of returns by the CPI-U. Finally, to obtain returns per acre we use data from the National Resources Inventories to estimate the total acreage allocated to agriculture in each period and divide total adjusted returns in each county by this amount.

Wage variables are constructed from County Business Patterns data and are proxied by dividing a county’s first-quarter payroll by total mid-March employees. The “wholesale wage” variable is for wholesale trade (SIC 5000/5100) as a whole. We do not focus on agricultural wholesale trade, due to the large number of missing values for this sector. The “manufacture wage” variable is for food manufacturing as a whole (SIC 2000). These wage variables are adjusted for inflation using theCPI-U(1982-1984 = 1).

Data on road mileage come from the Bureau of Transportation Statistics. The land area variable comes from the U.S. Census of Population and Housing. The USDA ERS is the source of data for the natural amenity scale, a variable that summarizes natural amenities on the basis of six factors: warm winter (average January temperature), winter sun (average number of sunny days in January), temperate summer (low winter-summer temperature gap), summer humidity (low average July humidity), topographic variation (topography scale), and water area (water area proportion of total county area).

One drawback of the ERS amenity measure is it aggregates across many dimensions such that one is uncertain what dimensions are responsible for a measured effect. Another is it does not capture the richness of local amenities, and recent research in the rural economic development literature suggests that the nature of amenity matters to local economic growth. In the present paper, however, the natural amenity variable is a control variable of only secondary interest, and it more than adequately controls for amenity influences in this circumstance. It is thus unnecessary to include disaggregated amenity variables, particularly since this would potentially introduce collinearity problems.

The compiled GIS database consisting of all variables described above is used for the empirical analysis. Table 1 presents descriptive statistics for these variables.

The estimation of the empirical models exploits the panel nature of the data and uses the random-effects specification. By treating the data as a panel, degrees of freedom are gained and estimation efficiency is thus improved. The basis of our choice to use a random-effects model rather than a fixed-effects model deserves mention.⁸ There are some important advantages of the random-effects approach relative to a fixed-effects model. One, random effects is more efficient, because it uses more information than does fixed effects. Two, unlike the fixed-effects model, time-invariant variables can be included in the random-effects model (Johnson 1995). Alongside its advantages, the random-effects model has an important drawback: it assumes the error term is uncorrelated with all explanatory variables (Hsiao 1986). Random effects are thus an appropriate specification to the extent that we have included the relevant explanatory variables in the models.⁹