The Economic Analysis of Corn Grain Optimization and Price Variation for Cattle on Feed in Texas

Abstract

Cattle placed on feed is a practice to maximize the amount of meat produced before being sent to slaughter, which has become a major agricultural industry. The optimization of input quantities, especially corn, is crucial to maximize production efficiency and ultimately profit. The objective of this research is to determine the optimal corn grain production rate for cattle on feed in Texas and estimated profit maximization under various price ratios for corn grain and live cattle. Utilizing data from various United States Department of Agriculture (USDA) sources, various different input production levels and prices were collected. Statistical Analysis System (SAS), procedures were used to estimate the different production functions. Sensitivity analysis were performed for the optimal production of corn grain rate, and consequent profit under various combinations of corn and live cattle prices for the four different functions. Additionally, a continuous form curve for optimal corn grain production rates under various price ratios was developed. Results indicated that the cubic model was the most accurate based upon the R² value. However, the continuous form model created for the sensitivity analysis concluded that the quadratic was the most accurate model under the different price ratios. The results of the study can be a useful tool for the decision-making process for producers and policymakers.

1. Introduction

1.1. Cattle Industry

Beef production in the U.S. has seen significant changes since the 2000s. The beef industry is known to be one of the most unique and complex compared to other food production industries, due to the variety of segments it involves. It takes roughly 2–3 years to evolve beef from farm to the dinner plate. Historically, cattle were moved based on the location of the available forage. Due to the decrease in available forage, cattle began to be placed on feed at an earlier age which results in higher-quality meat [1].

In 2017, there were 93.7 million head of cattle in the U.S., of which 13.1 million were on feed, an increase from the 11.6 million head on feed in 2002. Texas leads the nation in terms of total cattle numbers, producing around 12.5 million head, while Nebraska follows with 6.8 million head. However, Nebraska leads the nation in the number of cattle on feed with roughly 2.77 million head, while Texas is close behind at 2.65 million head. There were 30,219 feedlots in the U.S. in 2017, while those with a capacity less than 1000 head account for roughly 93% of the total. Texas accounts for six of the top 20 largest feedlots in the U.S., which are all located in the Northern Panhandle [1].

In Texas, the cattle feeding operations are predominately located in the Panhandle-Plains area, where more than 85% of state’s cattle are fed annually. Feedlots with a capacity of 16,000 head or more, generally enjoy a scale advantage over smaller-size feedlot operations [2]. Overall, larger businesses or feedlots benefit from economies of scale in order to be more efficient in the competition, while smaller businesses or feedlots experience higher profit risk as compared to larger feedlots [3].

In February 2002, feedlots with a capacity of 1000 or more head totaled 11.6 million head in the U.S., with Texas totaling 2.88 million head on feed. In December 2017, that number almost stayed the same with 11.5 million head in the U.S., and slightly decreased to 2.64 million in Texas. In July 2017, there were 2.35 million head of cattle and calves on feed in the Northern High Plains, accounting for 88% of the state’s total [4].

1.2. Risk

The cattle feeding operation has historically been a risky venture when returns can swing from profit to loss in a matter of weeks. The dramatic volatility in commodity prices can further increase the cattle feeding operations risk of return to above historic levels. Few standardized measures exist to measure and compare the current risk to historical levels, or to obtain forward-looking risk estimates. Updates and inclusion of additional risk elements could benefit those measures that do exist.

The main risk faced by feedlot managers stem from financial and biological risk. Financial risk refers to price volatilities of output (slaughter cattle) and inputs (feeder cattle, feedstuffs, etc.), while biological risk includes cattle mortality, weight gain variability, and the cost of veterinary supplies [5]. Seasonal effects can complicate the biological risks due to the animals’ performance during different climates. Currently, modeling risk in cattle feeding operations has focused either on financial or production risks and few studies have attempted to incorporate both elements [6,7,8,9].

The three primary sources of price risk that cattle feedlot managers face include variability in fed cattle price, feeder cattle price, and corn price. Fed cattle prices represent the largest (and often sole) impact to the feedlot’s revenue, while the prices of feeder cattle and corn directly impact the operating costs [8]. The “cattle crush” is the relationship among these three commodities, as these prices (fed cattle, feeder cattle, corn) comprise the primary components of the cattle feeding margin. Collins [10] and Tonsor and Schroeder [9] find there are known interactions among the prices of three commodities which justifies examining financial and profit risk under a multivariate framework. The study concludes that hedging only one component of the cattle feeding margin can expose feedlot managers to greater risk than unhedged, which means every risk component in the cattle feeding operation should be considered when analyzing the risk management strategy.

Another significant source of variability in cattle feeders’ profits is production risk, which includes: average daily gain (ADG), mortality, dry matter feed conversion (DMFC), veterinary care cost per head (VCPH), and physiologic differences between steers and heifers [11]. Belasco et al. [5] parameterize these risks and incorporated them into cattle feeding profit functions for Kansas and Nebraska feedlots. Although the production risk varies across feedlots, animals, and feeding locations, these components can be used to proxy cattle performance in a given area and be incorporated into cattle-feeding risk analysis.

Cattle feeding has been known to be a highly risky business and net returns to cattle feeders vary substantially over time. Factors that can significantly contribute to fluctuations in cattle finishing cost of gains and profits include changes in cattle prices, feed prices, and performance. Adduci et al. [12] reports that the practical approach for economic and technical evaluation of the food rationing allowed to identify, to highlight, and to confirm important criticalities caused by incorrect feeding management. Schroeder et al. [13] finds that approximately 93% of the variability in cost of gain over time can be explained by changes in corn prices, feed conversions, and daily gains. The study also explains that 93% to 94% of the variability in steer feeding profit over time can be explained by fed price, feeder steer price, corn price, interest rates, feed conversion, and average daily gain. Another factor that influences feedlot profits is the pen size.

This study will explore the relationships between different feed inputs for cattle feeding operations and examine the optimal corn grain yield for cattle on feed in Texas and the associated profit above feed expenses.

1.3. Objectives

The objective of this research is to economically analyze the impact of profit risk of cattle feeding operations in Texas. More specifically, the study will (1) develop and compare various forms of production function for cattle on feed with various inputs; (2) evaluate risk analysis of optimal corn grain and profit under different corn and live cattle price ratios; and (3) develop a continuous form model to estimate the optimal corn grain production rate under different price ratios of corn grain and cattle on feed.

2. Materials and Methods

2.1. Literature Review

Given the economic importance of the U.S. cattle feeding industry and the fluctuation in cattle feeding returns, there is a need to study the factors that affect the variability of cattle feeding profitability, which would help cattle feeders to determine where they need to focus attention when managing the risks.

Langemeier et al. [6] used a regression analysis to explain variability in profits per head for steers and the difference in the profit variability between steers and heifers. The study also used coefficients of separate determinations to measure the impact of each independent variable on the dependent variable. The following model was adopted to determine net return (or profit) per head:

$$Profit=f\left(IP, PERF, SP\right)$$

(1)

where, IP represents input price, PERF represents performance factors, and SP is the sale price. Input prices were expected to have negative relationships with profit, and sale price was expected to have positive relationships with profit, while the impact of performance on profit varied with the different performance factors. The study found that movement in fed cattle prices explained roughly 50% of the variability over time in cattle feeding profits, while feeder cattle price risk and corn prices accounted for approximately 25% and 22% respectively of the variability in profits. Cattle’s placement weight had a significant impact on the performance and cost factors on profits. The profit variability of cattle placed in the feeding operation at a lighter weight was strongly influenced by the average price of feed, while the profit variability of cattle placed at a heavier weight was influenced by the cost of the feeder cattle and ADG. Results indicated that cattle feeding operations should strongly consider their input price risk.

Kastens and Schroeder [14] set out to determine whether cattle feeder’s placement decisions were more strongly influenced by the expected profit based upon live cattle futures markets, or upon past cattle feeding profit. The study tried to understand why cattle feeders continue to place cattle on feed when the expected feeding profitability was negative. The study modeled the basis for feeder cattle placement to determine if cattle feeders use the expected profit-based futures markets as their expectation of the output price, or if they use a naïve expectation of the most recent profit. The study concluded that cattle feeders do in fact use naïve profit expectations to make placement decisions.

Mark et al. [8] further advanced the above study and performed a study to identify economic risk in cattle feeding operations focusing in Western Kansas. They set out to try and narrow down the effects of certain factors on profit variability in cattle feeding operations. Factors such as sex, placement weight, and placement month were found to have a significant impact on the profitability. The study concluded that feeder cattle prices had a greater impact on profit variability for spring and fall placements. This conclusion is also supported by Lawrence et al. [7], who used several different formulas and methodologies to compare the effects of different variables on the profitability of cattle feeders. Standardized beta coefficients were estimated and examined for differences among the factors that can affect the profit variability. The study concluded that fed cattle and feeder cattle prices are two of the most important factors in deciding the overall profitability of cattle feeding operations.

Current studies in the academic literature address profit risk and the factors influencing profit for cattle feeding operations. However, there is a paucity of studies on optimal production rates related to the different inputs, which is important in order for the industry to understand the impact of feed inputs on the number of head placed on feed. Most research on the cattle feeding industry is focused on Kansas and Nebraska operations, with relatively limited attention to Texas feedlots. Cattle feeding operations are predominately located in the Texas Panhandle, unlike other states, where the operations are distributed over the entire state. This creates a unique market that deserves additional research to help producers understand their profit risk when placing cattle on feed and the optimal levels needed to maintain economic viability.

2.2. Data

Production data for this study was collected from USDA National Agriculture Statistics Service (NASS) from 1985 to 2017. The production variables include cattle, corn grain, soybean, alfalfa, and coastal. Additionally, precipitation and temperature data were collected for the study’s location using PRISM [15]. Daily minimum and maximum temperatures were averaged for the daily average and condensed into monthly average temperatures. Likewise, daily precipitation totals were collected and summed together to get monthly totals for precipitation. A summary of all the production data can be found in

Table 1. Descriptive statistics for cattle Texas production data from 1985 to 2017.

Variable	Unit	Mean	Maximum	Minimum	Standard Deviation
Cattle on Feed	Millions	2.59	2.98	2.05	0.29
Corn Production	Millions	212.9	323.9	129.6	55.28
Soybean Production	Millions	177.8	292.5	112.1	42.39
Alfalfa Production	Thousands	5.94	8.93	3.66	1.34
Coastal Production	Millions	8.93	14.04	3.96	2.19
Rainfall	Millimeters	532.1	930.0	176.8	153.7
Winter Temperature	Celsius	3.64	5.25	1.67	0.99
Fall Temperature	Celsius	14.63	16.49	12.82	0.88
Spring Temperature	Celsius	13.84	16.73	12.54	1.06
Summer Temperature	Celsius	24.68	28.25	22.97	1.00

.

Cash price reports were collected weekly for the specified commodities obtained from various USDA resources for the time period January 1, 2002 to December 31, 2017. Feeder cattle, live cattle, and corn grain cash prices were gathered from select weekly USDA reports of Texas prices. The cash feeder cattle price was taken from USDA AMS Data Source for Tulia, TX (Weighted Average Prices) for Medium to Large Frame #1 steers weighting 750 to 800 pounds. In addition, live cattle cash prices were Weekly Weighted Average-Texas prices [16].

Soybean meal quotes were collected for Kansas City, Missouri for high protein soybean meal from USDA AMS on a weekly basis. This study is intended to reflect the market risk for placing cattle in Texas on feed. However, Texas cash prices for soybean meal were not available. Therefore, the soybean meal prices at Kansas City, Missouri were used as a substitute for Texas prices. A summary of all price data can be found in

Table 2. Summary of statistics for Texas cash prices from 2002 to 2017.

Variable	Unit	Mean	Maximum	Minimum	Standard Deviation
Live Cattle	$/cwt	105.1	172	61.96	24.92
Feeder Cattle	$/cwt	124.8	240.3	72	36.16
Corn	$/bu	4.11	8.52	2.03	1.63
Soybean	$/ton	305.5	594.5	150.3	104.9
Alfalfa	$/ton	185.1	277	120	39.78
Coastal	$/ton	111.4	190	63	27.08

.

2.3. Conceptual Framework

To help understand the physical relationships between inputs and outputs in agriculture, production functions are used to represent what output can be attained from various input levels, which can be expressed mathematically by the following function:

$$\mathrm{Y}=f\left(X_1, X_2\dots X_K|X_L\dots X_N\right)$$

(2)

where, Y represents total production of cattle on feed; X₁, X₂,… X_N are the inputs used in production, where X₁, X₂,… X_K represent variable inputs and X_L…X_N represents fixed inputs of production. An illustration of a production function is depicted in Figure 1a, which represents a single output (Y) is produced with a single input (X). Noting that in reality, there is usually more than one input used in the production of an output. For simplification, it will be assumed that only one input (X) is variable and other inputs are fixed.

When placing cattle on feed, the goal for managers is to increase production to maximize profit. The combination of inputs required to maximize the profit function will be calculated in this study, along with the optimal level of feed inputs to maximize profit will be estimated. Assuming a perfectly competitive market, the equation for profit-maximization is shown:

$$\pi =\left(P_{CAT}\ast Y\right)-\left(P_c\ast X\right)$$

(3)

where, π symbolizes profit; P_CAT represents output prices; Y represents the output; P_C denotes the input price; and X signifies the quantity of input used in production.

To calculate maximum level of profit, the first derivative of Equation (3) is taken with respect to C, or corn and set equal to zero:

$$\frac{\Delta \pi }{\Delta C}{=P}_{CAT}\ast \frac{\Delta Y}{\Delta C}-P_C=0$$

(4)

$$\frac{\Delta \pi }{\Delta C}{=P}_{CAT}\ast MPP-P_C=0$$

(5)

or

$$VMP=MIC$$

(6)

where, MPP = ∆Y/∆C is the marginal physical product of corn; VMP is the value of the marginal product at any given price so that VMP = P_CAT * MPP. MIC, or marginal input cost, is the additional or incremental cost of using one more unit of input, which is corn in this study. In a perfectly competitive market, MIC is equal to the input cost P_CAT. Solving Equation (6) with respect to corn, C, provides the profit-maximizing rate of input usage. The relationship given in Equation (5) is depicted in Figure 1b. Given the price of input (X) (P_C), the optimal level of input, X*, can be identified. By combining panels (a) and (b) together, the optimal level of output, Y*, can also be found. Therefore, point O on the production function represents the optimal point at which producers make their production decisions.

2.4. Production Function

Multiple regression models are used to show the relationship between dependent variables as the function of two or more variables. The mathematical format of a multiple regression model can be expressed as follows:

y = β₀ + β₁X₁ + β₂X₂ + ··· β_kX_k + ε

(7)

The coefficient of determination, R², is the explained sum of squares by the regression model to the total sum of squares. The coefficient of determination will be between 0 and 1 and denotes the strength of the linear association between x and y. The closer the coefficient of determination is to 0, the less the equation fits. Inversely, the closer R² is to 1, the better the equation fits [17].

Models are tested for accuracy by evaluating the coefficient of multiple determinations (R²) and the significance of t statistics for regression coefficient of each variable. The t-test indicates the degree of significance of a variable, which depends on how important the variable is to the estimated equation [17].

SAS 9.4 was used to run GLM and REG procedures to analyze various functional forms and determine the best forms that fit the data. The model initially included the following variables: cattle, corn, alfalfa, coastal, and soybean production. Additional variables to include average seasonal temperature and rainfall were added to the model. A time factor was also used to account for improvement in technology and other uncontrollable factors that change the yield over time. Different function forms were used to identify the most appropriate models. The following five function forms (Equations (8)–(13)) were used in this study.

The quadratic form can be represented as follows:

Y = a + bX + cX²

(8)

where x is the variable resource measured, y is the output, and c is a negative number which denotes diminishing marginal returns. This form allows for both declining and negative marginal productivity, but not both increasing and decreasing marginal products. A maximum total product is defined where the input magnitude or X is equal to 0.5bc⁻¹. The constant term a in Equation (8) represents the product forthcoming from the mix of fixed resources [18].

The cubic form is developed in the following form:

Y = a + bX + cX² + dX³

(9)

This production function embraces both increasing and decreasing marginal productivity and is seldom needed for a single variable resource. It has increasing marginal products until X is equal to 0.3333cd⁻¹, then, diminishing but positive marginal products until [18]:

X = 0.3333d⁻¹ [c ± (3db + c²)^.5]

(10)

A Cobb–Douglas, power function form can be expressed as follows:

Y = a ∗ X^b

(11)

where a is constant, and b represents the transformation ratio. In order to be used in the SAS, b is the elasticity of production, estimated in logarithmic form, and the marginal product can be expressed as follows:

$$\frac{dy}{dx}{=bax}^{b-1}=\frac{{bax}^b}{x}$$

(12)

Indicating that if b = 1, the marginal product is constant at the level a. When b > 1, the magnitude of marginal product increases as x increases, with respect to the magnitude of b. Then, if b < 1, the magnitude of the marginal product will decease as X increases [18].

The square root equation is developed in the equation as follows:

Y = a − b₁X₁ − b₂X₂ + b₃X₁⁵ + b₄X₂⁵ + b₅X₁⁵X₂⁵

(13)

where a simple compromise between the Cobb–Douglas and quadratic forms are provided. This equation allows a diminishing total product but also has marginal products which decline at a diminishing rate [18].

2.5. Sensitivity Analysis

Sensitivity analysis was utilized by conducting different price ratios between corn grain and live cattle prices. The corn price was initially set at $3.00/bushel (bu) and increased in increments of $1.00 until the price reached $7.00/bu. Live cattle prices began at $80/cwt (assuming an average selling weight of 1000 pounds (CWT)), which resulting in a starting price of $800 per 1000 pounds and increased by the increments of $5/cwt until reaching $120/cwt or $1200 per 1000 pounds. These increment values were chosen to capture the probable range of fluctuation for each commodity price. While assuming perfect competition, meaning MIC = P_C, Equation (5) was applied at each price ratio to determine the optimal corn production.

Equation (6) was applied to each price ratio to obtain the optimal corn quantity for that specific price ratio. Once this optimal quantity was determined, it was included in the estimated function to approximate the yield for cattle on feed for that specific price ratio. Calculated corn and cattle on feed yield values, along with respective prices, were then inserted into Equation (3) to estimate the profit for each price ratio. Due to the constant fluctuation in prices, this type of sensitivity analysis is crucial in making economic evaluations of the impact of corn prices when placing cattle on feed. According to Yu et al. [19], sensitivity analysis was based on theoretical scenarios and may not always properly represent actual situations. Estimating the optimal corn grain yield using continuous prices of corn grain and cattle, as defined in Equation (14), would be more beneficial:

C = b + mP_r

(14)

where C is the optimal corn grain yield; P_r represents the corn grain-cattle price ratio; b is the intercept; and m represents the slope. The construction of this graph allows estimation of the optimal corn grain quantity at all price ratios rather than the fixed ratios.

3. Results

3.1. Production Function and Profit Estimation

Utilizing PROC REG procedures by SAS, numerous functional forms were tested, including simple linear, multi-linear, and quadratic. Prior to model estimations, tests were performed to determine whether the error terms were autocorrelated. Tests using the PROG AUTOREG procedure in SAS indicated that the error terms were autocorrelated, but this was easily resolved by including the one-year lag of the dependent variable in each model. However, the results of the estimations were essentially preserved. Signs, magnitudes, and significance of all coefficients were largely preserved when the lagged dependent variable was introduced into each model. Consequently, the results presented here omit the lagged dependent variable for the sake of simplicity.

After analysis of all the attempted functions, several forms were particularly accurate in describing the observed data.

Table 3. Parameter estimates for all functional forms.

	Quadratic	Cubic	Cobb-Douglas	Square Root
Intercept	−3.8846 (−4.22) ***	−3.61123 (−5.15)***	−6.32575 (−5.48) ***	−3.83187 (−5.75) ***
Corn	0.02658 (6.68) ***	0.01393 (7.61) ***	0.24238 (3.60) ***	−0.01073 (−4.57) ***
Corn2	−5.908 × 10−5 (−6.37) ***	-	-	-
Corn3	-	−1.707 × 10−8 (−7.02) ***	-	-
Corn0.5	-	-	-	0.33018 (4.81) ***
Soybean	0.00188 (1.94) *	0.00212 (1.27) **	−0.13999 (−1.65) *	-
Coastal	−0.15625 (−1.82)*	-	-	-
Coastal2	0.00985 (2.21) **	−0.00832 (1.57)	-	-
Coastal3	-	6.7334 × 10−4 (13.61) *	-	-
Summer Temp	0.15798 (5.24) ***	0.16317 (6.19) ***	1.99115 (5.58) ***	-
Summer Temp0.5	-	-	-	0.57849 (6.70) ***
Alfalfa	-	-	0.18195 (2.87) ***	0.01205 (1.78) *
R2	0.8084	0.8203	0.6388	0.7557

T values are in parenthesis, ***, **, and * indicate at 1%, 5%, 10% significance level, respectively.

represents the estimated parameters for each functional form.

Table 4. Yield and profit-maximizing corn grain production rates, all models.

	Yield Maximization		Profit Maximization
Functional Form	Y (mil Head)	C (mil bu)	Profit (mil $)	Y (mil Head)	C (mil bu)
Quadratic	2.73	225	2517.23	2.69	200
Cubic	2.73	225	2504.94	2.68	200
Cobb-Douglas	2.94	350	2363.06	2.56	200
Square Root	2.10	225	1861.05	2.00	150

compares the yield maximization and profit maximization for each model. The quadratic, cubic, and square models reached yield maximization of around 225 million bushels of corn grain produced. The quadratic, cubic, and Cobb–Douglas models reached profit maximization of around 200 million bushels of corn grain produced, while the square root model was around 150 million bushels produced. Figure 2 and Figure 3 illustrate the estimated production functions and the profit maximization for all models, respectively.

Based on the above analysis, it can be found that first, all models agreed that corn, soybean, and summer temperature were factors in all models and coastal was included in the quadratic and cubic, while alfalfa was included in the Cobb–Douglas and square root models. Both the quadratic and cubic models resulted in a maximum yield of around 2.73 million head with 225 million bushels of corn grain produced. The models also concluded that the maximum profit was at 200 million bushels of corn grain produced with a yield of around 2.7 million head on feed, resulting in a profit of $2.51 billion. Corn grain and summer temperature had the highest t values within all models, proving the importance of these variables to the production function. Although, the quadratic, cubic, and Cobb–Douglas models disagreed on the yield of corn grain production needed to maximize the number of head on feed, all three models agreed that the profit above corn expense was maximum at about 200 million bushels of corn grain produced.

3.2. Sensitivity Analysis

Sensitivity analysis was performed for the optimal production of corn grain rate, and consequent profit under various combinations of corn and live cattle prices for the quadratic, cubic, Cobb–Douglas, and square root functional models. Analysis for the quadratic model is shown in

Table 5. Optimal corn grain production rate, yield and profit for various price ratios, quadratic model.

			Price of Cattle ($ per Head)
			800	850	900	950	1000	1050	1100	1150	1200
Optimal Corn Grain Production Rate	Price of Corn Grain ($/bu)	3.00	193.21	195.08	196.74	198.22	199.56	200.77	201.87	202.87	203.79
		4.00	182.63	185.12	187.34	189.32	191.10	192.71	194.17	195.51	196.74
		5.00	172.05	175.17	177.93	180.41	182.63	184.65	186.48	188.15	189.69
		6.00	161.48	165.21	168.51	171.50	174.17	176.59	178.79	180.79	182.63
		7.00	150.90	155.25	159.90	162.59	165.71	168.53	171.09	173.43	175.58
Yield at for Various Price Ratios	Price of Corn Grain ($/bu)	3.00	2.67	2.68	2.68	2.69	2.69	2.69	2.70	2.70	2.70
		4.00	2.62	2.63	2.65	2.65	2.66	2.67	2.67	2.68	2.68
		5.00	2.56	2.58	2.60	2.61	2.62	2.63	2.64	2.65	2.66
		6.00	2.49	2.52	2.54	2.56	2.58	2.59	2.60	2.61	2.62
		7.00	2.40	2.44	2.47	2.50	2.52	2.54	2.56	2.57	2.58
Profit for Various Price Ratios	Price of Corn Grain ($/bu)	3.00	1555.72	1689.35	1823.30	1957.50	2091.93	2226.56	2361.34	2496.27	2631.33
		4.00	1367.79	1499.25	1631.26	1763.73	1896.61	2029.82	2163.32	2297.08	2431.06
		5.00	1190.45	1319.10	1448.63	1578.87	1709.74	1841.14	1972.99	2105.25	2237.85
		6.00	1023.68	1148.92	1275.40	1402.92	1531.34	1660.52	1790.36	1920.77	2051.69
		7.00	867.50	988.69	1111.57	1235.88	1361.40	1487.40	1615.42	1743.66	1872.58

; Panel A presents the optimal corn grain production for different price ratios. Estimations conclude that optimal corn grain production is consistent at given price ratios. Such as, when the price ratio is 0.003 in the quadratic model, the optimal production level is 196.74 million bushels, whether the prices are $3.00 to $900 or $4 to $1200. As price ratios deceased, optimal corn grain production increased.

4. Conclusions and Discussion

Several functional forms were developed to represent the output–input relationship between cattle on feed and feed variables in cattle feeding operations. The four most accurate models, which are quadratic, cubic, Cobb–Douglas, and square root, were identified. The variables used to create the function forms included cattle on feed, corn grain production, soybean production, alfalfa production, coastal production, winter temperature, fall temperature, spring temperature, summer temperature, and rainfall. Of all the variables, each model agreed that corn production and summer temperature were the most influential variables included. The accuracy of these models was determined by the R² and t values for each functional form. Based strictly upon the R² value, the cubic model was the most accurate model derived, with an R² value of 0.8203 and high t values.

The results noted the three models (quadratic, cubic, and Cobb–Douglas) had yield maximization at 225 million bushels of corn grain produced but did not have the same yield of cattle on feed. Both quadratic and cubic reached a yield of 2.73 million head on feed, while the square root model was 2.10 million head. Furthermore, the quadratic and cubic models reached profit maximization at the same point. The similarities in results help to understand that the ideal profit maximization point is when 200 million bushels of corn grain were produced with a yield of 2.68 and 2.69 million head on feed. It is also worth noting that the quadratic model resulted in the largest profit, which was also proved by the sensitivity analysis.

The sensitivity models were created to help feedlot managers understand the optimal production of corn grain needed under price ratios to determine the optimal number of head that can be placed on feed. The models also symbolize estimated profit levels under those optimal production and yield values. This will be a useful tool when managers are purchasing feeder cattle to be placed on feed and to understand the profit above corn grain expenses.

The continuous form regression equations suggest that corn-grain–cattle price ratios impact the optimal corn grain production rate. Using the quadratic model, as price ratios ranged from $0.0025 to $0.00875, the profit-maximizing corn grain production rate varied from 150.9 to 203.79 million bushels. This created a profit above corn grain expenses ranging from $867.5 million to $2,631.33 million. While under the Cobb–Douglas model, price ratios again varied from $0.0025 to $0.00875 but profit above corn grain expenses ranged from $1116.17 million and $2499.66.

The impact of corn grain production on the livestock industry is significant due to its value as a feed additive. Due to the fact that 96% of Texas corn is being used for livestock feed, this signifies the importance of understanding the optimal levels of corn grain production. This can help not only farmers when trying to decide how much corn needs to be planted but also ranchers or feedlot managers. Feedlot managers need to understand the availability of corn grain and the price risk of using corn grain in their feed rations.

Although the model is the most accurate model within the statistical measures, it may not be the most accurate model in reality. The model suggests that once the optimal corn grain production rate is reached, the number of head on feed would decrease as corn grain production continued to increase, which is not realistic. The Cobb–Douglas model represents a more realistic model, showing that once you hit a certain rate of corn grain production, the number of head on feed will slowly continue to increase but not at the same rate as before it reached a certain production level.

Future research could set out to determine the impacts of other feed inputs and the relative impacts of those inputs on profitability when placing cattle on feed. Corn grain is only one of the many feed inputs feedlot managers use when placing cattle on feed and the importance of understanding the profit risk of using these inputs is vital to the industry. Other possible research can include the study of the changes in feed input efficiency, and the structure or scale economies of the industry, and the resulting implications for profitability of cattle feedlots.