2.1. Conceptual model

This analysis aims to show how input prices for feed have a conceptual link to cattle performance measures. We show the conceptual link under both profit maximization and cost minimization with implications for the empirical model for the feed conversion ratio. This allows us to estimate total effects, substitution effects, and feed substitution effects. We focus on the impact of feed prices on FCR more formally and only infer the expected effects on ADG and DOF. Our model is not a dynamic model and thus does not directly include time as a dimension. Therefore, formally analyzing the impact of feed prices on average daily gain and days on feed is not possible in this setup. We focus on how efficiency gains might extend to these other measures. Specifically, if FCR increases, cattle become less efficient, and ADG decreases. Further, if they gain less each day, then it takes longer to reach slaughter weight, and as a result DOF may increase – dependent on any changes in total weight gain.

2.1.1. Total effect: profit maximization approach

When feed prices change, the total effect of the price change on input levels consists of an expansion/contraction effect and a substitution effect. In turn, the adjusted input mix and the new output level impact technical efficiency. To examine the total effects of input price changes, we begin with a standard profit maximization framework with one output, two feedstuffs, and the placement weight of cattle as inputs.Footnote ¹ The production of fat cattle (measured by slaughter weight q) can be described by q = f(x ₁, x ₂, x _pw ), which is a twice continuously differentiable function of substitute feedstuffs, x ₁ and x ₂, placement weight, x _pw . Moreover, we assume diminishing marginal product, implying ${\partial f \over \partial x_{i}}\lt 0$ , where i ∈ {1, 2, pw}. Producers face fat cattle price p, feeder cattle price w _fc , feed prices w ₁ and w ₂, and fixed costs FC.

The producer’s problem can be defined as:

(1) $$\eqalign{ & \pi = {\max _{q,{x_1}, \ldots, {x_n}}}pq - {w_1}{x_1} - {w_2}{x_2} - {w_{fc}}{x_{pw}} - FC \cr & \ \ \ \ \ \ \ \ \ \ \ \ \ \ subject\ to{:}\ q = f\left( {{x_1},{x_2},{x_{pw}}} \right). \cr} $$

The first order conditions are as follows:

(2) $$pf_{{x_{1}}}-w_{1}=0,$$

(3) $$pf_{{x_{2}}}-w_{2}=0,\quad {\rm and}$$

(4) $$pf_{{x_{pw}}}-w_{fc}=0.$$

So long as the Hessian matrix is negative definite, then we can solve the first order conditions to obtain the following input demand functions:

(5) $$x_{1}=x_{1}^{*}\left(w_{1},w_{2},w_{fc},p\right),$$

(6) $$x_{2}=x_{2}^{*}\left(w_{1},w_{2},w_{fc},p\right), \ \ \ \ {\rm and}$$

(7) $$x_{pw}=x_{pw}^{*}\left(w_{1},w_{2},w_{fc},p\right).$$

The first derivatives with respect to input prices are:

(8) $${\partial x_{1}^{*} \over \partial w_{1}}\lt 0 {\partial x_{2}^{*} \over \partial w_{1}}\gt 0 {\partial x_{pw}^{*} \over \partial w_{1}}\gt 0 {\partial x_{1}^{*} \over \partial w_{2}}\gt 0 {\partial x_{2}^{*} \over \partial w_{2}}\lt 0 {\partial x_{pw}^{*} \over \partial w_{2}}\gt 0 {\partial x_{1}^{*} \over \partial w_{fc}}\gt 0 {\partial x_{2}^{*} \over \partial w_{fc}}\gt 0 {\partial x_{pw}^{*} \over \partial w_{fc}}\lt 0.$$

The diagonal first derivatives are all negative as we assume decreasing marginal product in each input. The signs for the off diagonals are positive because these are all substitute inputs.

The output supply is defined as:

(9) $$q^{*}=f\left(x_{1}^{*}\left(x_{1},x_{2},x_{pw},p\right), x_{2}^{*}\left(x_{1},x_{2},x_{pw},p\right),x_{pw}^{*}\left(x_{1},x_{2},x_{pw},p\right)\right).$$

From the duality between cost and production functions, it can be shown that ${\partial q^{*} \over \partial w_{i}}={-\partial x_{i}^{*} \over \partial p}$ . For normal inputs, ${\partial x_{i}^{*} \over \partial p}$ is positive. However, for inferior inputs, ${\partial x_{i}^{*} \over \partial p}$ is negative.Footnote ² Therefore, ${\partial q^{*} \over \partial w_{i}}$ is negative for normal inputs but positive for inferior inputs.

How does the FCR respond to a change in feed prices? First, FCR is defined as the sum of feed inputs over the weight gained in the feeding period as follows:

(10) $${\rm FCR}= {x_{1}^{*}+x_{2}^{*} \over q^{*}-x_{pw}^{*}}.$$

Then, the derivative of the feed conversion ratio with respect to feed prices is defined as:

(11) $${\partial {\rm FCR} \over \partial w_{i}}={\left(q^{*}-x_{pw}^{*}\right)\left({{\partial x_{1}^{*}} \over{\partial w_{i}}}+{{\partial x_{2}^{*}} \over{\partial w_{i}}}\right)-\left(x_{1}^{*}+x_{2}^{*}\right)\left({{\partial q^{*}}\over{\partial w_{i}}}-{{\partial x_{pw}^{*}} \over{\partial w_{i}}}\right) \over \left(q^{*}-x_{pw}^{*}\right)^{2}},$$

where i ∈ {1, 2}. Here, ${\partial x_{1}^{*} \over \partial w_{i}}$ represents the feed substitution effect of feedstuff x ₁, ${\partial x_{2}^{*} \over \partial w_{i}}$ represents the feed substitution effect of feedstuff x ₂, ${\partial q^{*} \over \partial w_{i}}$ represents the output effect, and ${\partial x_{pw}^{*} \over \partial w_{i}}$ represents the placement weight substitution effect. Based on previous definitions, we determine the sign for each term as follows:

(12) $${\partial {\rm FCR} \over \partial w_{i}}={\left(+\right)\left(?\right)-\left(+\right)\left(?\right) \over \left(+\right)}$$

The terms in question ultimately determine the sign of the response. First, $\left({\partial x_{1}^{*} \over \partial w_{i}}+{\partial x_{2}^{*} \over \partial w_{i}}\right)\lessgtr 0$ , depending on the relative magnitude of the derivatives, which is determined by the strength of substitutability and the rate of technical substitution between x ₁ and x ₂. Second, the sign of the term $\left({\partial q^{*} \over \partial w_{i}}-{\partial x_{pw}^{*} \over \partial w_{i}}\right)$ depends on the sign of ${\partial q^{*} \over \partial w_{i}}$ . If x _i is a normal input, then ${\partial q^{*} \over \partial w_{i}}\lt 0$ and $\left({\partial q^{*} \over \partial w_{i}}-{\partial x_{pw}^{*} \over \partial w_{i}}\right)\lt 0$ . However, if x _i is an inferior input, then ${\partial q^{*} \over \partial w_{i}}\gt 0$ and the sign of $\left({\partial q^{*} \over \partial w_{i}}-{\partial x_{pw}^{*} \over \partial w_{i}}\right)$ depends on the relative magnitude of the slaughter weight and placement weight responses, being positive if the change in slaughter weight is larger than the change in placement weight, and vice versa. The various scenario combinations are summarized in Table 1.

Table 1. Direction of the impact of feed price changes on feed conversion ratio

The results of the profit maximization analysis shown in Table 1 demonstrate that little can be said about the mechanisms of the effect from the sign of the effect. However, under the assumption that the input is normal, the sign comes down to the relationship between the own- and cross-price effects. Yet, we still cannot determine the relative own- and cross-price effects given the sign of the total effect of a price change on FCR. Moreover, assuming inputs are normal is potentially restrictive. In summary, the sign and magnitude of the total effect is dependent on the following three sub-effects: (1) the output effect ${\partial q^{*} \over \partial w_{i}}$ , (2) the placement weight substitution effect ${\partial x_{pw}^{*} \over \partial w_{i}}$ , and (3) the feed substitution effect ${\partial x_{j}^{*} \over \partial w_{i}}$ . However, it is difficult to disentangle these effects under assumptions of profit maximization.

2.2. Input substitution effects: cost minimization approach

To isolate the input substitution effects we simply need to hold output constant at q = q ⁰, so FCR is redefined as follows:

(13) $${\rm FCR}= {x_{1}^{*}+x_{2}^{*} \over q^{{\rm o}}-x_{pw}^{*}}.$$

Note that this is equivalent to cost minimization. However, rather than show the entire optimization problem a second time, we can simply hold q constant in the derivative of FCR with respect to feed input prices. Under these updated assumptions, the derivative of the feed conversion ratio with respect to feed prices is calculated as:

(14) $${\partial {\rm FCR} \over \partial w_{i}}={\left(q^{{\rm o}}-x_{pw}^{*}\right)\left({{\partial x_{1}^{*}} \over {\partial w_{i}}}+{{\partial x_{2}^{*}} \over {\partial w_{i}}}\right)-\left(x_{1}^{*}+x_{2}^{*}\right)\left({{\partial x_{pw}^{*}} \over {\partial w_{i}}}\right) \over \left(q^{{\rm o}}-x_{pw}^{*}\right)^{2}},$$

where i ∈ {1, 2}. Based on previous definitions, we determine the sign for each term as follows:

(15) $${\partial {\rm FCR} \over \partial w_{i}}={\left(+\right)\left(?\right)-\left(+\right)\left(+\right) \over \left(+\right)}.$$

This case is much simpler than the profit maximization case, with only two potential scenarios. First, if ${\partial x_{i}^{*} \over \partial w_{i}}\gt {\partial x_{j}^{*} \over \partial w_{i}}$ , then ${\partial {\rm FCR} \over \partial w_{i}}\lt 0$ . Second, if ${\partial x_{i}^{*} \over \partial w_{i}}\lt {\partial x_{j}^{*} \over \partial w_{i}}$ , then the sign of ${\partial {\rm FCR} \over \partial w_{i}}$ is unknown. As a result, there is little that can be inferred about the relative own- and cross-price effects given the sign of the effect of input substitution on FCR. The sign and magnitude of the substitution effect are dependent on the placement weight substitution effect ${\partial x_{pw}^{*} \over \partial w_{i}}$ , and the feed substitution effect ${\partial x_{j}^{*} \over \partial w_{i}}$ .

2.3. Feed substitution effects: cost minimization approach

To isolate the feed substitution effects, we need to hold placement weight constant at x _pw = x _pw ⁰, in addition to holding constant output at q = q ⁰. FCR is redefined as follows:

(16) $${\rm FCR}= {x_{1}^{*}+x_{2}^{*} \over q^{{\rm o}}-x_{pw}^{{\rm o}}}.$$

Then, the derivative of the feed conversion ratio with respect to feed prices is:

(17) $${\partial {\rm FCR} \over \partial w_{i}}={\left({{\partial x_{1}^{*}} \over {\partial w_{i}}}+{{\partial x_{2}^{*}} \over {\partial w_{i}}}\right) \over q^{0}-x_{pw}^{0}}$$

Signing this derivative is straightforward, as the denominator is strictly positive. Therefore, if ${\partial x_{i}^{*} \over \partial w_{i}}\gt {\partial x_{j}^{*} \over \partial w_{i}}$ , then ${\partial {\rm FCR} \over \partial w_{i}}\lt 0$ . On the other hand, if ${\partial x_{i}^{*} \over \partial w_{i}}\lt {\partial x_{j}^{*} \over \partial w_{i}}$ , then ${\partial {\rm FCR} \over \partial w_{i}}\gt 0.$ In this case we can make inference about the relative own- and cross-price effects given the sign of the effect of input substitution on FCR. The sign is determined by the relationship between the own- and cross-price effects, and by extension the relative marginal products.

(18) $${\partial {\rm FCR} \over \partial w_{i}}\gt 0 \Longleftrightarrow MP_{x_{i}^{*}}\gt MP_{x_{j}^{*}}\quad {\rm and}$$

(19) $${\partial {\rm FCR} \over \partial w_{i}}\lt 0 \Longleftrightarrow MP_{x_{i}^{*}}\lt MP_{x_{j}^{*}}.$$

In other words, if the substitute feed is less efficient, then cattle become less technically efficient, and vice versa. The conclusion is that substituting to a lower marginal product feed may indeed be the optimal decision in terms of cost minimization.

2.4. Summary and empirical implications

In summary, this framework links feed price variation to technical efficiency outcomes in feedlot cattle production by explicitly modeling producer decision making under profit maximization and cost minimization assumptions. We highlight the complex nature of this relationship and provide as much economic intuition as possible to interpret the results. However, for the total effect the series of various economic conditions that determine the sign of the effect are impossible to determine only from the sign of the effect. Even so, our analysis does describe several scenarios to consider depending on the direction of output and substitution effects. With regards to the input substitution effect, we note that holding output constant, the producer will optimize based on the trade-offs between different feeds and placement weight. Yet, the changing placement weights make it impossible to definitively infer the economic conditions, unless the effect on FCR is positive, indicating the substitute feedstuffs have a lower marginal product. The feed substitution effect is the most straightforward to interpret: if the marginal effect of a feed price increase on FCR is negative (improved efficiency) the substitute feedstuffs have a higher marginal product, and vice versa.

The intuition under cost minimization is that when a feed price increases, producers reduce the quantity of that feed and substitute at least one other feed in its place to reach the output goal. The new feeds that are substituted will have a different marginal product, which determines the effect of the substitution on technical efficiency. The intuition of this framework must remain attached to the key assumption that producers are making optimal decisions. Substitution toward a lower marginal product feedstuff may seem counterintuitive at first, but this type of behavior illustrates the difference between optimizing profit (or cost) versus maximizing output. Furthermore, a feedstuff with a lower marginal product may not necessarily be a feed of lower feed value, per se. The marginal product of each specific feed is dependent on the production function for cattle growth, which is determined by biological and environmental factors. Moreover, assuming diminishing marginal product, commodities which are fed at high rates may have a relatively low marginal product, despite having a high feed value. Therefore, for feeds that commonly make up significant portions of feedlot diets, a price increase may have the surprising effect of improving efficiency. As a result, hypothesizing a priori the direction of the effect of feedstuff prices on efficiency is difficult.

The implications of this conceptual analysis for empirical estimation are straightforward. First, feed prices have a clear theoretical connection to technical efficiency. Accordingly, regressions of efficiency on feed prices are justified and the coefficients contain information about producer decision making. Second, the effect may differ based on whether placement weight and slaughter weight are constant or not. In a regression analysis, holding a something “constant” is accomplished by adding it as a covariate. Therefore, we can estimate the three effects discussed above by the inclusion or omission of variables for placement weight and slaughter weight in our regression models.

2.5. Empirical model

The conceptual model in the previous section has implications for our empirical approach. The first specification aims to estimate the total effect of prices on efficiency, including via slaughter weight changes, placement weight substitution for feed, and substitution among feedstuffs. This specification is based on equation (10), where all the variables are functions of input prices and the output price. This specification is written:

(20) $$y_{it}=\alpha +{\boldsymbol{\beta }}{\bf \Delta}\ {\boldsymbol ln } \left({\boldsymbol w}_{\boldsymbol{it}}\right)+\gamma \Delta \mathit{\ln } \left(p_{it}\right)+{\boldsymbol{\rho }\boldsymbol{\kappa }_{it}}+\theta y_{it-1}+\varepsilon _{it},$$

where y _it is a measure of technical efficiency at time t. Here, y ∈ {FCR, ADG, DOF}. There are two observations per period, one each for heifers and steers, represented by the subscript i. On the right-hand side of equation (20), α is an intercept term, β is a vector of coefficients to be estimated, $\bf \Delta\ \boldsymbol{\ln } (\boldsymbol{w}_{\boldsymbol{it}})$ is a vector of log-differenced input prices, γ is a coefficient to be estimated, Δln (p _it ) is the log-differenced price of fat cattle, ρ is a vector of coefficients to be estimated, and κ _it is a vector of controls. Most of the feed prices and control variables do not vary across i, although some do. The control vector includes the following variables: interest rate (cost of working capital), a linear trend, and the following binary variables: heifer (=1 if heifer data, 0 otherwise), the renewable fuels standard (=1 if placement month is January 2008 or after, 0 otherwise), COVID years (=1 if closeout month is March 2020 or after and placement month is April 2022 or before, 0 otherwise), and placement monthly seasonality variables. To address serial correlation of the error, y _{it − 1} is also added to the right-hand side with θ representing a coefficient to be estimated. Lastly, ϵ _it is a normally distributed error term. The input prices used are described in the data section of this paper. All feed prices were deflated by a feed cost index (further discussed in the next section) so that the interpretation is changes in relative prices as opposed to absolute changes.

The second specification aims to estimate the substitution effect of prices on efficiency, via placement weight substitution for feed and substitution among feedstuffs. This specification aims to hold slaughter weight constant as in equation (13). Therefore, we include slaughter weight as a control so that the effect of feed price changes is independent of changes in slaughter weight. The equation is defined as follows:

(21) $$y_{it}=\alpha +{\boldsymbol{\beta }} {\bf \Delta} {\boldsymbol ln } \left({\boldsymbol w}_{\boldsymbol it}\right)+\gamma \Delta \mathit{\ln } \left(p_{it}\right)+\delta q_{it}+{\boldsymbol{\rho \kappa }}_{it}+\theta y_{it-1}+\varepsilon _{it},$$

where all the elements are explained previously except we now include δ, a coefficient to be estimated, and cattle slaughter weight, q _it .

The third specification aims to estimate the feed substitution effect of prices on efficiency, via substitution among feedstuffs in response to a price change. This specification aims to hold both placement weight and slaughter weight constant as in equation (16). Therefore, we include both placement weight and slaughter weight as controls so that the effect of feed price changes is independent of both. The equation is defined as follows:

(22) $$y_{it}=\alpha +{\boldsymbol{\beta }} {\bf \Delta} {\boldsymbol ln } \left({\boldsymbol w}_{\boldsymbol it}\right)+\gamma \Delta \mathit{\ln } \left(p_{it}\right)+\delta q_{it}+ \lambda x_{it}^{pw}+ {\boldsymbol{\rho \kappa }}_{it}+\theta y_{it-1}+\varepsilon _{it},$$

where all the elements are explained previously except we now include λ, a parameter to be estimated, and cattle placement weight, x _t ^pw .

One potential limitation to identifying average feed price substitution patterns and the effect on technical efficiency is omitted variable bias. This bias arises if there are unobservable factors, such as animal genetics, feed rations, management practices, or operation size, that impact feedlot technical efficiency and are also correlated with feed prices. In our setting, this arises as a potential issue because we do not observe individual feedlot characteristics in our aggregate dataset. However, because we use aggregate market-level prices and technical efficiency indicators, it is unlikely that unobserved feedlot-level factors are systematically correlated with feed prices. For example, consider a feedlot’s growth-promoting implant protocol. This practice can affect technical efficiency by influencing average daily gain and the feed conversion ratio, but it is typically determined by animal characteristics, ownership arrangements, and marketing strategies. While it is possible that feedlots adjust their implant protocols in response to broader market prices, it is unlikely that these decisions are systematically correlated with short-run variation in aggregate feed prices. In this context, feed prices in our model can be treated as exogenous to any individual feedlot’s management. Therefore, while unobserved management practices may contribute to variation in technical efficiency, they are unlikely to bias our coefficient estimates due to lack of correlation with the explanatory variables.