differences, especially for WP and lagged WP, and the explanatory power is smaller (0.442 instead of 0.458).
We have explored the idea of modeling baseball production under the assumption that winning percent and attendance are multiple or joint outputs as alternatives to modeling each production function separately. The same data set and set of inputs was used in three production models so that estimates from those models can be compared.
Estimation of two separate production functions using OLS implies that we expect no causal relationship between the outputs that would lead to simultaneity. Also, it implies that inputs are not shared, that they are shared with no exhaustion, or that we have correctly entered the amount of input used by each output. If we assume that increased team performance, measured by winning percent, has a causal effect on attendance, but increased attendance does not cause increased winning percent, then the model is recursive. The exception is contemporaneous correlation between the disturbances of the two equations. Theoretically we expect a positive correlation between the disturbances because omitted variables may affect winning percent and attendance in the same direction. However, our estimate of this correlation is negative and insignificant. We conclude that there is no need to use a simultaneous estimation method. The fact that our 3SLS estimates are very similar to the OLS estimates confirms this conclusion.
Canonical regression was used to estimate a joint production function model that allows inputs to be shared by the two outputs. The estimates of marginal elasticities from this model are quite different from the elasticities estimated by the OLS and 3SLS models, and, upon closer examination, we find some possible explanations of these differences. One drawback of the joint production function model for baseball production is that all inputs must be assumed to be used in the production of all outputs. This is not true in the case of baseball. In our model, we expect ticket price and population to affect attendance but not winning percent. When using a joint production function, all inputs must be included, but they cannot yield zero marginal elasticities for the output they are not expected to effect. In fact, for each input, the ratio of its marginal elasticity in the WP equation to its marginal elasticity in the ATT equation must be the same.
The joint production model implies that each input contributes in the same
proportion to production of the two products. We can see this implication by
considering the marginal elasticities from this model given in equations (12) and (13).
Using these marginal elasticities we find that
for all k = 1, 2,. . ., m. This fixed relationship between the marginal elasticities means that it is not possible for one input to have a large impact on WP and a small impact on ATT while a different input has a large impact on ATT but a small impact on WP.