Ray Production Function
Estimates for the Maize-Bean
Mixed Crop System in Madzuu
Doug Brown and Chris Barrett
Cornell University
March 15, 2004
BASIS CRSP Project Annual Team Meeting
Nyeri, Kenya
Core Issues
Estimating production response when
(i) multiple outputs share common inputs
- which inputs are limiting?
- what’s the optimal crop mix?
(ii) there may be non-convexities
associated with use of inputs inaccessible
to some farmers (e.g., livestock, inorganic
fertilizers)
Method of Analysis
Ray Production Function (Löthgren 1997,2000)
Use the Euclidian norm – the multi-dimensional distance –
║y║as dependent variable and the polar-coordinate
angles – the direction in multi-output space – i, as
regressors.
For a two-crop (maize-bean) system, one estimates the
multi-output production function: y  f ( x, 1 ,  )
Allow for interactions that can create local nonconvexities
and exact second-order approximation of true prod’n fn:
we use a generalized quadratic, w/ normalized variables.
Try with directions in both output and seed space.
Data
Plot-level input and output data, long rains
2002, from Madzuu, Vihiga District
129 maize-bean intercrop plots, 112 hhs
Mean yields: 0.97 t/ha maize
0.46 t/ha beans
input-output θ correlation = 0.077
Very preliminary results
Using θ based on outputs, yields are:
- convex in livestock holdings and joint NP application rates
- declining in plot size
- concave in N application rate
- monotone increasing in labor
- increasing as reduce maize/beans ratio
- increasing in soil quality at acquisition
Very preliminary results
Using θ based on seed, yields are:
- declining in plot size
- concave in N and P application rates with
complementarity between them
- monotone increasing in labor
- increasing as reduce maize/beans ratio
- increasing in soil quality at acquisition
(livestock no longer have significant effect)
Conclusions and Practical Implications
Madzuu farmers err in direction of maize cultivation
Soil quality very important, with P more limiting than N, but the
two complement each other
Labor availability limits output
Although inverse yield-size relation, total output grows with size
Ability to apply fertilizer, keep labor and perhaps livestock
increases yields … access to finance and human health central to
productivity and incomes, consistent with poverty trap idea.