A State-Dependent Production Function: An Economist’s Apology Charles B. Moss Food and Resource Economics Department 1

Introduction  In this paper, I am using apology in a classical sense:  Apology comes from a Greek word apologia which means to speak in defense of.  Some years ago, I read a book by G.S. Hardy titled A Mathematician’s Apology in which the author tried to defend or explain the way mathematicians view the world.  In this presentation, I want to defend or explain the way that economists use production functions and to renew the conversation on the estimation and use of production functions. 2/17/20162

Food and Resource Economics Department What is the Role of Economics?  Returning to the work of the Austrian Economist Ludwig von Mises, economics is a science of human action.  Specifically, economics is the science of human action regarding the allocation of goods and services.  In this historical approach, the most basic datum is the market transaction – the quantity of any good purchased at a specific price.  This market price is determined in part by what consumers are willing to pay for any given quantity of goods and services – the demand.  The other half of the scissors is the quantity that producers are willing to supply a given quantity of goods and services – the supply.  Production economics is primarily interested in the supply of output. 2/17/20163

Food and Resource Economics Department Two Approaches to Production Economics  Primal Approach  Specification dates back to Wicksteed’s definition of the production function.  Steps:  Estimate a production function using input-output data.  Given these estimated function, firm level supply function and demand for each input can be derived.  Dual Approach  Early work in the area dates back to Hotelling, but its recent popularity started with the work of Shephard, Diewert, and McFadden.  Steps:  Assume that agents are making optimizing decisions based on a production technology they know.  Estimate the optimizing relationships directly (i.e., the supply and derived demand functions). 2/17/20164

Food and Resource Economics Department Typical Primal Estimation 2/17/20165  Gather production data  This table comes from the USDA Chemical Use Survey.  Data and the economic question is a significant opportunity for collaboration.  Specify the production function.  Statistical estimation of the function. NitPhosPotCorn

Food and Resource Economics Department Specifying the Production Function 2/17/20166  Using the Cobb-Douglas Production Function  Estimating the coefficients using ordinary least squares  Solve for the economic relationships

Food and Resource Economics Department Deriving the Implication of the Primal 2/17/20167  Economic results  Factor Demands  Output Supply

Food and Resource Economics Department Economic/Policy Questions Asked 2/17/20168  Both the primal and the dual approach can be used to answer questions such as:  What is the supply response to a change in input or output prices?  The dual approach requires the assumption that the researcher can observe people making optimal decisions.  Hence, it is difficult to address the impact of new technologies (ex ante).  The approach may also obfuscate the impact of risk and uncertainty on production.

Food and Resource Economics Department Production Function  My program in production economics focuses on how individuals decide to employ factors of production (land, labor and capital) in an effort to create production which is offered to the market.  The essence of this question is again one of constraint. If we envision a N x M space where there are N inputs and M outputs there must be an constraint (or envelope) which limits the combination of inputs and outputs which are feasible.  It may be possible for the producer to use 250 pounds of fertilizer per acre to produce one bale of cotton;  However, it is impossible for that producer to choose to produce 5 bales of cotton per acre with the same 250 pounds of fertilizer. 2/17/20169

Food and Resource Economics Department Graphical Definition of Production Function Cotton Nitrogen Feasible /17/201610

Food and Resource Economics Department The Technology Set  In general terms the production technology is mathematically depicted as  Economics theory suggests a set or conditions on this technology which make the economic question interesting.  Economics requires the technology be defined so that the individual can optimize some objective function (usually profit).  The technology should be bounded (so that an infinite amount of output cannot be produced from a finite bundle of inputs),  Concave (so that a unique optimal exists),  Inputs should be weakly essential (so that a positive quantity of a least one input is required), and  Continuous. 2/17/201611

Food and Resource Economics Department The Production Mapping  Given that the production technology meets these criteria a production map (or production function) can be defined which depicts the level of outputs resulting from the application of any fixed set of inputs  This formulation is consistent with the objection that production scientists have levied against simplified economic applications.  Life is complicated so reducing the input space could negate the economic implications of the production function. May 27,

Food and Resource Economics Department 2/17/  To address some of these shortcomings this analysis starts with a production function where combinations of controllable inputs (pounds of nitrogen applied to each acre) are combined with uncontrollable inputs (such as rainfall, which I will use as a stochastic variable such as rainfall) to produce output.  This transformation can be written as

Food and Resource Economics Department 2/17/  Approximating this production function with a second- order Taylor series expansion:

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Food and Resource Economics Department 2/17/  As a starting point, we formulate a quadratic production function where production is a function of two controllable inputs and one uncontrollable input.

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Food and Resource Economics Department 2/17/  To estimate the state-dependent production function, I use a quantile regression approach: where P(Y i < y) denotes the probability of the observed variable (Y i ) less than some target value (y), F(.) is a known cumulative probability density function, x i are observed independent variables, and  is a vector of estimated coefficients.

Food and Resource Economics Department 2/17/  Koenker and Bassett demonstrate that the regression relationship at the  th quantile can be estimated by solving

Food and Resource Economics Department 2/17/  To examine the possibility of this specification, I formulated a stochastic production function consistent with the general specification above Next, I generate a dataset assuming that the stochastic factor of production (  ) is distributed normally with mean of zero and a variance of 400.

Food and Resource Economics Department 2/17/  In addition, I also considered a negative exponential error term  Finally, I applied the specification to wheat production on the Great Plains

Food and Resource Economics Department Results for Great Plains 2/17/ Quantile0.50 Quantile0.80 Quantile Intercept Nitrogen Phosphorous Nitrogen Phosphorous Nit*Phos Missouri Nebraska Kansas

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