a state-dependent production function: an economist’s apology charles b. moss food and resource...
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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. 2/17/20163TRANSCRIPT
A State-Dependent Production Function: An
Economist’s Apology
Charles B. MossFood and Resource
Economics Department
1
Food and Resource Economics Department
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.
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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.
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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).
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Food and Resource Economics Department
Typical Primal Estimation
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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.
Nit Phos Pot Corn127.0 60.0 90.0 140.0202.0 104.0 120.0 110.088.0 24.0 90.0 61.0150.0 69.0 120.0 138.0200.0 0.0 0.0 150.0153.3 52.6 126.6 102.0139.0 35.0 90.0 160.0150.0 60.0 120.0 115.0160.0 40.0 50.0 165.0180.0 37.0 120.0 140.0160.0 30.0 60.0 135.0182.7 76.8 127.7 160.0
Food and Resource Economics Department
Specifying the Production Function
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Using the Cobb-Douglas Production Function
Estimating the coefficients using ordinary least squares
Solve for the economic relationships
1 2 3 1 2 3ln ln ln ln lny Ax x x y A x x x
0 1 1 2 2 3 3ln ln ln lny x x x
1 2 1 1 2 2
11 1
22 2
max
0
0
Y
Y
Y
p x x w x w xYp w
x xYp w
x x
Food and Resource Economics Department
Deriving the Implication of the Primal
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Economic results Factor Demands
Output Supply
1
111* 1 11 1 2
2
, ,Y Ywx p w w p
w
1
1 11* 1 22 1 2
1
, ,Y Ywx p w w p
w
1 1
* 11 2
1 2
, ,Y YY p w w pw w
Food and Resource Economics Department
Economic/Policy Questions Asked
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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.
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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.
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Food and Resource Economics Department
Graphical Definition of Production FunctionCotton
Nitrogen
Feasible
250
1
5
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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.
, : ,N Mx y T x R y R
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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.
, : N Mf x y R R
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Food and Resource Economics Department
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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
1:f X R
Food and Resource Economics Department
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Approximating this production function with a second-order Taylor series expansion:
00 0 0 0
0
0 020 0
0 0
20 0
, , ,
1 ,2
,
x
x
xxf x f x f x
x xx xf x
O x
Food and Resource Economics Department
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0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
20 0 0 0 0 0
0 0 0 0 0 0 0 0 0
20 0
, , ,
1, , ,2
, , ,1 , ,2
1, , ,2
,
x xx
x
x
f x g x h x
g x x x x x x A x x x
h x x x x A x
A x O x
x x x A x A x
O x
Food and Resource Economics Department
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As a starting point, we formulate a quadratic production function where production is a function of two controllable inputs and one uncontrollable input.
1 1 1
1 2 2 2 2
0.15 0.001 0.0005 0.0021, , 1.5 0.25 0.0005 0.0015 0.0012
0.10 0.002 0.001 0.009
x x xf x x x x x
Food and Resource Economics Department
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1 1 11 2
2 2 2
1 1 11 2
2 2 2
0.1163 0.001 0.00051, , 16.83 1.45820.2332 0.0005 0.00152
0.15 0.001 0.00051, ,0.00 1.500.25 0.0005 0.00152
x x xf x x
x x x
x x xf x x
x x x
1 1 11 2
2 2 2
0.1837 0.001 0.00051, , 16.83 1.90830.2668 0.0005 0.00152
x x xf x x
x x x
Food and Resource Economics Department
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To estimate the state-dependent production function, I use a quantile regression approach:
where P(Yi < y) denotes the probability of the observed variable (Yi) less than some target value (y), F(.) is a known cumulative probability density function, xi are observed independent variables, and is a vector of estimated coefficients.
i iP Y y F y x
Food and Resource Economics Department
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Koenker and Bassett demonstrate that the regression relationship at the th quantile can be estimated by solving
: :
min 1k
i i i i
i i i iR i i y x i i y x
y x y x
Food and Resource Economics Department
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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.
1 1 1
1 2 2 2 2
0.15 0.001 0.0005 0.0021, , 1.5 0.25 0.0005 0.0015 0.0012
0.10 0.002 0.001 0.009
x x xf x x x x x
Food and Resource Economics Department
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In addition, I also considered a negative exponential error term
Finally, I applied the specification to wheat production on the Great Plains
exp , ,0z
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Results for Great Plains
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0.20 Quantile
0.50 Quantile
0.80 Quantile
Intercept 19.713 9.647 32.906Nitrogen 0.278 0.325 0.484Phosphorous -0.544 0.975 -1.217Nitrogen2 -0.0057 -0.0059 -0.0039Phosphorous2
-0.0048 -0.0131 -0.0089
Nit*Phos 0.0151 0.0149 0.0066Missouri 7.727 8.028 9.106Nebraska 10.241 10.665 11.181Kansas 13.504 11.486 13.233
Food and Resource Economics Department
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1 0 2 0 3 0 4 0 5 0N i t r o g e n p o u n d s a c r e 1 5
2 0
2 5
3 0
3 5
W h e a t b u s h e l s a c r e