# Introduction to Calibration Theory Richard Howitt UC Davis

Introduction to Calibration Theory Richard Howitt UC Davis and ERA Economics California Water and Environmental Modeling Forum Technical Workshop Economic Modeling of Agricultural Water Use and Production January 31, 2014 Computational Economics Econometrics and Programming approaches Historically these approaches have been at odds, but recent advances have started to close this gap Advantages of Programming over Econometrics Ability to use minimal data sets

Ability to calibrate on a disaggregated basis Ability to interact with and include information from engineering and bio-physical models, Where do we apply programming models? Explain observed outcomes Predict economic phenomena Influence economic outcomes Economic Models Econometric Models Often more flexible and theoretically consistent, however not often used with disaggregated empirical microeconomic policy models of agricultural production

Constrained Structural Optimization (Programming) Ability to reproduce detailed constrained output decisions with minimal data requirements, at the cost of restrictive (and often unrealistic) constraints Positive Mathematical Programming (PMP) Uses the observed allocations of crops and livestock to derive nonlinear cost functions that calibrate the model without adding unrealistic constraints Positive Mathematical Programming Behavioral Calibration Theory We need our calibrated model to reproduce observed outcomes without imposing

restrictive calibration constraints Nonlinear Calibration Proposition Objective function must be nonlinear in at least some of the activities Calibration Dimension Proposition Ability to calibrate the model with complete accuracy depends on the number of nonlinear terms that can be independently calibrated PMP Cost-Based Calibration Let marginal revenue = \$500/acre Average cost = \$300/acre Observed acreage allocation = 50 acres

PMP Cost-Based Calibration Define a quadratic total cost function: TC x 0.5 x MC x 2 AC 0.5 x Optimization requires: MR=MC at x=50

We can calculate 2 MC AC and sequentially, 2 MC AC 0.5 x 2 * 8 and 300 0.5*8*50 x PMP Cost-Based Calibration

We can then combine this information into the unconstrained (calibrated) quadratic cost problem: max 500 x x 0.5 x 2 500 x 100 x 4 x 2 Standard optimization shows that the model calibrates when: * 0 x 50 x PMP With Multiple Crops Empirical Calibration Model Overview Three stages: 1) Constrained LP model is used to derive the dual

values for both the resource and calibration constraints, 1 and 2 respectively. 2) The calibrating constraint dual values (2 ) are used, along with the data based average yield function, to uniquely derive the calibrating cost function parameters (i )and (i). 3) The cost parameters are used with the base year data to specify the PMP model. PMP Example 2 Crops: Wheat and Oats Observe: 3 acres of wheat and 2 acres of oats Wheat (w) Crop prices

Pw = \$2.98/bu. Variable cost/acre ww = \$129.62 (Oats) (o) Po = \$2.20/bu. Average yield/acre o = 65.9 bu. w = 69 bu. wo = \$109.98 PMP Graphical Example

PMP Example (Stage 1) We can write the LP problem as: max (2.98*69 130) xw (2.20*65.9 110) xo subject to xw xo 5 xw 3 xo 2 Note the addition of a perturbation term to decouple resource and calibration constraints PMP Example (Stage 2) We again assume a quadratic total land cost function and

now solve for i and i First: 22 k f ( xk ) 2 k ; 0.5k xk 2 k ; k xk Second: Therefore: w a

ij ij ci i 0.5i xi ij i ci 0.5i xi PMP Example (Stage 3) After some algebra we can write the calibrated problem as and verify calibration in VMP and acreage: max (2.98*69) xw (2.20*65.9) xo (88.62 0.5*27.33xw ) xw 109.98xo subject to xw xo 5

PMP Graphical Example SWAP PMP Calibration We will consider a multi-region and multi-crop model where base production may be constrained by water or land CES Production Function Constant Elasticity of Substitution (CES) productions allow for limited substitutability between inputs Exponential Land Cost Function

We will use an exponential instead of quadratic total cost function CES Production Function PMP Calibration Linear Calibration Program CES Parameter Calibration Exponential Cost Function Calibration

Fully Calibrated Model Step 1: Linear Program Regions: g Crops: i Inputs: j Water sources: w Step II: CES Parameter Calibration

Assume Constant Returns to Scale Assume the Elasticity of Substitution is known from previous studies or expert opinion. In the absence of either, we find that 0.17 is a numerically stable estimate that allows for limited substitution CES Production Function 1/ i i i i y gi gi gi1 xgi1 gi 2 xgi 2 ... gij xgij Step II: CES Parameter Calibration

Consider a single crop and region to illustrate the sequential calibration procedure: 1 Define: And we can define the corresponding farm profit maximization program: max j x j xj

j / x . j j j Step II: CES Parameter Calibration Constant Returns to Scale requires:

j 1. j Taking the ratio of any two first order conditions for optimal input allocation, incorporating the CRS restriction, and some algebra yields our solution for any share parameter: 1 x1 1 1 1

l 1 x1 1 1 1 letting l all j 1 l 1 l xl l x1 1 1

. 1 l 1 xl 1 l xl Step II: CES Parameter Calibration As a final step we can calculate the scale parameter using the observed input levels as: ( yld / xland ) xland

j x j j / i . Step III: Exponential PMP Cost Function We now specify an exponential PMP Cost Function TC ( xland ) e Quadratic xland Exponential

3000 2500 2000 Cost 1500 1000 500 0 0 20 40 60

80 100 -500 Acres 120 140 160 180 200 Step III: Exponential PMP Cost

Function The PMP and elasticity equations must be satisfied at the calibrated (observed) level of land use The PMP condition holds with equality The elasticity condition is fit by leastsquares Implied elasticity estimates New methods Disaggregate regional elasticities Step IV: Calibrated Program

The base data, functions, and calibrated parameters are combined into a final program without calibration constraints The program can now be used for policy simulations Production Function Models: Extensions Theoretical Underpinnings of SWAP Crop adjustments can be caused by three things : 1. Amount of irrigated land in production can change with water availability and prices

2. Changing the mix of crops produced so that the value produced by a unit of water is increased 3. The intensive margin of substitution Intensive vs. Extensive Margin

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