overview of optimization in ag economics lecture 2

Overview of Optimization Overview of Optimization in Ag Economicsin Ag Economics

Lecture 2

Interregional and Spatial Economics

Leibnitz’s Rule

,

,, ,

B r

A r

B r

A r

V r f x r dx

V r B r A r f x rV r f B r r f A r r dx

r r r r

In equilibrium

pd(z) is the consumer’s inverse demand curve and ps(z) is the producer’s supply curve.

0 0

maxx x

d s

xp z dz p z dz

Applying Leibnitz’s Rule

0d sp z p z

1 2

1 21 2 1

, ,0 0 0

1 2

1 1 1 21

2 2 1 22

max

. .

0

0

T

T

x x xd d s

x x x

T

d s

d s

p z dz p z dz p z dz tx

s t x x x

Sp x p x x

x

Sp x p x x t

x

Econometrics and Statistical Applications

Historically, econometrics relied on closed form solutions made possible by linear models of normally distributed random variables. More complicated models that introduce factors such as concavity constraints and nonnormality do not imply closed form solutions.

Concavity constrained cost functions:

Basic cost function formulation:

0

min,

. . , 0

1 1, 2 2

xw x

C w ys t F x y

C w y w w w y y y w y

In this formulation, both the A and B matrices are symmetric by Young’s theorem. In addition, if the optimization conditions are met (as we will describe in this course) the cost function is concave in input prices, implying that the A matrix is negative semi-definite.

Fitting the cost function: We estimated this cost function using concentrated maximum likelihood. Specifically, the residual vector based on any set of parameter estimates is

0

1 1 11 1 12 2 13 3 11 1 21 2 31 31

2 2 21 1 22 2 23 3 12 1 22 2 32 31

3 3 31 1 32 2 33 3 13 1 23 2 33 31

1 12 2i i i i i i i i i

i i i i i ii

i i i i i i

i i i i i i

c w w w y y y w y

x w w w y y yex w w w y y y

x w w w y y y

These parameters are estimated by maximizing

While this formulation is complex, it can be estimated using iterative generalized least squares without resorting to complex mathematical programming algorithms.

1

max ln2

1. .

T

i ii

T

s t e eT

While this formulation is complex, it can be estimated using iterative generalized least squares without resorting to complex mathematical programming algorithms. However, one approach to estimating a concavity constrained const function is to estimated the Cholesky decomposition of the A matrix. Specifically,

11 11 12 13

12 22 22 23

13 23 33 33

11 11 11 12 11 13

11 12 12 12 22 22 12 13 22 23

11 13 12 13 22 23 13 13 23 23 33 33

0 0

0 0

0 0

Policy Analysis–Reduction of Price Floor

Good 1 Good 2

Good 1 Good 2

General Equilibrium

Modeling the interaction between the two markets involves moving from a partial equilibrium analysis to a general equilibrium analysis.

Early work on general equilibrium analysis involves the concept of a Walrasian equilibrium. The primary idea of the Walrasian equilibrium was the concept that some price vector could be found for any endowment that equated the supply and demand, or resulted in zero excess demand:

I(p) is the excess demand for good i, it is a function of the price vector. Demand is determined by the initial endowment of goods W.

pi(i(p)-Wi) is the complementary slackness conditions. This condition implies that either the price of the ith good is zero, or its excess demand is zero.

0

0

,,

ii

iii

i

Wp

Wpp

wpSWpDp

Nonparametric Efficiency Analysis

Economic efficiency of farms and agribusinesses has been analyzed by first estimating a parametric cost or profit function as developed in Featherstone and Moss.However, the efficiency results in these studies are conditioned on the choice of structural form used to estimate the parametric production function as well as the distributional assumptions used in estimation.

An alternative approach is to allow the most efficient firms to form an efficient technological envelope. Assuming that a firm produces a vector of m output from n inputs, a vector of outputs, , could be produced using some combination of outputs from firms in the sample.

Data Envelope Analysis

1

,1

min

1

*

1

1

*

n

ii

j

n

iiij

n

iii

z

z

mjyzyst

zcc