## What are the implications of the linear homogeneity and symmetry properties of the cost function for the coefficient values in the above equation?

1. You have been presented with the following cost function representing the relationship between the prices of the three inputs used in production (wi,i = 1,2,3), output (Y) and the minimum cost of production (C):

(1)

• Write out the formula for the output elasticity of cost (
• What are the implications of the linear homogeneity and symmetry properties of the cost function for the coefficient values in the above equation?
• Use Shephard’s lemma to derive the three cost share equations. Write down the equations.
• Part-1

Briefly discuss how you would econometrically estimate the parameters of the function if you had data on input prices, input use levels, output and cost of production. Your answer should include a description of the equations you would estimate, parameter restrictions you would impose, the number of coefficients estimated and the econometric technique you would employ.

Part-2

Now, suppose you wanted to check if a simpler version with constant returns to scale (CRS) was sufficient to model the data. Write out the CRS version of the cost function and then describe how you would econometrically test the hypothesis of a CRS technology.

1. Given a production process that can be described by a CobbDouglas production technology, (), do the following:

(a) Describe the technology set T (the set of all feasible input and output combinations)

T = {(x1,x2,y) :                                                                                                           }

• Provide the definition for an input distance function using the set T

D(x1,x2,y) =                   {θ :                                                                              }

• Show, step by step, how you would derive the equation for the input distance function, D(x1 ,x2, y).
• Pick any three properties of input distance functions and show that your equation satisfies those (e.g. linear homogeneity in inputs, quasi-concavity in output, etc.)
• If A = 2, α = 0.4, β = 0.4, what are the input-oriented efficiency (TE) scores of the following input-output combinations: