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Agricultural and Applied Economics 641
Homework 1
A technology is represented by the production function y = f(x), where y is output and x = (x1, ..., xn) is a vector of n inputs.
1. Consider the CES (constant elasticity of substitution) production function
y = g ∙ [a x1-r + (1-a) x2-r]-v/r
1.1. Briefly discuss the properties of the CES production function.
1.2. Using data from a Corn Fertility study (see below), estimate the parameters of the CES production function.
1.2.1. What is the elasticity of substitution between the two inputs (nitrogen fertilizer and potassium).
1.2.2. What is the scale elasticity?
1.3. Test the validity of the Cobb-Douglas and Leontief specifications.
1.4. Test for constant returns to scale.
1.5. Interpret the results.
2. Consider the following specifications of production function
- quadratic function: y = a0 + Si bi xi + Si³j cij xi xj
- square-root function: y = a0 + Si bi (xi).5+ Si³j cij (xi).5 (xj).5
- Cobb-Douglas: ln(y) = a0 + Si bi ln(xi)
- translog function: ln(y) = a0 + Si bi ln(xi) + Si³j cij ln(xi) ln(xj)
2.1. Using data from a Corn Fertility study (see below), estimate the parameters of each production function. Interpret the results.
2.2. Test the validity of the Cobb-Douglas specification. Interpret the results.
2.3. Choose the production function that gives the best fit (e.g., using the adjusted R-square).
2.3.1. Discuss the properties of the chosen production function and implications for the nature of the underlying technology.
2.3.2. Given a price of corn of $5.20/bu, a nitrogen fertilizer price of $.20/lb, and a potassium price of $.25/lb, find the profit maximizing optimal inputs and output.
2.3.3. Evaluate the price elasticities of supply and demands (with respect to both output price and input prices).
2.4. Interpret the results.
Note: The empirical analysis involves a data set containing 43 observations from a Corn Fertility study conducted at the Arlington Agricultural Experiment Station, University of Wisconsin. The data are in the file “CornY.csv” on the class WEB site. The variables include corn yield (Yield, bu/acre), available nitrogen (NN, lb/acre) and available potassium (KK, lb/acre). A program containing the R code to support the empirical analysis is available on the class WEB site.