AAE 575 Problem Set # 1: Technical Aspects of Production Functions
Example:
Suppose you have the following production function y = α x β , where y output, x is input use
and α and β are parameters. What restrictions on the parameters α and β ensure that the
production function is “well-behaved” in terms of its level, slope and curvature?
Answer:
Level: α ≥ 0 ensures that y ≥ 0 for all x ≥ 0.
dy
Slope:
= βα x β −1 ≥ 0 for all x ≥ 0 if α ≥ 0 and β ≥ 0.
dx
d2y
Curvature:
=
β ( β − 1)α x β − 2 ≤ 0 (concave) for all x ≥ 0 if α ≥ 0 and 0 ≤ β ≤ 1.
2
dx
Summary: This production function is “well-behaved” for all x ≥ 0 if α ≥ 0 and 0 ≤ β ≤ 1.
1) Suppose you have a negative-exponential production function y = ymax (1 − exp(α + β x) ) ,
where y output, x is input use and y max , α and β are parameters. What parameter restrictions
ensure a “well-behaved” production function in terms of its level, slope and curvature?
αx
, where y output, x is
1 + (α x / β )
input use and α and β are parameters. What restrictions on the parameters ensure that the
production function is “well-behaved” in terms of its level, slope and curvature?
2) Suppose you have a hyperbolic production function y =
Extra Credit: Suppose you have quadratic production function y =a + bx + cx 2 , where y output, x
is input use, and a, b and c are parameters. What restrictions on the parameters ensure that
the production function is “well-behaved” in terms of its level, slope and curvature?
3) Suppose you have a cubic production function y = 1000 − 200 x + 20 x 2 − 0.4 x 3 , where y
output and x is input use. Is this production function quasi-concave for all 0 ≤ x ≤ 25? How
do you know? Hint: Use the spreadsheet on the class web page to plot the function.
4) For each of the following production functions, what is the equation for the marginal product
and average product, assuming y is output, x is input use and other variables are parameters?
(Hint: Use the spreadsheet on the class web page).
a. Quadratic: y =a + bx + cx 2
b. Negative-Exponential: y = ymax (1 − exp(α + β x) )
αx
1 + (α x / β )
α + β x if α + β x < ymax
=
= min (α + β x, ymax )
y 
d. von Liebig:
 ymax if α + β x ≥ ymax
c. Hyperbolic: y =
5) For the generalized Cobb-Douglas production function y = α x1β1 x2β2 , where y is output, x 1 and
x 2 are inputs and α and β 1 , and β 2 are parameters, what is the equation for the marginal
product x 1 , the marginal product x 2 , the average product of x 1 and the average product of x 2 ?
6) Assume a generalized negative-exponential production function
y = ymax (1 − exp(α1 + β1 x1 )(1 − exp(α 2 + β 2 x2 ) , where y is output, x 1 and x 2 are inputs and
y max , αi and β i are parameters.
a. What is the marginal product of x 1 ? Does it depend on x 2 ? What is the average product
of x 1 ? Does it depend on x 2 ?
b. What is the equation for the isoquant (i.e., solve for x 2 as a function of x 1 and everything
else)? Are isoquants linear for this production function?
Extra Credit: Suppose you have a generalized hyperbolic model of the following form:
 Ax + A x 
=
y ymax 1 − 1 1A1x1 + A22 x22  , where y is output, x 1 and x 2 are inputs and y max , A i and b i are

1 + b1 +b2 

parameters (based on Swinton et al. 1994, equation 4).
a. What is the marginal product of x 1 ? Does it depend on x 2 ? What is the marginal product
of x 2 ?
b. You will not be able to write down an explicit equation for the isoquant (x 2 as a function
dx
MP1
of x 1 ). However, you can still find the slope of the isoquant using 2 = MRTS = −
.
dx1
MP2
Use your answers to part a to obtain an expression for the isoquant slope.
7) Again, assume a generalized negative-exponential production function
y = ymax (1 − exp(α1 + β1 x1 )(1 − exp(α 2 + β 2 x2 ) , where y is output, x 1 and x 2 are inputs and
∂2 y
? What is its
∂x1∂x2
sign? Does this mean that x 1 and x 2 are technically substitutes (competitive) or technically
complementary? (Hint: build on your answers in 1 and 6a.)
y max , αi and β i are parameters. What is the cross partial derivative f12 =
References
Swinton, S.M., D.D. Buhler, F. Forcella, J.L. Gunsolus and R.P. King. Estimation of crop yield
loss due to interference by multiple weed species. Weed Science 42: 103-109.