Agricultural Production Economics – Week2:
Profit max. and Duality
(Note the data used in below exercises is hypothetical)
I) Graph the production function y = 0.4x + 0.09x2 – 0.003x3 and use a value of x between
0 and 20 (use Excel, R or pen and paper). For this production function:
a) Derive and graph the TPP, APP and MPP.
b) Derive the sign, slope and curvature of the MPP analytically.
c) Specify the inflection point analytically.
II) Suppose that the coefficients or parameters of a production function of the polynomial
form are to be found. The production function is y = ax + bx2 + cx3
where y = corn yield in bushels per acre
x = nitrogen application in pounds per acre
a, b and c are coefficients or unknown parameters. The production function should produce a
corn yield of 150 bushels per acre when 200 pounds of nitrogen is applied to an acre. This should
be the maximum corn yield (MPP = 0). The maximum APP should occur at a nitrogen application
rate of 100 pounds per acre. Find the parameters a, b and c for a production function meeting
these restrictions.
Hint: First find the equation for APP and MPP, and the equations representing maximum APP
and zero MPP. Then insert the correct nitrogen application levels in the three equations
representing TPP, maximum APP and zero MPP. There are three equations in three unknowns
(a, b, and c). Solve this system for a, b, and c.
III) A new crop variety is developed in the institute of crop science. The production function of
this crop is y = 2x0.5 dt/acre
Economists in this institute are given a task to evaluate economic profitability of this crop. We know
that the price of variable cost for input for crop is 2 EUR/kg and price of crop yield is 10 EUR/dt.
Task:
• Derive corresponding VMP, AVP and MFC functions;
• Solve the profit-maximizing level for input use.
• Graph the function, and its VMP, AVP and MFC.
IV) A farmer who has livestock wants to produce triticale to feed the livestock. The farmer
has obtained information on the production function of triticale. It is assumed to be as
follows:
y = 0.75x + 0.0042x2 – 0.000023x3 , where x is fertilizer input measured in dt/ha
The output price of triticale is p=4 EUR/dt,
and its input price that includes fertilizer is v=0.15 EUR/dt
Profit max. and Duality
(Note the data used in below exercises is hypothetical)
I) Graph the production function y = 0.4x + 0.09x2 – 0.003x3 and use a value of x between
0 and 20 (use Excel, R or pen and paper). For this production function:
a) Derive and graph the TPP, APP and MPP.
b) Derive the sign, slope and curvature of the MPP analytically.
c) Specify the inflection point analytically.
II) Suppose that the coefficients or parameters of a production function of the polynomial
form are to be found. The production function is y = ax + bx2 + cx3
where y = corn yield in bushels per acre
x = nitrogen application in pounds per acre
a, b and c are coefficients or unknown parameters. The production function should produce a
corn yield of 150 bushels per acre when 200 pounds of nitrogen is applied to an acre. This should
be the maximum corn yield (MPP = 0). The maximum APP should occur at a nitrogen application
rate of 100 pounds per acre. Find the parameters a, b and c for a production function meeting
these restrictions.
Hint: First find the equation for APP and MPP, and the equations representing maximum APP
and zero MPP. Then insert the correct nitrogen application levels in the three equations
representing TPP, maximum APP and zero MPP. There are three equations in three unknowns
(a, b, and c). Solve this system for a, b, and c.
III) A new crop variety is developed in the institute of crop science. The production function of
this crop is y = 2x0.5 dt/acre
Economists in this institute are given a task to evaluate economic profitability of this crop. We know
that the price of variable cost for input for crop is 2 EUR/kg and price of crop yield is 10 EUR/dt.
Task:
• Derive corresponding VMP, AVP and MFC functions;
• Solve the profit-maximizing level for input use.
• Graph the function, and its VMP, AVP and MFC.
IV) A farmer who has livestock wants to produce triticale to feed the livestock. The farmer
has obtained information on the production function of triticale. It is assumed to be as
follows:
y = 0.75x + 0.0042x2 – 0.000023x3 , where x is fertilizer input measured in dt/ha
The output price of triticale is p=4 EUR/dt,
and its input price that includes fertilizer is v=0.15 EUR/dt
