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Applied Economics for Business
Management
Lecture #10
Lecture Outline
• Review
• Homework Set #8
• Continue Production Theory:
▫ Generalized Production Function
• Exam
Generalized Production Function
• In the general case of m products and n variable factors of
production, the production function in implicit form is:
• For profit maximization:
Generalized Production Function
• We assume that production occurs on this production
frontier called
• So to tie in profit maximization subject to this production
frontier, we have the following general form:
1st order conditions:
└m equations
└n equations
Generalized Production Function
• In the general case there are m + n + 1 equations and m +
n + 1 variables, so the system of equations is solvable.
• The first order conditions would determine the levels of
outputs and inputs that maximize profits while operating
on the production frontier.
• Of course, the second order conditions (not shown) should
verify the critical values for the maximum.
• What are the economic relationships which we can derive
from this general case?
▫ Relationship between 2 factors of production.
Relationship between 2 factors of
production
Take inputs j and j+1.
Recall optimal allocation of inputs is where the isoquant is
tangent to the isocost line.
• What are the economic relationships which we can derive
from this general case?
▫ Relationship between 2 factors of production.
▫ Relationship between 2 products
Relationship between 2 products
• Take 2 products i and i + 1:
• What are the economic relationships which we can derive
from this general case?
▫ Relationship between 2 factors of production.
▫ Relationship between 2 products
▫ Relationship between a variable factor of production and
a product
Relationship between a variable factor
of production and a product
• Take input j and product i.
Multiply both sides by -1.
Profits are maximized where the value of the MPP of xj is equal
to its factor price.
Demand for Nitrogen, Phosphorus and
Potash Fertilizer Nutrients….
• A study done by Hoy Carmen from the University of
California at Davis.
• Objective of study: To empirically estimate aggregate state
demands for nitrogen, phosphorus and potash for the
Western United States.
Demand for Nitrogen, Phosphorus and
Potash Fertilizer Nutrients….
• Carmen’s model:
▫ Uses production economic theory, specifically derived
demand theory.
▫ Carmen states in his article:
 “To derive the input demand function, one forms the profit
function in terms of output price, the production function, and
costs associated with the inputs.
 Maximization of profits with respect to the quantity of inputs
by taking the partial derivatives of the profit function with
respect to the inputs, setting the partial derivatives equal to
zero and solving these equations for the quantity of inputs,
yields the input demand functions.”
• He chose the Cobb-Douglas form for estimating fertilizer
demand functions:
• Question: Are the derived demand functions
homogeneous of degree zero in prices?
▫ Yes. All prices are deflated by the wholesale price
index.
Regression results for nitrogen fertilizer demand:
• Since the input demand is estimated in Cobb-Douglas form,
the coefficient on PN is the nitrogen price elasticity of
demand for that particular state.
• Which states have elastic nitrogen price elasticities of
demand that are statistically significant at the 95%
confidence level?
• The production processes for commercial fertilizers are
highly energy intensive.
▫ Assume a period of extremely high energy prices
necessitating fertilizer rationing by increasing its real price.
• Given these estimated results for nitrogen demand,
which states would you suspect would have greater
success in reducing nitrogen application?
Application of Production Economic
Theory to Lease Arrangements
• There are generally two types of lease arrangements:
▫ Cash lease
▫ Share lease
• Landlords can charge cash rent for the use of the land by
the tenant or elect to have a lease arrangement in which
returns and costs are shared by the tenant and landlord.
• The cash lease simply involves a direct cash payment from
the tenant to the landlord for the use of the land.
Application of Production Economic
Theory to Lease Arrangements
• The tenant decides:
▫ What to grow on the land
▫ How to grow these crops
▫ How to market and sell the crop
• The landlord is no longer concerned with the success or
failure of the tenant except that crop failure might
jeopardize the tenant’s ability to cover cash rent.
Application of Production Economic
Theory to Lease Arrangements
• Tenants on cash rent leases tend to underutilize inputs that
have no immediate effect (such as soil conservation
methods where the benefits occur over a longer period of
time).
• Unless there is a guarantee from the landlord of a longer
term arrangement, the tenant chooses not to make longer
term investments or improvements.
Share Rental Leases
• In our standard profit maximization framework, we have
• 1st order conditions:
• For the first type of share lease arrangement, consider the
following:
▫ assume the lease arrangement whereby the landlord
gets back a share of gross revenues from the tenant
and pays none of the expenses (costs).
Share Rental Leases
• For example: the landlord gets 25% of the crop and the
tenant gets 75% of the crop.
• Assume that both the tenant and landlord are profit
maximizers:
▫ landlord’s profits:
▫ tenant’s profits:
Share Rental Leases
• To maximize profits, the landlord chooses to maximize
output.
▫ (why? because he/she does not pay any of the costs)
•
producer (tenant) should produce where MPP = 0 or
output is maximized.
• Since the landlord doesn’t pay any costs, his/her
suggestion to the tenant is to maximize output.
Share Rental Leases
• Since the landlord doesn’t pay any costs, his/her
suggestion to the tenant is to maximize output.
• What about the tenant?
▫ To maximize profits, the tenant plans to utilize inputs as
follows:
tenant uses less x1 than the profit
maximizing level
Share Rental Losses
• Likewise, using x2 such that
choose to underutilize x2
… tenant would
• So the tenant would choose to use less x1 and x2 than what
the landlord would choose to use.
• Lease arrangements where the landlord receives a share of
the crop but pays none of the costs can result in conflict
between tenant and landlord.
• Perhaps a better lease arrangement could be drawn
whereby the landlord and tenant share both returns and
costs.
Share Rental Leases
• Suppose the landlord gets k% of the revenue and pays s%
of the cost. Then the tenant gets (1 – k)% of the revenue
and pays (1 – s)% of the cost. The values of k and s are
negotiated.
• The landlord’s profit function:
• The tenant’s profit function:
1st order conditions for landlord:
1st order conditions for tenant:
1st order conditions for tenant:
 Landlord prefers to overutilize inputs (as
compared to the usual profit max usage case).
so the tenant favor underutilizing inputs than would be the
usual profit max level.
Suppose now that we take the case where
For the landlord:
 landlord prefers to underutilize inputs (than would be the
usual profit max level)
 tenant favors overutilizing inputs (than would be
the usual profit max usage)
• These cases illustrate that even if the landlord and tenant
share costs and revenue, there is potential for conflict.
• Only if k and s are equal will the landlord’s and tenant’s
profit max conditions agree.
 landlord:
Tenant:
Share Rental Leases
• So the landlord and tenant would not tend to overutilize or
underutilize input usage.
• Note that these actions by tenant and landlord are not to
“cheat” the other but are consistent with the perceptions of
profit maximization.
• This lease illustration comes from David Debertin,
Agricultural Production Functions, 1986 pp. 142 – 145.