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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? ▫R▫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? ▫R▫Relationship between 2 factors of production. ▫R▫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? ▫R▫Relationship between 2 factors of production. ▫R▫Relationship between 2 products ▫R▫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 x j 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? ▫Y▫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 P N 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: ▫C▫Cash lease ▫S▫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.
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 1 st 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 x 1 than the profit maximizing level
Share Rental Losses Likewise, using x 2 such that … tenant would choose to underutilize x 2 So the tenant would choose to use less x 1 and x 2 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:
1 st order conditions for landlord:
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.
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.