Posted: November 6th, 2015

Mathematics

Mathematics

Economic Context
In building economic models, researchers quickly face the issue of what functional form to use to represent the relationships in their models;
particularly if they wish to estimate the relationship using econometric techniques. In class we use several standard mathematical functions
(e.g., exponentials, natural logarithms, polynomials) to model simple relationships but, in practice, these can be very restrictive in terms of
how economic variables are related.
An extensive literature exists in economics that addresses the specification of functional forms in a wide variety of economic models:
production functions, cost functions, utility functions, equilibrium models, to name just a few. In some cases, specific functional forms have
been developed (e.g., Cobb-Douglas, CES) because they have well known properties that can be easily understood and/or estimated. Others
have pursued an approach called flexible functional forms whereby the economic relationship is approximated by using concepts from
Taylor series expansion of a function. Neither approach has been entirely satisfactory for any particular type of economic model.
In this homework, you will begin your first exploration of the issues surrounding functional forms by looking at how economic models
represent cost functions.
Model Exploration
Consider a basic cost function model as taught in principles and intermediate microeconomics, c(y), where y denotes the level of output and
c(y) denotes total cost. Note that we are ignoring the role of input prices in the model. The typical textbook model illustrates the cost
function in terms of average and marginal costs measures:
atc
avc
mc
y
$
atc
avc
mc
y
$
Figure 1. Cost Functions
? Issues to explore
In exploring these basic “standard” cost functions using Mathematics, consider the following issues:
1. What class of functions can allow for the relationships illustrated above?
2. Are there any restrictions on the relevant parameters of the class of functions?
3. What issues, if any, might arise in trying to estimate such cost functions?
4. Can the class of functions for given parameter values reflect the following phenomenon? If so, how?
(a) high or low fixed cost industries
(b) long run versus short run costs
5. Can the class of functions handle the following observed long run phenomenon?
many industries have an output range over which they experience decreasing average costs, but then at some output (minimum efficient
scale), average costs become relatively constant for large ranges of output
? How to explore
Before starting to explore a class of functions in Mathematical, conduct two types of searches. Your most productive search route will be a
general search via Google. You’ll find many results; focus on the academic materials by faculty using cost functions in research and
teaching.

Use Mathematical to computationally explore the class of functions you selected: what shapes of average total cost, average variable cost,
marginal cost can be generated by the class of functions? how do the shapes vary with parameters specified in the function? can all the
issues raised above “be addressed” with varying parameters ranges and restrictions on the parameters?

? Document guidelines
Develop a write up of your explorations in an interactive Mathematical notebook that meets professional standards:
! The discussion of your computational analysis should be 750-1250 words and the writing should meet professional standards.
The audience for the discussion is other economists who have had training in mathematical economics. Thus, you should use the
language and terminology of mathematical economics to convey your findings.

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