Properties of the equilibrium price for a demand function
Not what you're looking for?
The demand function for a certain good is D(p,m) and the supply function is S(p). For a given m, the equilibrium price p* is given by p* = f(m).
a) Show that
f'(m) = [dD(p*,m)/dm] / [S'(p*) - dD(p*,m)/dp*]
b) Verify this result when S(p) = 2p and D(p,m) = 6m^2p^(-1) + m
c) By using the implicit function rule, or otherwise, show how the result in (a) may be modified if the supply function depends on both p and m.
Purchase this Solution
Solution Summary
This solution answers questions regarding properties of the equilibrium price are proved when the demand function for a certain good is D(p,m). step by step calculations are given.
Solution Preview
Note that an equilibrium price is such that the demand and the supply are the same, so such p* satisfies
D(p*,m)=S(p*) ----(1)
Now we are told that p*=f(m). We want to calculate f'(m) or equivalently dp*/dm at the equilibrium p*. Using (1)
d/dm(D(p*,m))=d/dm(S(p*)).
Use the chain rule: d/dm(S(p*))=d/dp*(S(p*)) dp*/dm=S'(p*) f'(m).
And the chain rule for two variables:
d/dm(D(p*,m))=dD/dp* dp*/dm +dD/dm dm/dm=dD/dp* ...
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Probability Quiz
Some questions on probability
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.