Chain Rule & Demand Function
Please show all the steps to calculate the answers to the attached questions about using the chain rule on an American population function and a demand function for desk lamp prices.
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Solution Summary
The solution gives detailed steps on determining derivative and using the derivative (chain rule) to find the rate of change. All formula and calculations are shown and explained.
Properties of the equilibrium price for a demand function
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.
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