Please solve the screenshot. In particular, please explain the steps. Thank you! Suppose that, p* is a market equilibrium price such that Qª = Q* = Q*. So the system ofequations of interest isQ* = D(p°),Q* = S((1 – 0)p").Suppose D and S are differentiable with derivatives D' < 0 and s' > 0. Evaluate thederivatives dQ" /dê® and dp" /d0.

Consider the provided question, We have to find the derivative dQ*dθ and dp*dθ. Since , at…

Answered: Suppose that, p* is a market…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please solve the screenshot. In particular, please explain the steps. Thank you!

Suppose that, p* is a market equilibrium price such that Qª = Q* = Q*. So the system of
equations of interest is
Q* = D(p°),
Q* = S((1 – 0)p").
Suppose D and S are differentiable with derivatives D' < 0 and s' > 0. Evaluate the
derivatives dQ" /dê® and dp" /d0.

Expert Solution

Step 1

Consider the provided question,

We have to find the derivative $\frac{d Q^{*}}{d θ} and \frac{d p^{*}}{d θ} .$

Since , at Equilibrium,

$D (p^{*}) = S ((1 - θ) p^{*}) Differentiating w . r . to θ . \frac{dD (p^{*})}{dθ} = \frac{dS ((1 - θ) p^{*})}{dθ} D^{'} (p^{*}) \frac{{dp}^{*}}{dθ} = S^{'} ((1 - θ) p^{*}) [(1 - θ) \frac{{dp}^{*}}{dθ} + p^{*} (- 1)] D^{'} (p^{*}) \frac{{dp}^{*}}{dθ} = S^{'} ((1 - θ) p^{*}) (1 - θ) \frac{{dp}^{*}}{dθ} - p^{*} S^{'} ((1 - θ) p^{*}) \frac{{dp}^{*}}{dθ} [D^{'} (p^{*}) - S^{'} ((1 - θ) p^{*}) (1 - θ)] = - p^{*} S^{'} ((1 - θ) p^{*}) \frac{{dp}^{*}}{dθ} = \frac{- p^{*} S^{'} ((1 - θ) p^{*})}{D^{'} (p^{*}) - S^{'} ((1 - θ) p^{*}) (1 - θ)} Thus, \frac{{dp}^{*}}{dθ} = \frac{- p^{*} S^{'} ((1 - θ) p^{*})}{D^{'} (p^{*}) - S^{'} ((1 - θ) p^{*}) (1 - θ)}$

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