Part (a) - Implicit Differentiation of Equation $P = C'(Q^*)$ with respect to $P$

We are given the equation: $P = C'(Q^*)$ To find the expression for $\frac{dQ^*}{dP}$, we differentiate both sides of this equation with respect to $P$. Since $P$ is the independent variable and $Q^*$ is implicitly a function of $P$, we need to apply the chain rule to $C'(Q^*)$.

Differentiating both sides:

[ \frac{d}{dP}\left[P\right] = \frac{d}{dP}\left[C'(Q^*)\right] ]

The left-hand side is straightforward:

$\frac{d}{dP}[P] = 1$

For the right-hand side, we apply the chain rule, noting that $C'(Q^*)$ depends on $Q^*(P)$:

$\frac{d}{dP}\left[C'(Q^*)\right] = C''(Q^*) \cdot \frac{dQ^*}{dP}$

Thus, we have the equation:

$1 = C''(Q^*) \cdot \frac{dQ^*}{dP}$

Solving for $\frac{dQ^*}{dP}$, we get:

$\frac{dQ^*}{dP} = \frac{1}{C''(Q^*)}$

Part (b) - Sign of $\frac{dQ^*}{dP}$

Given that $C''(Q^*) > 0$ (since we are told $C'(Q)$ is increasing and the cost function is convex), we know that:

$\frac{dQ^*}{dP} = \frac{1}{C''(Q^*)} > 0$

This means that $\frac{dQ^*}{dP}$ is positive, indicating that as the price $P$ increases, the optimal quantity $Q^*$ that maximizes profit also increases. This is intuitive because higher prices make it profitable to produce and sell more units, assuming the marginal cost function is increasing.

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Do you want more details on any step or further clarification?

Here are 5 related questions for you:

1. How does the convexity of the cost function affect the firm's production decisions?
2. What is the economic interpretation of $C'(Q^*)$ being equal to the price?
3. What would happen if $C'(Q)$ was not increasing?
4. Can $dQ^*/dP$ ever be negative in this context? Why or why not?
5. How does the second derivative of the cost function, $C''(Q)$, affect the responsiveness of $Q^*$ to changes in $P$?

Tip: When doing implicit differentiation, always remember to use the chain rule for variables that depend on other variables!