a.) Consideraprofit-maximizingfirmproducingasinglecommodity.Ifthefirmgetsafixedprice 𝑃 per unit sold, its profit from selling Q units is 𝜋(𝑄) = 𝑃𝑄 − 𝐶(𝑄), where 𝐶(𝑄) is the cost function. Assume that 𝐶0(𝑄) > 0 and 𝐶00(𝑄) > 0. Note that, 𝑄 = 𝑄∗ > 0 maximizes profits provided
𝑃 = 𝐶′(𝑄∗) ..... (1)
Thus, at the optimum, marginal cost must equal the price per unit (see equation 1 above). By
implicit differentiation of equation 1 with respect to 𝑃, find an expression for 𝑑𝑄∗/𝑑𝑃.
b.) Comment on the sign of 𝑑𝑄∗/𝑑𝑃.

### 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.

---

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!

Consider a profit-maximizing firm producing a single commodity. If the firm gets a fixed price P per unit sold, its profit from selling Q units is π(Q) = PQ − C(Q), where C(Q) is the cost function. Assume that C'(Q) > 0 and C''(Q) > 0. Find an expression for dQ*/dP through implicit differentiation of the condition P = C'(Q*). Comment on the sign of dQ*/dP.

Explore the process of implicit differentiation to find the optimal quantity response (dQ*/dP) in a profit-maximizing firm scenario. Understand the relationship between price and marginal cost, with detailed mathematical steps. Suitable for university students studying economics or calculus.

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