Problem 16 - Applied Calculus

Question

A reasonably realistic model of a firm's costs is given by the short-run Cobb- Douglas cost curve $$ C(q)=K q^{1 / a}+F $$ where \(a\) is a positive constant, \(F\) is the fixed cost, and \(K\) measures the technology available to the firm. (a) Show that \(C\) is concave down if \(a>1\). (b) Assuming that \(a<1\), find what value of \(q\) minimizes the average cost.

Step-by-Step Solution

Answer

(a) \( C(q) \) is concave down if \( a > 1 \). (b) Minimum average cost occurs at \( q = \left( \frac{F}{K(1/a - 1)} \right)^a \) for \( a < 1 \).

Step 1: Understand the Structure of C(q)

The cost function provided is \( C(q) = K q^{1/a} + F \), where \( K \) is a constant related to technology, \( F \) is the fixed cost, and \( q \) is the quantity of goods produced. The term \( q^{1/a} \) indicates how costs change with production level.

Step 2: Determine Concavity for a > 1

To show that \( C(q) \) is concave down, we need to demonstrate that its second derivative is non-positive when \( a > 1 \). First, find the first derivative with respect to \( q \):\[C'(q) = \frac{d}{dq} \left( K q^{1/a} + F \right) = \frac{K}{a} q^{1/a - 1}\]Next, find the second derivative:\[C''(q) = \frac{d}{dq} \left( \frac{K}{a} q^{1/a - 1} \right) = \frac{K(1/a - 1)}{a} q^{1/a - 2}\]Since \( a > 1 \), \( 1/a - 1 < 0 \), making \( C''(q) \leq 0 \), proving \( C(q) \) is concave down for \( a > 1 \).

Step 3: Calculate Average Cost

The average cost function \( AC(q) \) is given by \( \frac{C(q)}{q} \). Thus,\[AC(q) = \frac{K q^{1/a} + F}{q} = K q^{1/a - 1} + \frac{F}{q}\]

Step 4: Find the Minimum Average Cost for a < 1

To find the minimum average cost, take the derivative of \( AC(q) \) with respect to \( q \), set it to zero, and solve for \( q \):\[AC'(q) = \frac{d}{dq} \left( K q^{1/a - 1} + \frac{F}{q} \right) = \frac{K(1/a - 1)}{q^{2 - 1/a}} - \frac{F}{q^2}\]Set \( AC'(q) = 0 \) and solve:\[\frac{K(1/a - 1)}{q^{2 - 1/a}} = \frac{F}{q^2}\]Multiply both sides by \( q^{2} q^{2 - 1/a} \) to isolate terms:\[K(1/a - 1) q^{1/a} = F\]Thus, solve for \( q \):\[q = \left( \frac{F}{K(1/a - 1)} \right)^a\]This is only valid when \( a < 1 \).

Step 5: Verify the Conditions

Double-check that each step, especially solving \( AC'(q) = 0 \), is consistent with conditions \( a < 1 \) and assumptions made in the problem. This verifies that \( q = \left( \frac{F}{K(1/a - 1)} \right)^a \) indeed gives the minimum average cost.

Key Concepts

Concavity

The concept of concavity is central in understanding the behavior of the cost function, particularly when working with the Cobb-Douglas cost curve in microeconomic models. In our exercise, the cost function is given as \( C(q) = K q^{1/a} + F \). Here, "concave down" means that the graph of this function curves downwards, resembling an upside-down bowl. To accurately determine concavity, we use the second derivative test.

Taking the first derivative with respect to \( q \), we have \( C'(q) = \frac{K}{a} q^{1/a - 1} \). To check for concavity, we further differentiate to find the second derivative: \( C''(q) = \frac{K(1/a - 1)}{a} q^{1/a - 2} \).

For the function to be concave down, \( C''(q) \) must be non-positive (
zero or less), which occurs when \( a > 1 \). This is because \( 1/a - 1 \) becomes negative for values of \( a \) greater than one, ensuring \( C''(q) \leq 0 \). Hence, the cost function is concave down when \( a > 1 \). Understanding concavity helps in analyzing how costs change relative to production levels.

Average Cost

Average cost is a key metric in assessing production efficiency and making pricing decisions. It is calculated by dividing the total cost by the quantity produced, giving a per-unit cost.

For the Cobb-Douglas cost function \( C(q) = K q^{1/a} + F \), the average cost function is \( AC(q) = \frac{C(q)}{q} \), leading to \( AC(q) = K q^{1/a - 1} + \frac{F}{q} \).

This expression reveals two components:

• The term \( K q^{1/a - 1} \) reflects the variable cost per unit, adjusting according to the scale of production.


• The fraction \( \frac{F}{q} \) represents the fixed cost distributed across the quantity produced; as \( q \) increases, this component decreases, showing economies of scale.
  

Analyzing the average cost helps firms determine the most efficient production level, where per-unit costs are minimized.

Derivative

Derivatives are mathematical tools used to find the rate at which quantities change. In economics, they are vital for understanding how variables such as cost, revenue, and demand react to changes in production levels.

In the Cobb-Douglas cost function \( C(q) = K q^{1/a} + F \), the derivative with respect to \( q \) helps us assess how costs evolve as we produce more goods. The first derivative \( C'(q) = \frac{K}{a} q^{1/a - 1} \) describes the instantaneous rate of change of cost per unit.

To determine key features like concavity or the minimum point of average cost, we also consider the second derivative \( C''(q) = \frac{K(1/a - 1)}{a} q^{1/a - 2} \). Negative second derivatives confirm concavity down, thus helping optimize production decisions. Mastery of derivatives enables firms to strategically navigate cost functions.

Cost Minimization

In economic theory, cost minimization is a central goal for firms aiming to enhance profitability while maintaining production efficiency. The Cobb-Douglas cost scenario involves determining the output level \( q \) that minimizes average cost, particularly when \( a < 1 \).

To achieve this, we analyze the average cost function \( AC(q) = K q^{1/a - 1} + \frac{F}{q} \) and its derivative. Setting \( AC'(q) \) to zero gives the quantity \( q \) at which average cost is minimized. Solving, we find \( q = \left( \frac{F}{K(1/a - 1)} \right)^a \).

This formula helps decide the optimal production level by balancing fixed and variable costs. Key to successful cost minimization is ensuring the assumptions of \( a < 1 \) hold, as this condition affects the entire calculation process. This knowledge allows firms to optimize their expenditures and maximize profit potential.