Question

Suppose that a firm's fixed proportion production function is given by \\[ q=\min (5 K, 10 L) \\] and that the rental rates for capital and labor are given by \(v=1, w=3\) a. Calculate the firm's long-run total, average, and marginal cost curves. b. Suppose that \(K\) is fixed at 10 in the short run. Calculate the firm's short-run total, average, and marginal cost curves. What is the marginal cost of the 10 th unit? The 50 th unit? The 100 th unit?

Short answer

Question: Calculate the long-run average cost, long-run marginal cost, short-run average cost, and short-run marginal cost for the given production function and rental rates. Also, find the marginal cost of the 10th, 50th, and 100th units when K is fixed at 10. Answer: The long-run average cost (LAC) and long-run marginal cost (LMC) are both 1/2. The short-run average cost (SAC) is (10/q) + (3/10), and the short-run marginal cost (SMC) is 3/10. The marginal cost for the 10th, 50th, and 100th units when K is fixed at 10 is 3/10.

Step-by-step solution

Calculate Isoquant

In the long-run, the firm chooses the optimal combination of K and L to minimize costs. To solve this problem, we need to first find the isoquant of the given production function. An isoquant is the combination of K and L for which output, q, is constant. In this case, we have: \\[ q = \min(5K, 10L) \\]

Calculate Long-Run Total Cost Function

The total cost of production in the long-run is equal to the cost of using capital (vK) plus the cost of using labor (wL): \\[ LTC = vK + wL \\] Given the rental rates of capital and labor, v=1 and w=3, we plug in these values: \\[ LTC = K + 3L \\]

Find the Optimal Combination of K and L for Each Output Level

To minimize costs, we need to find the optimal combination of K and L for each output level. Since q = min(5K, 10L), we can express K and L in terms of q: \\[ K = \frac{q}{5}\\ L = \frac{q}{10} \\]

Plug the Expressions for K and L into the Total Cost Function

Replace K and L in the total cost function to get the long-run total cost function: \\[ LTC(q) = \frac{q}{5} + 3(\frac{q}{10}) = \frac{1}{5}q + \frac{3}{10}q = \frac{1}{2}q \\]

Calculate Long-Run Average and Marginal Cost Functions

Long-run average cost (LAC) is obtained by dividing the total cost by the output (q): \\[ LAC(q) = \frac{LTC(q)}{q} = \frac{1}{2} \\] Long-run marginal cost (LMC) is the derivative of the total cost function with respect to q: \\[ LMC(q) = \frac{d LTC(q)}{dq} = \frac{1}{2} \\] #b. Short-run Cost Curves (K=10)#

Calculate Short-Run Total Cost Function

In the short-run, the total cost of production is equal to the cost of using capital (vK) plus the cost of using labor (wL): \\[ STC = vK + wL \\] Plug in the fixed value of K and the rental rates for capital and labor: \\[ STC = 10 + 3L \\]

Express L in terms of q

Since K is fixed at 10 and q = min(5K, 10L), we can express L in terms of q: \\[ L = \frac{q}{10} \\]

Plug the Expression for L into the Total Cost Function

Replace L in the total cost function to get the short-run total cost function: \\[ STC(q) = 10 + 3(\frac{q}{10}) = 10 + \frac{3}{10}q \\]

Calculate Short-Run Average and Marginal Cost Functions

Short-run average cost (SAC) is obtained by dividing the total cost by the output (q): \\[ SAC(q) = \frac{STC(q)}{q} = \frac{10}{q} + \frac{3}{10} \\] Short-run marginal cost (SMC) is the derivative of the total cost function with respect to q: \\[ SMC(q) = \frac{d STC(q)}{dq} = \frac{3}{10} \\]

Calculate the Marginal Cost of Specific Units

We are asked to find the marginal cost of the 10th, 50th, and 100th units. Since the short-run marginal cost (SMC) is constant, the marginal cost for all these units will be the same: Marginal cost for the 10th unit: SMC(10) = 3/10 Marginal cost for the 50th unit: SMC(50) = 3/10 Marginal cost for the 100th unit: SMC(100) = 3/10

Key concepts

Fixed Proportion Production Function

A fixed proportion production function, also known as a Leontief production function, is characterized by a rigid relationship between inputs. It represents a scenario where inputs must be used in exact proportions to produce output.

The function given in the problem, \( q = \text{min}(5K, 10L) \), is an example of a fixed proportion production function. It suggests that output \( q \) is limited by the input in the smallest quantity when adjusting for the production ratio. In other words, if there's more of one input than is required in proportion to the other, the excess cannot be used to increase production. This type of production often occurs in situations where inputs are complementary and cannot substitute for one another, for instance, a single machine operated by a single worker.

Understanding the fixed nature of this production function is key to determining the most cost-efficient combination of inputs, which is where the concepts of isoquants and cost curves come in.

Isoquant

An isoquant represents all the combinations of two inputs that yield the same level of output. It is comparable to an indifference curve in consumer theory but applies to production. The isoquant is pivotal in helping firms decide the most efficient combination of inputs that will produce a given level of output.

In the context of the given exercise, we look at the isoquant for the fixed proportion production function \( q = \text{min}(5K, 10L) \). Because of the 'min' operator, the isoquant in this case forms a right angle, a characteristic feature of Leontief production functions. The right angle reflects the fixed input proportions: Increasing one input without increasing the other does not lead to increased output. Hence, the firm's goal is to find a point on the isoquant where the costs of inputs are minimized.

A deeper understanding of isoquants can aid in visualizing the possible trade-offs between inputs within the production process, although no trade-off exists for fixed proportion production functions.

Marginal Cost

Marginal cost is the additional cost incurred from producing one more unit of a good or service. It is a foundational concept in economics because it helps businesses in decision-making processes about production levels.

In the exercise provided, marginal cost is derived from differentiating the total cost function with respect to output \( q \). Long-run marginal cost (LMC) corresponds to the constant rate at which total costs increase as output increases in the long-run, given by \( LMC(q) = \frac{1}{2} \). In the short-run, where capital \( K \) is fixed, short-run marginal cost (SMC) represents the rate at which total costs increase with output, calculated to be \( SMC(q) = \frac{3}{10} \).

These calculations assist in understanding the cost structure of a firm. For instance, the constancy of marginal cost in this case is due to the linear nature of the total cost function, a trait that stems from the fixed proportions in the production technology.