Question

Suppose that a firm's fixed proportion production function is given by \[ q=\min (5 K, 10 \mathrm{L}) \] 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 Xis fixed at 10 in the short run. Calculate the firm's short- run total, aver age, and marginal cost curves, What is the marginal cost of the 10 th unit? The 50 th unit? The 100 th unit?

Short answer

Answer: The long-run total cost function is $TC(K)=2.5K$, the average cost function is $AC(q)=\frac{1.25}{L}$, and the marginal cost function is $MC(q)=1.25$. Question: What are the short-run total, average, and marginal cost curves for the given firm when capital is fixed at 10? Answer: The short-run total cost function is $STC(L)=25+3L$, the average cost function is $SAC(q)=\frac{2.5+0.6L}{L}$, and the marginal cost function is $MC(q)=3$.

Step-by-step solution

a. Long-Run Total, Average, and Marginal Cost Curves

1. Find the total cost function in the long run: Since the production function is given by $q=min(5K,10L)$ and the rental rates for capital and labor are $v=1$ and $w=3$, we can find the total cost as follows: \[ TC(K, L) = vK + wL \] 2. Solve for L in terms of K using the production function: Since the production function has fixed proportions, the amount of capital and labor needed to produce a certain level of output is fixed. We can write: \[ 5K = 10L \] \[ L = 0.5K \] 3. Substitute L in terms of K in the total cost function: Substitute for L gives: \[ TC(K) = K + 3(0.5K) = 2.5K \] 4. Calculate the average cost function: The average cost function is given by: \[ AC(q) = \frac{TC(K)}{q} \] Substitute the total cost function to get: \[ AC(q) = \frac{2.5K}{\frac{10L}{5}} = \frac{1.25}{L} \] 5. Calculate the marginal cost function: Since the production function is in fixed proportions, the marginal cost is constant. The constant marginal cost can be found by taking the derivative of the total cost function with respect to q: \[ MC(q) = \frac{d(TC(K))}{dq} = \frac{d(2.5K)}{dq} = \frac{d(10L)}{dq} = \frac{d(5K)}{dq} = 1.25 \] So, the long-run total cost function is $TC(K)=2.5K$, the average cost function is $AC(q)=\frac{1.25}{L}$, and the marginal cost function is $MC(q)=1.25$.

b. Short-Run Total, Average, and Marginal Cost Curves

1. Find the short-run total cost function: Given that capital is fixed at 10 in the short-run, substitute this value into the total cost function we derived above: \[ STC(L) = 2.5(10) + 3L = 25 + 3L \] 2. Calculate the short-run average cost function: The short-run average cost function is given by: \[ SAC(q) = \frac{STC(L)}{q} \] Substitute the short-run total cost function to get: \[ SAC(q) = \frac{25 + 3L}{\frac{10L}{5}} = \frac{2.5 + 0.6L}{L} \] 3. Calculate the short-run marginal cost function: The short-run marginal cost function can be found by taking the derivative of the short-run total cost function with respect to L: \[ SMC(q) = \frac{d(STC(L))}{dq} = \frac{d(25 + 3L)}{dq} = 3 \] 4. Calculate the marginal cost of the 10th, 50th, and 100th units: Since the short-run marginal cost is constant, the marginal cost of the 10th, 50th, and 100th units will all be 3. So, the short-run total cost function is $STC(L)=25+3L$, the average cost function is $SAC(q)=\frac{2.5+0.6L}{L}$, and the marginal cost function is $MC(q)=3$.

Key concepts

Understanding Long-Run Cost Curves

In economic theory, long-run cost curves are crucial for understanding how costs behave when all input factors are variable. This allows firms the flexibility to alter all inputs according to production needs over time. With the production function, $q=min(5K,10L)$, introducing long-run analysis entails looking at costs when the firm can adjust both labor ($L$) and capital ($K$).
The total cost (TC) function for the long run can be expressed as:

• $TC(K,L)=vK+wL$

where $v=1$ is the rent for capital, and $w=3$ is the wage for labor. Once we substitute values from the fixed input relationship $L=0.5K$, we redefine total costs:

• $TC(K)=2.5K$

The average cost ($AC$) is the total cost per unit of output, calculated as $AC(q)=\frac{2.5K}{q}$. A crucial insight is that marginal cost ($MC$), which measures the cost of producing one more unit, is constant here because of the fixed input proportions. The calculation simplifies as $MC(q)=1.25$. This simplicity emerges from the linear relationship in production levels and input use.

Exploring Short-Run Cost Analysis

In the short-run, some inputs are fixed. For this exercise, capital ($K$) is fixed at 10 units. This change in flexibility affects the cost analysis, notably differing from the long-run approach. The short-run total cost (STC) becomes:

• $STC(L)=25+3L$

This clearly shows a fixed cost part due to capital ($25$), plus a variable cost part dependent on labor's cost ($3L$).
Similarly, the average cost in short-run (SAC) evaluates as $SAC(q)=\frac{25+3L}{q}$. Each additional unit adds the cost $3$, keeping the short-run marginal cost (SMC) constant. This uniformity persists regardless of whether you’re calculating it for the 10th, 50th, or 100th unit as $SMC(q)=3$. This constancy in marginal cost highlights constraints imposed by a fixed input, leading to uniform cost increases per additional output unit.

Deciphering Marginal Cost Calculation

Marginal cost (MC) calculation is fundamental in assessing the cost-effectiveness of producing additional units. It examines incremental costs tied to the last unit produced.
In the long-run, the marginal cost was deduced alongside fixed proportions; it's the derivative of total costs with respect to output, simplified to $MC(q)=1.25$. This constant reflects consistent effort to increase output when adjustments in all input levels are possible.
Conversely, in the short-run setting with fixed capital $K=10$, the marginal cost rises directly from the labor component. Calculated differently from the long-run, which assumes no fixed inputs, short-run MC, here $SMC(q)=3$, manifests from deriving total short-run costs relative to variable inputs. This figure remains constant for any unit change due to the dependency solely on labor adjustments without modifications in capital input.Understanding these computations offers insight for firms in decision-making about production levels, guiding efficient allocation of resources for minimal cost impact.