Question

Suppose, as in Problem $12.6,$ a firm produces hockey sticks with a production function of $q=2\mathrm{V}KL$. Capital stock is fixed at $K$ in the short run. a. Calculate the firm's total costs as a function of $q,w,v,$ and $K$ b. Given $q,w,$ and $v,$ how should the capital stock be chosen to minimize total cost? c. Use your results from part (b) to calculate the long-run total cost of hockey stick production d. For $w-\mathrm{\$}4,v-\mathrm{\$}1,$ graph the long-run total cost curve for hockey stick production. Show that this is an envelope for the short-run curves computed in part (a) by examin ing values of $K$ of $100,200,$ and 400

Short answer

The firm's total cost function is: $TC(q,w,v,K)=wL+vK=w\frac{q}{2VK}+vK$ b) Given q, w, and v, what is the optimal capital stock to minimize total cost? The optimal capital stock to minimize total cost is infinite. c) What is the long-run total cost of hockey stick production? The long-run total cost of hockey stick production is 0. d) Describe and analyze the long-run total cost curve for hockey stick production and the short-run curves with K values of 100, 200, and 400. The long-run total cost curve is a horizontal line at 0 on the vertical axis. The short-run total cost curves for K values of 100, 200, and 400 form a tangential "envelope" to the long-run total cost curve, illustrating how the short-run cost curves approach the long-run cost curve as K increases.

Step-by-step solution

Calculate the firm's total costs as a function of q, w, v, and K

Firstly, we need to rewrite the production function for labor L, as we are given the values of q, K, and V. $q=2VKL$ $L=\frac{q}{2VK}$ Now we can express the total costs, which consist of the cost of labor wL and the cost of capital vK. $TC(q,w,v,K)=wL+vK=w\frac{q}{2VK}+vK$

Given q, w, and v, find the optimal capital stock to minimize total cost

To find the optimal capital stock K, we need to take the derivative of the total cost function with respect to K and set it equal to 0. $\frac{\partial TC(q,w,v,K)}{\partial K}=0$ Now we'll solve for K to find the optimal level. $\frac{\partial }{\partial K}(w\frac{q}{2VK}+vK)=0$ $\frac{-wq}{2V(K^2)}=0$ Since w, q, and V are all non-zero, K^2 must be infinity, and K must be infinity.

Calculate the long-run total cost of hockey stick production

We've found that the optimal K is infinite, so we now need to find the minimum long-run total cost. To do this, we will find the minimum of the total cost function for any given level of q, assuming that V and w are fixed (v = $1andw=$4). We know that: $TC(q,w,v,K)=wL+vK=4\frac{q}{2V(\mathrm{\infty })}+1(\mathrm{\infty })=0$ Therefore, the long-run total cost is 0.

Graph the long-run total cost curve for hockey stick production and analyze short-run curves

Since the long-run total cost is always 0 regardless of the quantity of hockey sticks produced, the long-run total cost curve is a horizontal line at the 0 value on the vertical axis. For the short-run curves with K values of 100, 200, and 400, we can calculate the total cost using the equation from Step 1. $TC(q,w,v,K)=wL+vK=4\frac{q}{2V(K)}+1(K)$ For K = 100: $TC(q,4,1,100)=4\frac{q}{2V(100)}+1(100)$ For K = 200: $TC(q,4,1,200)=4\frac{q}{2V(200)}+1(200)$ For K = 400: $TC(q,4,1,400)=4\frac{q}{2V(400)}+1(400)$ We can then plot these on the same graph with the long-run total cost curve. These will form a tangential "envelope" to the long-run total cost curve, illustrating how the short-run cost curves approach the long-run cost curve as K increases.

Key concepts

Short-run and Long-run Costs

Understanding the distinction between short-run and long-run costs is crucial in production theory. In the short run, at least one factor of production is fixed. In our example, the capital stock $K$ is fixed. Short-run costs, therefore, depend on adjusting the variable factor, which is labor $L$ in this context. The total cost in the short run is the sum of variable and fixed costs:

$$TC(q,w,v,K)=w\times L+v\times K$$

where $w$ is the wage rate, $L$ the amount of labor, $v$ the cost of capital, and $K$ the capital stock. Since $K$ is a constant in the short run, firms adjust labor to produce desired output levels leading to variable costs depending on the quantity produced. In contrast, the long run allows all inputs to become variable, giving the firm flexibility to choose the optimal combination of labor and capital. In this scenario, costs are minimized by adjusting both $L$ and $K$, potentially driving fixed costs to zero, as the firm can change all its resources and adapt to optimal levels for any production quantity.

Cost Minimization

Cost minimization involves finding the most cost-effective combination of input factors to produce a certain level of output. For the exercise at hand, the objective is to determine the optimal level of capital stock $K$ that minimizes total costs given production level $q$, wage rate $w$, and cost of capital $v$.To find this optimal point, we differentiate the total cost function with respect to $K$ and set the derivative equals to zero. This reveals the rate at which costs change relative to the change in capital input, helping identify the point where any increase in $K$ would no longer reduce total costs:

$$\frac{\partial TC(q,w,v,K)}{\partial K}=0$$

This mathematical method ensures that the derived level of $K$ maximizes efficiency given the conditions. In many real-world scenarios, however, other constraints, such as budget or input availability, might affect the practicality of achieving this theoretical optimal level.

Total Cost Function

The Total Cost Function represents the entire cost to produce a certain quantity of goods and is an essential part of production theory analysis. It captures both the variable and fixed costs associated with production.

In our case, the production function $q=2VKL$ provides a foundation to derive this function. The total cost can be expressed as:

$$TC(q,w,v,K)=w\times \frac{q}{2VK}+v\times K$$

This equation integrates the costs of labor ($wL$) and capital ($vK$), adjusted for the efficiency input factor governed by $V$ and the fixed nature of $K$ in the short run.

Graphically, as $K$ changes, the short-run total cost curve shifts. In the long run, with $K$ optimal and variable, the curve forms a baseline, called the long-run curve. This curve represents the minimal costs across various output levels, essentially forming an envelope that scenario points fit under when the short-run adjustments are computed for fixed $K$ levels.