Question

Suppose there are 1,000 identical firms producing diamonds. Let the total cost function for each firm be given by \\[ C(q, w)=q^{2}+w q \\] where \(q\) is the firm's output level and \(w\) is the wage rate of diamond cutters. a. If \(w=10\), what will be the firm's (short-run) supply curve? What is the industry's supply curve? How many diamonds will be produced at a price of 20 each? How many more diamonds would be produced at a price of \(21 ?\) b. Suppose the wages of diamond cutters depend on the total quantity of diamonds produced, and suppose the form of this relationship is given by \\[ w=0.002 Q \\] here \(Q\) represents total industry output, which is 1,000 times the output of the typical firm. In this situation, show that the firm's marginal cost (and short-run supply) curve depends on \(Q\). What is the industry supply curve? How much will be produced at a price of \(20 ?\) How much more will be produced at a price of \(21 ?\) What do you conclude about the shape of the short-run supply curve?

Short answer

Based on the step-by-step solution, a short answer to the problem would be: To find the short-run supply curve for a single firm and the industry's supply curve when \(w=10\), we first calculated the firm's marginal cost (MC) and found it to be \(2q + w\). Then, the industry's supply curve is equal to 1,000 times the firm's MC. The total number of diamonds produced at prices 20 and 21 were 5,000 and 5,500, respectively. In part b, after being given the relationship between the wage rate, \(w\), and total industry output, \(Q\), we found the firm's MC in terms of Q. We then derived the industry supply curve and calculated the quantity produced at prices 20 and 21, which were 5,000 and 5,250, respectively. Lastly, we analyzed the shape of the short-run supply curve, which was upward-sloping with a linear slope of 4.

Step-by-step solution

Find the firm's marginal cost

In order to find the firm's (short-run) supply curve, we need to find the firm's marginal cost (MC). The marginal cost is the derivative of the total cost function with respect to the output level, \(q\). The total cost function is given by \(C(q,w)=q^{2}+wq\). Taking the derivative with respect to \(q\) gives us the marginal cost: MC = \(\frac{dC(q,w)}{dq} = 2q + w\)

Find the industry's supply curve

Given that there are 1,000 identical firms producing diamonds, the industry's supply curve is the sum of the individual firms' supply, which is equal to 1,000 times the firm's marginal cost. Industry Supply Curve = \(MC \times 1000 = (2q + w) 1000\)

Calculate the number of diamonds produced at a price of 20

When the price (\(P\)) is 20 and \(w=10\), we can equate the price with the firm's marginal cost: \(P = 2q + w = 20\) Now, we can solve for the output level \(q\): \(2q = 20 - 10\) \(2q = 10\) \(q = 5\) Since there are 1,000 identical firms, the total number of diamonds produced at the price of 20 is: Total Diamonds = \(Q = 1000 \times 5 = 5000\)

Calculate the number of diamonds produced at a price of 21

When the price is 21, we can equate the price with the firm's marginal cost: \(P = 2q + w = 21\) Solving for the output level \(q\): \(2q = 21 - 10\) \(2q = 11\) \(q = 5.5\) Since there are 1,000 identical firms, the total number of diamonds produced at the price of 21 is: Total Diamonds = \(Q = 1000 \times 5.5 = 5500\)

Calculate the firm's marginal cost with respect to Q

For part b, we are given the relationship between the wage rate, \(w\), and total industry output, \(Q\) as \(w=0.002Q\). Since \(Q = 1000 \times q\), we can write the wage rate as a function of \(q\): \(w = 0.002(1000q) = 2q\) Now, we can rewrite the firm's marginal cost function in terms of \(Q\): MC = \(2q + 2q\) MC = \(4q\)

Find the industry supply curve

The industry supply curve is the sum of the individual firms' supply, which is equal to 1,000 times the firm's marginal cost: Industry Supply Curve = \(MC \times 1000 = (4q) 1000\)

Calculate the quantity produced at prices 20 and 21

We can equate the price with the firm's marginal cost and find the quantities produced at prices 20 and 21: At \(P = 20\): \(20 = 4q \implies q = 5\) Total Diamonds = \(Q = 1000 \times 5 = 5000\) At \(P = 21\): \(21 = 4q \implies q = 5.25\) Total Diamonds = \(Q = 1000 \times 5.25 = 5250\)

Analyze the shape of the short-run supply curve

From the calculations, we can conclude that the short-run supply curve is upward-sloping. As the price increases, the quantity of diamonds supplied also increases. The shape of the short-run supply curve is linear, with a slope of 4.

Key concepts

Total Cost Function

Understanding the total cost function is fundamental in economics as it helps determine a firm's production expenses at different output levels. In the scenario given, the total cost function for a firm producing diamonds is expressed as \( C(q, w) = q^2 + wq \). Here, \( q \) is the quantity of diamonds produced by the firm, while \( w \) is the wage rate paid to diamond cutters.

The quadratic term \( q^2 \) represents the variable costs that increase with the square of the output level, capturing elements like the cost of raw materials which can increase exponentially with production. The linear term \( wq \) reflects the direct proportionality of wages to the quantity produced. Understanding this function allows a firm to compute the marginal cost, which is the increase in total cost when one additional unit of output is produced.

Short-Run Supply Curve

The short-run supply curve is crucial in representing how much of a good a firm is willing to offer at different price levels, keeping some inputs fixed due to short-term constraints. The firm’s supply curve in the exercise is derived from calculating the marginal cost (MC), which is the increase in total costs associated with a one-unit increase in production.

The relationship between marginal cost and the supply curve is direct because firms offer more goods for sale when the price is higher than the marginal cost. For the given cost function, the MC is computed as \( MC = \frac{dC(q, w)}{dq} = 2q + w \), which is a key determinant of a firm's short-run supply decision. When multiple identical firms are in the market, the industry supply curve can be constructed by aggregating individual supply curves, indicating how the entire market responds to price changes in the short run.

Industry Output

Industry output is a measure of the total production within a specific industry. It is the aggregate of all outputs from the individual firms. In our diamond industry example, industry output is symbolized as \( Q \), and since all firms are identical, \( Q \) is simply 1,000 times the output of a typical firm.

When analyzing industry output in relation to the wage rate or input prices, it's important to consider how increased production can affect these variables. For instance, if wage rates depend on output levels, this interdependence can impact firms' production decisions since it affects the total cost and, consequently, the supply curve of the industry as a whole. Therefore, the concept of industry output is intertwined with market supply and the equilibrium price levels.

Wage Rate

The wage rate is a critical factor in the production process as it constitutes the payment to labor, which is one of the key inputs into production. In our case, the wage rate \( w \) plays a direct role in the total cost function \( C(q, w) \) of diamond-cutting firms, illustrating how changes in wages impact production costs and supply decisions.

In many instances, the wage rate is considered fixed, but in our more complex scenario, it varies with the industry output. We're given that the wage rate for diamond cutters is determined by the formula \( w=0.002Q \), which means that as industry output increases, the wage rate increases linearly. This dynamic characteristic of the wage rate highlights the interrelatedness of labor costs and market supply, shaping the nature of the short-run supply curve and affecting how much firms are willing to produce at different price points.