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 cach? 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

a) Firm's supply: \( q = \frac{p-10}{2} \); Industry supply: \( Q = 500p-5000 \). At price 20: 5000 diamonds, additional at 21 p: 500. b) Industry supply: \( Q = 250p \) with wage dependency. At price 20: 5000; additional at 21 p: 250. The supply curve is less elastic when wages depend on output.

Step-by-step solution

Determine the Individual Firm's Short-Run Supply Curve for w=10

The supply curve in the short run for a firm is determined by its marginal cost (MC) curve above the average variable cost (AVC). First, find the MC by differentiating the cost function: \[ MC = \frac{dC(q, w)}{dq} = 2q + w. \]Substitute \( w=10 \):\[ MC = 2q + 10. \]So, the firm's supply curve is \( p = 2q + 10 \). Rearrange to express \( q \) as a function of \( p \):\[ q = \frac{p - 10}{2}. \]

Determine the Industry's Supply Curve at w=10

For the industry supply curve, multiply the individual firm’s supply by the number of firms (1000):\[ Q = 1000 \cdot q = 1000 \cdot \frac{p - 10}{2} = 500(p - 10). \]Thus, the industry supply curve can be written as:\[ Q = 500p - 5000. \]

Calculate Industry Output at Price 20 for w=10

Use the industry supply curve to find \( Q \) when \( p = 20 \):\[ Q = 500 \cdot 20 - 5000 = 10000 - 5000 = 5000. \]So, 5000 diamonds are produced at price 20.

Calculate Additional Diamonds Produced at Price 21 for w=10

Find \( Q \) at \( p = 21 \) and subtract the quantity at \( p = 20 \):\[ Q = 500 \cdot 21 - 5000 = 10500 - 5000 = 5500. \]So, 500 more diamonds are produced when the price increases to 21.

Express Wages as a Function of Industry Output

Given \( w = 0.002Q \), substitute this into the firm’s MC equation:\[ MC = 2q + 0.002Q. \]

Derive Firm's Short-Run Supply Curve with Variable Wages

Set firm's marginal cost equal to price to derive short-run supply curve:\[ p = 2q + 0.002Q. \]

Derive Industry Supply Curve with Variable Wages

Express total industry output \( Q \) in terms of price:Using the relationship \( Q = 1000q \) and substituting for \( q \), solve:\[ Q = 1000 \left( \frac{p - 0.002Q}{2} \right), \]\[ Q = 500p - Q. \]Rearrange to find \( Q \):\[ 2Q = 500p \]\[ Q = 250p. \]This is the industry's supply curve with wages dependent on industry output.

Calculate Industry Output at Price 20 with Variable Wages

Use the new industry supply curve to find \( Q \) for \( p = 20 \):\[ Q = 250 \cdot 20 = 5000. \]So, 5000 diamonds are produced at price 20.

Calculate Additional Diamonds Produced at Price 21 with Variable Wages

Find \( Q \) at \( p = 21 \) using the new supply curve:\[ Q = 250 \cdot 21 = 5250. \]So, 250 more diamonds are produced when the price increases to 21, indicating the supply curve is less responsive.

Key concepts

Supply Curve

In microeconomic theory, the supply curve is a fundamental concept that illustrates the relationship between the price of a good and the quantity supplied by producers. The curve typically slopes upward, indicating that as prices increase, producers are willing and able to supply more of the good. This occurs because higher prices can justify the higher marginal costs involved in producing additional units.

When we look at a firm, its supply curve is derived from its marginal cost curve, specifically the part above the average variable cost. For example, if a diamond-producing firm has a cost function with output \( q \) and a fixed wage \( w \), such as \( C(q, w) = q^2 + wq \), the firm’s supply curve can be expressed by setting the price \( p \) equal to the marginal cost \( MC \).

Thus, the firm's supply curve becomes an equation where price changes lead directly to changes in output, reflecting the supply curve's foundational concept in economics.

Marginal Cost

Marginal cost is the additional cost incurred by producing one more unit of a good. It's an essential part of a firm's decision-making because it influences the supply curve and determines the quantity the firm will produce at different price points.

To calculate the marginal cost, we differentiate the total cost function with respect to quantity. For our diamond-producing firm, this involves differentiating \( C(q, w) = q^2 + wq \) with respect to \( q \). The result, \( MC = 2q + w \), shows how the cost of producing an additional diamond depends on both \( q \) and \( w \).

By understanding marginal cost, firms can set their production levels to ensure they are maximizing their profits. They aim to produce up to the point where the price of the good equals the marginal cost.

Short-run Industry Supply

The short-run industry supply is the aggregate of individual firms’ supply curves within a market. In this context, it represents the total quantity of a good that all producers in the industry are willing to supply at each price level within the short period.

For many identical firms, like the 1,000 diamond producers mentioned, the industry supply curve is constructed by multiplying the individual firm's supply curve by the number of firms. For instance, if each firm’s supply curve is given by \( q = \frac{p - 10}{2} \), then the industry supply, \( Q \), is \( 1000 \times q = 500(p - 10) \).

This means that changes in the price affect the short-run supply of the entire industry, as depicted in the industry supply curve \( Q = 500p - 5000 \). Understanding this helps determine how the industry’s total output responds to price changes and wage variations in the short run.

Production Theory

Production theory revolves around understanding how goods and services are created using input factors within the economy. It examines the process that transforms factor inputs like labor and capital into outputs of goods and services, focusing on efficiency, costs, and output levels.

In microeconomics, understanding production functions (like \( C(q, w) = q^2 + wq \)) allows firms to determine the optimal level of production. These functions help in analyzing the impact of input costs, like wages, on the total cost of production.

By analyzing these aspects, production theory provides valuable insights into how firms can operate efficiently to maximize profits. This includes choosing the right combination of inputs and calculating costs and output levels that align with market demand and pricing strategies.