Question

Assume the quantity of envelopes licked per hour by Sticky Gums, Inc., is \(Q=10,000 \sqrt{L}\) where \(L\) is the number of laborers hired per hour by the firm. Assume further that the envelope-licking business is perfectly competitive with a market price of \(\$ .01\) per envelope. a. How much labor would be hired at a competitive wage of \(\$ 10\) ? \(\$ 5\) ? \(\$ 2\) ? Use your results to sketch a demand curve for labor. b. Assume that Sticky Gums hires its labor at an hourly wage of \(\$ 10 .\) What quantity of envelopes will be licked when the price of a licked envelope is \(\$ .10, \$ .05, \$ .02 ?\) Use your results to sketch a supply curve for licked envelopes.

Short answer

Answer: The labor demand curve indicates that when the hourly wage is $10, the firm hires 25 laborers; when the wage is $5, the firm hires 100 laborers; and when the wage is $2, the firm hires 625 laborers. At a labor wage of $10 per hour, the quantity of envelopes licked remains constant at 50,000 for all given market prices ($0.10, $0.05, and $0.02 per envelope).

Step-by-step solution

Calculate Labor Demand for Given Competitive Wages

To find the labor demand for each given competitive wage, we first need to determine the profit-maximizing quantity of labor that the firm would hire. As the firm is in a perfectly competitive market, its total revenue is given by the price per envelope multiplied by the total quantity of envelopes licked. The total revenue will be: $$ TR = P \times Q $$ With P being the envelope price (\(\$.01\)), the total revenue function becomes: $$ TR = 0.01 \times 10,000 \sqrt{L} $$ Labor costs must also be calculated. The total cost of hiring labor depends on both the hourly wage (W) and the number of laborers (L). For the given hourly wages, the labor cost (C) can be represented as: $$ C = WL $$ The firm's profit function will be the difference between total revenue and labor costs: $$ \pi = TR - C $$ Substituting the total revenue and labor cost functions into the profit function, we get: $$ \pi = 0.01 \times 10,000 \sqrt{L} - WL $$ To find the optimal quantity of labor, we take the first derivative of the profit function with respect to L and set it equal to zero: $$ \frac{d\pi}{dL} = 0 $$

Calculating Optimal Labor

We will calculate the optimal labor for each wage separately. Case 1: The wage is \(\$10\) per hour. $$ \pi = 100 \sqrt{L} - 10L $$ $$ \frac{d\pi}{dL} = \frac{100}{2\sqrt{L}} - 10 $$ Setting the derivative equal to zero, to find the optimal labor: $$ \frac{100}{2\sqrt{L}} - 10 = 0 $$ $$ L = 25 $$ Case 2: The wage is \(\$5\) per hour. $$ \pi = 100 \sqrt{L} - 5L $$ $$ \frac{d\pi}{dL} = \frac{100}{2\sqrt{L}} - 5 $$ Setting the derivative equal to zero, to find the optimal labor: $$ \frac{100}{2\sqrt{L}} - 5 = 0 $$ $$ L = 100 $$ Case 3: The wage is \(\$2\) per hour. $$ \pi = 100 \sqrt{L} - 2L $$ $$ \frac{d\pi}{dL} = \frac{100}{2\sqrt{L}} - 2 $$ Setting the derivative equal to zero, to find the optimal labor: $$ \frac{100}{2\sqrt{L}} - 2 = 0 $$ $$ L = 625 $$ To sketch the labor demand curve, plot the optimal labor values against the corresponding wage rates.

Calculate the Quantity of Envelopes Licked for Given Market Prices and Labor Wage \(\$10\)

We will now calculate the total quantity of licked envelopes produced by Sticky Gums, Inc., for each given market price when the labor wage is \(\$10\) per hour. At a wage of \(\$10\) per hour, the profit-maximizing quantity of labor is \(L=25\). Case 1: The market price is \(\$.10\) per envelope. $$ Q = 10,000\sqrt{L} = 10,000\sqrt{25} = 50,000 $$ Case 2: The market price is \(\$.05\) per envelope. $$ Q = 10,000\sqrt{L} = 10,000\sqrt{25} = 50,000 $$ Case 3: The market price is \(\$.02\) per envelope. $$ Q = 10,000\sqrt{L} = 10,000\sqrt{25} = 50,000 $$ At a labor wage of \(\$10\) per hour, the quantity of envelopes licked remains constant at \(50,000\) for all the given prices. To sketch the supply curve for licked envelopes, plot the quantity of envelopes licked for each market price.

Key concepts

Understanding the Competitive Market

In economic theory, a competitive market is a concept that describes a market structure where many firms offer products that are similar and none of them has significant market power to influence prices on their own. The price in a competitive market is determined by the intersection of the market demand curve and supply curve.

In the case of Sticky Gums, Inc., the company operates in a perfectly competitive envelope-licking business. This means that the firm is a price taker - it must accept the market price for licked envelopes. Given this environment, the quantity of labor that the firm decides to employ is aimed at profit maximization, considering the fixed market prices for both labor (wages) and the output (price per licked envelope). The interaction among multiple firms in such a market leads to the most efficient allocation of resources, where consumer demand is met at the lowest possible cost.

Aim for Profit Maximization

When it comes to profit maximization, firms in competitive markets aim to maximize the difference between their total revenues and total costs. The key to understanding this concept is examining how changing the input, such as labor in our current example, affects the firm’s profits.

Calculating Optimal Labor Quantity

By taking the derivative of the profit function concerning labor and finding where this derivative equals zero, Sticky Gums, Inc. can determine the optimal number of laborers to hire to maximize profits. In the exercise, we see different scenarios based on various wages that reflect the firm’s responsiveness to labor costs. The goal is always to adjust inputs to reach the point where the cost of the last unit of input equals the revenue it generates, which is known as the profit-maximizing condition.

The Role of the Supply Curve

The supply curve shows the relationship between the price of a product or service and the quantity of it that a firm is willing to supply. In the provided exercise, the supply curve for licked envelopes is determined by the quantity of output that Sticky Gums, Inc. can produce, which is dependent on the labor employed.

Supply Curve Dynamics

The supply curve is typically upward sloping, meaning that as prices rise, the quantity supplied also increases. It's shaped by the cost of production, which includes labor. By calculating the quantity of licked envelopes at different prices while keeping the wage at $10, we observe that the supply of envelopes remains constant, showcasing a vertical supply curve in this specific scenario. This unusual circumstance is due to the exercise's assumption that the quantity of labor does not change with the price of the product, which is a simplification for the sake of learning the basic concepts.