Question

Consider a city that has a number of hot dog stands operating throughout the downtown area. Suppose that each vendor has a marginal cost of \(\$ 1.50\) per hot dog sold and no fixed cost. Suppose the maximum number of hot dogs that any one vendor can sell is 100 per day. a. If the price of a hot dog is \(\$ 2,\) how many hot dogs does each vendor want to sell? b. If the industry is perfectly competitive, will the price remain at \(\$ 2\) for a hot dog? If not, what will the price be? c. If each vendor sells exactly 100 hot dogs a day and the demand for hot dogs from vendors in the city is \(Q=4400-1200 P,\) how many vendors are there? d. Suppose the city decides to regulate hot dog vendors by issuing permits. If the city issues only 20 permits and if each vendor continues to sell 100 hot dogs a day, what price will a hot dog sell for? e. Suppose the city decides to sell the permits. What is the highest price that a vendor would pay for a permit?

Short answer

a. Each vendor wants to sell 100 hot dogs at a price of $2. \nb. In a perfectly competitive market, the price will fall to $1.5. \nc. There are approximately 20 vendors in the city. \nd. The price of a hot dog will rise to $2 after the city issues 20 permits. \ne. The highest price a vendor would pay for a permit would be $50.

Step-by-step solution

Calculate Quantity at Given Price

In a perfectly competitive market, vendors will sell hot dogs until their marginal cost equals to the price they receive. So, at a price of \( $2 \) per hot dog, given the marginal cost of \( $1.50 \), each vendor would want to sell as many hot dogs as they can, which is 100 hot dogs a day, so that they can maximize their profit.

Determine the Price in a Perfectly Competitive Market

In a perfectly competitive market, the price will be equal to the marginal cost, otherwise new vendors would enter the market and increase the supply, thus bringing down the price. Therefore, the price of a hot dog would drop to \( $1.5 \) in a perfectly competitive market.

Calculate the Number of Vendors

Using the given demand equation, \( Q = 4400 - 1200P \), we substitute \( P=$1.5 \) and \( Q=100 \)hot dogs per vendor into the equation to calculate the number of vendors. We get approximately 20 vendors in the city.

Determine the New Price After Permit Issuing

Given that the city issues only 20 permits and each vendor continues to sell 100 hot dogs a day, the number of hot dogs sold remains at 2000. Using the demand equation \( Q = 4400 - 1200P \) and substituting \( Q=2000 \), we can calculate the new price that a hot dog will sell for. The price will be approximately \( $2 \).

Calculate the Highest Price for a Permit

The highest price a vendor would pay for a permit should be no more than the profit they would earn from selling hot dogs. The profit per vendor is \( (P - MC) * Quantity \), substituting the values \( P=$2, MC=$1.5, Q=100 \), the profit would be \( $50 \). Therefore, the highest price a vendor would pay for a permit would be \( $50 \).

Key concepts

Marginal Cost

Marginal cost is a crucial concept in understanding how businesses make decisions regarding the quantity of goods they produce. It represents the additional cost incurred by producing one more unit of a good. In the context of a perfectly competitive market, each vendor operates under the pressure to minimize costs and maximize profits.
To better understand marginal cost, consider the hot dog stand exercise. Each vendor has a marginal cost of \( \\(1.50 \) per hot dog. This means, for every additional hot dog a vendor chooses to sell, it costs them \( \\)1.50 \).

• The vendor's objective is to produce and sell hot dogs as long as the price they receive is equal to or greater than the marginal cost.
• To maximize profit, vendors will keep increasing output until the cost of producing an extra hot dog equals the price they sell it for.

This decision-making process ensures that resources are used efficiently and helps vendors in perfectly competitive markets stay in business.

Demand Equation

The demand equation is a mathematical expression used to describe the relationship between the quantity of a good that consumers are willing to buy and the price of that good. It is a crucial tool for vendors as it helps them understand how market conditions affect sales.
In the hot dog market scenario, the demand equation is given by \( Q=4400-1200P \). The letter \( Q \) represents the quantity of hot dogs demanded, while \( P \) represents the price of a hot dog.

• This equation shows that as the price of hot dogs increases, the quantity demanded decreases.
• Conversely, a lower price results in a higher demand.

By using this demand equation, vendors can forecast how many hot dogs they are likely to sell at different price points. It allows them to strategize effectively, knowing that price changes directly influence consumer buying behavior.

Market Price

The market price in a perfectly competitive market is determined by the intersection of supply and demand. Vendors in this setting are price takers; they accept the market price as it is because no single vendor can influence it.
From the exercise, at first, the market price was \( \\(2 \) per hot dog. However, in a perfectly competitive milieu, this price would adjust to the marginal cost, which is \( \\)1.50 \).

• When the price is above the marginal cost, new vendors enter the market, increasing supply and driving down the price.
• If the price drops below the marginal cost, vendors will exit the market, reducing supply and pushing the price back up.

Thus, the market naturally gravitates towards equilibrium, where the quantity supplied equals the quantity demanded, and the price stabilizes at the profit-maximizing level.

Profit Maximization

Profit maximization is the goal of every vendor, especially in competitive markets. It involves adjusting output to ensure costs remain lower than revenues, thus maximizing financial returns.
For the hot dog vendors, maximizing profit involves producing up to the point where the price received for each hot dog equals the marginal cost of \( \\(1.50 \). At this equilibrium price, each vendor is selling 100 hot dogs per day, as this amount optimizes their profit given the constraints.
When permits limit the number of vendors, the situation changes. With only 20 permits, the reduced number of vendors selling 100 hot dogs each leads to a different market dynamic.

• With restricted supply due to permits (2000 hot dogs total), the price adjusts to \( \\)2 \).
• The profit per vendor becomes \( (P - MC) \times Q = (2 - 1.5) \times 100 = \$50 \).

Thus, the permit system influences market dynamics by altering supply equilibrium and thus affects prices and vendor profitability.