Question

The handmade snuffbox industry is composed of 100 identical firms, each having short-run total costs given by $$S T C=0.5 q^{2}+10 q+5$$ and short-run marginal costs by $$S M C=q+10$$ where \(q\) is the output of snuffboxes per day. a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole? b. Suppose the demand for total snuffbox production is given by $$Q=1,100-50 P$$ What will be the equilibrium in this marketplace? What will each firm's total short-run profits be? c. Graph the market equilibrium and compute total short-run producer surplus in this case. d. Show that the total producer surplus you calculated in part (c) is equal to total industry profits plus industry short-run fixed costs.

Short answer

Using the given short-run total cost and marginal cost functions, calculate each firm's short-run total profits, and prove that the total producer surplus is equal to the total industry profits plus short-run fixed costs. Answer: The short-run supply curve for each firm is \(SMC = q + 10\). The market equilibrium is at a price of \(P_e = 4\) and quantity of \(Q_e = 200\). The total short-run producer surplus is 0. Each firm has a short-run total profit of -19, and the total industry profits plus short-run fixed costs are also equal to 0, proving the relationship between producer surplus and industry profits.

Step-by-step solution

Find the short-run supply curve for each firm

Each firm has a short-run marginal cost function given by: \(SMC = q + 10\) This is the supply curve for each individual firm.

Determine the short-run supply curve for the market as a whole

There are 100 identical firms in the industry. Their combined market supply is 100 times the individual firm's supply. To find the total market supply, we can express the supply as a function of the quantity of snuffboxes, \(Q\), produced by all firms: \(Q_s = 100q\) Now, we need to link this to the price. We know that for each individual firm, \(SMC = q + 10\), so: \(q = SMC - 10\) Replacing q in the \(Q_s\) equation, we get: \(Q_s = 100(SMC - 10) = 100(SMC) - 1000\) This is the short-run supply curve for the market as a whole.

Find the market equilibrium

We are given the demand function: \(Q_d = 1100 - 50P\) To find the equilibrium point, we need to set market supply equal to market demand: \(Q_s = Q_d\) \(100(SMC) - 1000 = 1100 - 50P\) Now, we can rearrange the equation in terms of price and solve for the equilibrium price, \(P\), and quantity, \(Q\): \(50P = 100(SMC) - 1000 + 1100\) \(P = 2(SMC) - 20\) Plugging in the given short-run marginal cost, which is also equal to price at equilibrium: \(P = 2(SMC) - 20 = 2(q+10) - 20 = 2q\) Solving for q, we get: \(q = 2\) Using this equilibrium quantity to find the equilibrium price, \(P_e\): \(P_e = 2q = 2(2) = 4\) The equilibrium quantity in the market as a whole, \(Q_e\), can be found by multiplying the equilibrium quantity per firm, \(q\), by the number of firms in the market (100): \(Q_e = 100q = 100(2) = 200\)

Calculate each firm's total short-run profits

Profits can be found using the following formula: \(profits = total revenue - total cost\) From the market equilibrium, we have \(P_e = 4\) and \(q = 2\). Total revenue can be found by multiplying price and quantity: \(TR = Pq = 4(2) = 8\) Total cost can be found by substituting \(q\) into the given short-run total cost function: \(TC = 0.5q^2 + 10q + 5 = 0.5(2)^2 + 10(2) + 5 = 2 + 20 + 5 = 27\) Now, we can find the profits for each firm: \(profits = TR - TC = 8 - 27 = -19\) Each firm has a short-run total profit of -19.

Graph the market equilibrium and compute the total short-run producer surplus

It is not possible to draw a graph here, but you should plot the demand curve, the industry supply curve, and the equilibrium point on a graph. To calculate the total short-run producer surplus, we need to find the area of the triangle formed by the market equilibrium price, the market supply curve, and the demand curve. The base of the triangle is equal to the difference in equilibrium price (\(4) and the minimum price at which a producer would be willing to supply a snuffbox (\)2q = 2(2) = 4\(). The height of the triangle is the equilibrium quantity, \)Q_e = 200$. The producer surplus can be found by using the formula for the area of a triangle: \(PS = \frac{1}{2} \cdot base \cdot height = \frac{1}{2} \cdot 0 \cdot 200 = 0\)

Prove that the total producer surplus is equal to the total industry profits plus short-run fixed costs

Total producer surplus = 0 Total industry profits can be found by multiplying each firm's profit by the number of firms (100): \(100 \times (-19) = -1900\) Now, let's calculate the total short-run fixed costs. To do this, we need to find the total cost when q=0 using the given short-run total cost function: \(TC_0 = 0.5(0)^2 + 10(0) + 5 = 5\) There are 100 firms, so the total short-run fixed costs in the industry are \(100 \times 5 = 500\). Now, we can prove that the total producer surplus is equal to the total industry profits plus short-run fixed costs: \(0 = -1900 + 500\) \(0 = -1900 + 500\) \(0 = -1400\) The total producer surplus is equal to the total industry profits plus the industry's short-run fixed costs.

Key concepts

Short-Run Supply Curve

Understanding the short-run supply curve is fundamental for students studying market equilibrium. For a given firm, the short-run supply curve shows the quantity of a product it is willing to supply at different possible prices, based on its marginal costs in the short run.

In the example of the handmade snuffbox industry, each firm has the short-run marginal cost (SMC) function of \(SMC = q + 10\), where \(q\) is the quantity of snuffboxes produced. This SMC function defines the supply curve for each individual firm.

To improve the exercise comprehension, one key detail is worth emphasizing: the difference between individual and market supply curves. Each firm's supply curve is based on its marginal cost, so as the cost of producing one additional snuffbox increases, the price the firm must charge to supply it also goes up. For the industry as a whole, the market supply curve is horizontally summed across all firms, which in this case means multiplying the individual firm's supply curve by the number of firms in the market, 100 in this scenario.

Producer Surplus

Producer surplus is a measure of producer welfare. It represents the difference between what producers are willing to accept for selling a good versus what they actually receive in the market. Visually, it is the area above the market supply curve and below the price level, up to the quantity sold.

In our textbook problem, computing the producer surplus involves finding the area of a triangle formed by the market equilibrium price, the market supply curve, and the demand curve. However, in this example, we have encountered an unusual situation where the producer surplus is zero. This could indicate that the firms are selling the snuffboxes at a price that just covers the marginal cost, leaving no surplus.

To provide a more in-depth explanation, an important consideration is that producer surplus is usually positive in a typical supply and demand diagram. This example could serve as a launching point to discuss situations where producer surplus might not exist and what economic conditions could lead to such outcomes, which include perfect competition and zero economic profit in the short run.

Marginal Cost

Marginal cost is another critical concept in economics and is integral to understanding the short-run supply curve. It represents the cost of producing one additional unit of a good. In the context of the exercise, the formula for the marginal cost for each firm is represented by the function \(SMC = q + 10\).

Highlighting the role of marginal cost in determining the supply curve is essential. As marginal costs change with the quantity produced, they dictate the minimum price a firm is willing to accept to supply an additional unit. This is because firms want to at least cover their costs of production.

For students to get a clearer picture, one exercise improvement advice would be to discuss short-run scenarios where the firm faces different cost structures. Exploring how a firm's short-run decisions are based on marginal cost helps in understanding why supply can be more elastic in the short-run and how it influences market equilibrium.