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 given 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. e. Suppose the government imposed a \(\$ 3\) tax on snuffboxes. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between snuffbox buyers and sellers? g. Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short-run profits in the snuffloox industry. Why don't fixed costs enter into this computation of the change in short-run producer surplus?

Short answer

a: Individual firm: q = P - 10; Market: Q = 100P - 1000. b: P = 14, Q = 400, profit = 3 per firm. c & d: Total producer surplus = 800. e: Tax shifts supply curve, equilibrium price rises. f: Consumers bear $2, producers $1 of tax. g: Loss of surplus matches profit change excluding fixed costs.

Step-by-step solution

Determine the Individual Firm's Supply Curve

The short-run supply curve for a firm is derived from its Short-Run Marginal Cost (SMC) curve above the average variable cost. Given the SMC function \(S M C = q + 10\), this is the firm's supply curve because it reflects the marginal cost of production, which a competitive firm equates to the market price (P). Therefore, each firm's supply curve is \(q = P - 10\) when \(P \geq ATC\) (using the given cost function, assume price covers variable cost in the short run).

Market Supply Curve

There are 100 identical firms. Thus, the market supply curve is the horizontal summation of all individual firm supply curves. Therefore, if each firm supplies \(q = P - 10\), then the market supply \(Q_s = 100q = 100(P - 10)\). Simplifying gives the market supply curve: \(Q_s = 100P - 1000\).

Find Market Equilibrium

Set the demand equal to the supply to find the equilibrium. Given \(Q_d = 1100 - 50P\), set \(Q_s = Q_d\):\[100P - 1000 = 1100 - 50P\]Solving for \(P\):\[150P = 2100\]\[P = 14\]Substitute \(P = 14\) back into the demand equation to find equilibrium quantity \(Q\):\[Q = 1100 - 50 \cdot 14 = 1100 - 700 = 400\]

Calculate Individual Firm's Output and Profit

Each firm produces at \(q = P - 10 = 14 - 10 = 4\). Total cost (TC) from STC function is \(0.5\cdot 4^2 + 10 \cdot 4 + 5 = 8 + 40 + 5 = 53\). Total revenue (TR) is \(P \cdot q = 14 \cdot 4 = 56\). Profit is \(TR - TC = 56 - 53 = 3\).

Graph the Market Equilibrium

The graph would feature the supply curve \(Q_s = 100P - 1000\) and the demand curve \(Q_d = 1100 - 50P\) intersecting at \(P = 14\) and \(Q = 400\). The area above the supply curve up to the price line reflects the producer surplus.

Calculate Total Producer Surplus

Producer surplus in one firm is the difference between TR and variable costs. Total producer surplus is the sum across all firms. Since profit sums were already calculated, and since SMC starts exactly at \(q = 0\),Producer surplus = Profit + Fixed Cost = 3 per firm, and each firm has fixed costs of 5,so Each firm’s producer surplus = 3 + 5 = 8; Total producer surplus = 100 firms × 8 = 800.

Impose a Tax on Snuffboxes and Revise Equilibrium

Impose a \(\$3\) tax, modifying SMC: \(q + 10 + 3 = q + 13\). New supply \(q = P - 13\), leading to the new market supply \(Q_{s_{new}} = 100(P - 13) = 100P - 1300\). Equilibrium is found by setting \(Q_{s_{new}} = Q_d\):\[100P - 1300 = 1100 - 50P\]\[150P = 2400\]\[P = 16\] (tax-inclusive, consumer price).Substitute in \(Q_d\) for quantity \(Q = 400\). The effective producer price is \(P - 3 = 13\).

Determine the Tax Burden

With new prices, the burden of the \\(3 tax is shared among consumers, who face a price of \\)16, and producers receiving \\(13. The price increase from \\)14 to \\(16 shares the burden such that consumers pay \\)2 more, producers receive \$1 less, and effectively share the tax burden.

Calculate Loss of Producer Surplus and Relation to Profits

Firm adjusts to new profit: new firm output \(q = P - 13 = 3\). New TR = \(3 \times 13 = 39\), TC = \(0.5\cdot 3^2 + 10 \cdot 3 + 5 = 39.5\), new profit \(= 39 - 39.5 = -0.5\), loss from \(3\) to \(-0.5\). Loss in producer surplus per firm is \(2.5\), corresponding directly with tax, affirming that change in producer profits without adding fixed costs equals the change.

Key concepts

Short-run Supply Curve

The short-run supply curve in microeconomic theory represents the quantity of a good that a firm is willing to produce and sell at a given price, holding all other factors constant. This curve is crucial because it shows how much firms are willing to supply based on the prevailing market price. In the short run, the supply curve is closely related to the firm's marginal cost (MC) because it is assumed that firms will produce additional units of output as long as the marginal cost of production is below the market price.

For individual firms, like those in the handmade snuffbox industry, the supply curve is derived from the short-run marginal cost curve. The formula given, \(SMC = q + 10\), is the firm's supply curve for production above the average variable cost. This implies that each firm's supply curve starts at the point where the firm covers its variable costs, ensuring that production is profitable. In our example, each firm will produce such that \( q = P - 10 \), where \(P\) is the market price.

To find the market supply curve when there are multiple identical firms, like the 100 snuffbox firms in our problem, we sum the individual firms' supply curves horizontally. This results in the market supply curve formula: \(Q_s = 100(P - 10) = 100P - 1000\), showing the total output that all firms in the market will supply at any given price.

Equilibrium

Equilibrium in a market occurs at the price and quantity where the supply of an item perfectly matches its demand. It represents a state of balance where there are no forces causing the price or quantity to change. Determining equilibrium involves equating the supply and demand functions.

In our example with the snuffbox industry, the demand is given by \(Q_d = 1,100 - 50P\). To find the equilibrium price and quantity, we set the market demand equal to the market supply: \(100P - 1000 = 1,100 - 50P\). Solving this gives us an equilibrium price of \(P = 14\). At this price, the equilibrium quantity can be found by substituting back into the demand equation, resulting in \(Q = 400\).

This equilibrium indicates efficiency in the market, where the exact amount produced is also the amount consumed. Any deviation from this price and quantity would lead to either a surplus or a shortage, prompting adjustments back to equilibrium.

Producer Surplus

Producer surplus is a key concept in microeconomics, representing the difference between what producers are willing to accept for a good versus what they actually receive. It's essentially the net benefit that producers get from selling at the market price, over and above their minimum acceptable price.

In practical terms, producer surplus can be visualized as the area above the supply curve and below the price level, up to the quantity sold. For the snuffbox market, the total producer surplus is calculated using the profit of the rational firm plus fixed costs. In our model, each firm's profit before any tax is \(3\), and with fixed costs of \(5\), the total producer surplus per firm is \(8\). Given 100 firms, this results in a total producer surplus of \(800\).

This surplus is an important measure because it reflects the economic wellbeing of producers, indicating how much producers are benefiting from the market beyond their production costs.

Tax Incidence

Tax incidence refers to how the burden of a tax is divided between buyers and sellers. It determines the change in the economic welfare of the market participants and can shift the market equilibrium. Understanding who bears the tax burden helps evaluate government policies and their effects on the economy.

In the scenario with a \(\\(3\) tax on snuffboxes, the supply curve shifts as the tax increases the marginal cost effectively. The new supply curve is \(q = P - 13\), leading to a new equilibrium with a higher consumer price and a lower producer price. The market adjusts, so consumers end up paying more while producers receive less. With the tax, equilibrium price becomes \(P = 16\), but producers effectively get \(P = 13\) after the tax.

The incidence is shared between consumers and producers, with consumers facing a \(\\)2\) increase and producers receiving \(\$1\) less than before. This split illustrates the concept of tax incidence, showing the allocation of tax burden between stakeholders in a market.