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Chapter 12: Problem 3

A perfectly competitive market has 1,000 firms. In the very short run, cach of the firms has a fixed supply of 100 units. The market demand is given by \\[Q=160,000-10,000 P.\\] a. Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one firm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that, in the short run, each firm has a supply curve that shows the quantity the firm will supply \(\left(q_{i}\right)\) as a function of market price. The specific form of this supply curve is given by \\[q_{i}=-200+50 P.\\] Using this short-run supply response, supply revised answers to (a)-(d).

Short Answer

Expert verified
Equilibrium price is 6; firm demand is perfectly elastic at 6. Short-run and very-short-run prices remain about 6 with elasticity -0.6 (industry) and infinity (firm).

Step by step solution

01

Calculate total market supply in the very short run

Each firm supplies 100 units and there are 1,000 firms, so the total market supply is calculated as: \( Q_s = 1,000 \times 100 = 100,000 \text{ units} \).
02

Establish the equilibrium condition

In equilibrium, the total market supply must equal the total market demand. We set the supply equal to the demand equation: \( 100,000 = 160,000 - 10,000P \).
03

Solve for equilibrium price in the very short run

Subtract 160,000 from both sides to get: \( -60,000 = -10,000P \). Then, solve for \( P \) by dividing both sides by \(-10,000\): \( P = 6 \). So, the equilibrium price is 6.
04

Determine the demand schedule for one firm

In a perfectly competitive market, individual firms are price takers. At equilibrium price \( P = 6 \), each firm faces a perfectly elastic demand curve at this price, with demand \( q_i = 100 \) if they choose to supply 100 units.
05

Analyze scenarios with one firm changing supply

If one seller decides to sell nothing, the supply becomes \( Q_s = 100,000 - 100 = 99,900 \) units. If one seller decides to sell 200 units, supply becomes \( Q_s = 100,000 + 100 = 100,100 \). For very small changes, market price remains close to the equilibrium price of \( P = 6 \) due to large total market supply.
06

Calculate elasticity of industry demand curve

At the equilibrium point \((P = 6, Q = 100,000)\), the derivative of the demand curve with respect to price is \(-10,000\). Thus, elasticity is \( E_d = (P/Q)(dQ/dP) = (6/100,000)\times (-10,000) = -0.6 \).
07

Calculate elasticity of demand facing one seller

For a single seller, the demand curve is perfectly elastic (horizontal) at the market price \( P = 6 \), making the elasticity infinity.
08

Short-run market supply with supply curve per firm

Using \(q_i = -200 + 50P\), the total market supply is \( Q_s = 1,000(-200 + 50P) = -200,000 + 50,000P \).
09

Solve for equilibrium price in the short run

Set short-run supply equal to demand: \(-200,000 + 50,000P = 160,000 - 10,000P\). Solving, we get: \( 60,000P = 360,000 \), so \( P = 6 \).
10

Short-run scenarios with one seller changing supply

Changes in supply by one seller translate differently in short-run versus very short run. However, individually negligible impact in a large market keeps \( P \) roughly constant at 6.
11

Short-run revised elasticity calculations

Industry demand elasticity remains \(-0.6\), and the demand curve for an individual seller remains perfectly elastic at \( P = 6 \), regardless of short-run changes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilibrium Price
The equilibrium price in a market is where the total market supply equals the total market demand. In this exercise, we're working with a perfectly competitive market consisting of 1,000 firms. Each firm initially supplies 100 units in the very short run, resulting in a total market supply of 100,000 units.
To find the equilibrium price, we substitute this supply into the demand equation given by \( Q = 160,000 - 10,000 P \). By setting the supply equal to the demand, we solve for \( P \) as follows:
  • Subtract 160,000 from both sides: \( -60,000 = -10,000 P \)
  • Divide both sides by -10,000 to get: \( P = 6 \)
Thus, the equilibrium price in the very short run is 6 units of currency.
Short Run Supply Curve
In the short run, firms have more flexibility to adjust their output in response to market conditions. Each firm now supplies according to the equation \( q_i = -200 + 50P \). This represents a scenario where the supply is dependent on the price, allowing firms to change their output when the market price changes.
For the entire market, the short run supply curve becomes:
  • \( Q_s = 1,000(-200 + 50P) = -200,000 + 50,000 P \)
To find the new short-run equilibrium price, we set the short-run supply equation equal to the demand equation:
  • \( -200,000 + 50,000 P = 160,000 - 10,000 P \)
  • By simplifying, we find \( 60,000 P = 360,000 \), giving us an equilibrium price \( P = 6 \)
Thus, under short-run conditions, the equilibrium price remains steady at 6 despite the more flexible supply.
Market Demand
Market demand indicates the total quantity of a good that all buyers in the market are willing and able to purchase at each price level. Here, the demand equation is given by \( Q = 160,000 - 10,000 P \). It shows how demand decreases as the price increases.
By understanding the demand curve, firms in a competitive market can anticipate how changes in price may affect the quantity of goods sold.

How does demand affect market equilibrium?

In equilibrium, demand and supply are equal, stabilizing the market price. If, for example, the demand increases due to external factors, the equilibrium price might rise unless matched by a similar supply increase. This balance ensures that prices remain stable and allocates resources efficiently in competitive markets.
Elasticity of Demand
Elasticity of demand refers to how sensitive the quantity demanded of a good is to a change in price. In this scenario, the elasticity of the industry demand curve is calculated at the equilibrium point \((P = 6, Q = 100,000)\).
The derivative of the demand curve with respect to price is \(-10,000\). Using the elasticity formula \( E_d = (P/Q)(dQ/dP) \), we compute:
  • \( E_d = (6/100,000) \times (-10,000) = -0.6 \)
This negative elasticity indicates that demand is relatively inelastic around the equilibrium point. For any single seller in a perfectly competitive market, however, the demand curve they face is perfectly elastic (horizontal) at the market price of 6, implying an infinite elasticity. This reflects the idea that a single firm cannot influence market prices and must accept the prevailing market conditions.

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Most popular questions from this chapter

A perfcctly competitive industry has a large number of potential entrants. Each firm has an identical cost structure such that long-run average cost is minimized at an output of 20 units \(\left(q_{i}=20\right) .\) The minimum average cost is \(\$ 10\) per unit. Total market demand is given by \\[Q=1,500-50 P.\\] a. What is the industry's long-run supply schedule? b. What is the long-run equilibrium price \(\left(P^{*}\right)\) ? The total industry output \(\left(Q^{*}\right)\) ? The output of each firm \(\left(q^{*}\right)\) ? The number of firms? The profits of each firm? c. The short-run total cost function associated with each firm's long-run equilibrium output is given by \\[C(q)=0.5 q^{2}-10 q+200.\\] Calculate the short-run average and marginal cost function. At what output level does shortrun average cost reach a minimum? d. Calculate the short-run supply function for each firm and the industry short-run supply function. c. Suppose now that the market demand function shifts upward to \(Q=2,000-50 P\). Using this new demand curve, answer part (b) for the very short run when firms cannot change their outputs. f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry?

Suppose that the demand for stilts is given by \\[Q=1,500-50 P\\] and that the long-run total operating costs of each stilt-making firm in a competitive industry are given by \\[C(q)=0.5 q^{2}-10 q.\\] Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by \\[Q_{s}=0.25 w,\\] where \(w\) is the annual wage paid. Suppose also that each stilt-making firm requires one (and only one) entrepreneur (hence, the quantity of entrepreneurs hired is equal to the number of firms). Long-run total costs for each firm are then given by \\[C(\boldsymbol{q}, \boldsymbol{w})=0.5 q^{2}-10 q+w.\\] a. What is the long-run equilibrium quantity of stilts produced? How many stilts are produced by each firm? What is the long-run equilibrium price of stilts? How many firms will there be? How many entrepreneurs will be hired, and what is their wage? b. Suppose that the demand for stilts shifts outward to \\[Q=2,428-50 P.\\] How would you now answer the questions posed in part (a)? c. Because stilt-making entrepreneurs are the cause of the upward-sioping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the increase in rents between parts (a) and (b). Show that this value is identical to the change in long-run producer surplus as measured along the stilt supply curve.

The domestic demand for portable radios is given by \\[Q=5,000-100 P,\\] where price \((P)\) is measured in dollars and quantity \((Q)\) is measured in thousands of radios per year. The domestic supply curve for radios is given by $$Q=150 P.$$ a. What is the domestic equilibrium in the portable radio market? b. Suppose portable radios can be imported at a world price of $$\$ 10$$ per radio. If trade were unencumbered, what would the new market equilibrium be? How many portable radios would be imported? c. If domestic portable radio producers succeeded in having a $$\$ 5$$ tariff implemented, how would this change the market equilibrium? How much would be collected in tariff revenues? How much consumer surplus would be transferred to domestic producers? What would the deadweight loss from the tariff be? d. How would your results from part (c) be changed if the government reached an agreement with foreign suppliers to "voluntarily" limit the portable radios they export to \(1,250,000\) per year? Explain how this differs from the case of a tariff.

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?

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?
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