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Chapter 14: Problem 4

Suppose the demand for frisbees is given by \\[ g=100-2 P \\] and the supply by \\[ Q=20+6 P \\] a What will be the equilibrium price and quantities for frisbees? b. Suppose the government levies a tax of \(\$ 4\) per frisbee. Now what will be the equilibrium quantity, the price consumers will pay, and the price firms will receive? How is the bur den of the tax shared by buyers and sellers? c. How would your answers to parts (a) and (b) change if the supply curve were instead \\[ Q=70+P ? \\] What do you conclude by comparing these two cases?

Short Answer

Expert verified
Answer: When the government imposes a tax on frisbees, the new equilibrium price that consumers pay increases to $13, while the price firms receive becomes $9. The new equilibrium quantity decreases to 74 frisbees. Buyers bear $3 of the $4 tax, and sellers bear $1. With an alternative supply function, the equilibrium price and quantity remain at $10 and 80 frisbees, respectively. However, the tax burden still remains the same, with buyers bearing $3 and sellers bearing $1 of the tax.

Step by step solution

01

Set the Demand and Supply Functions Equal to Each Other

Since equilibrium occurs when supply equals demand, we need to set the two functions equal to each other: \\[100 - 2P = 20 + 6P\\]
02

Solve for the Equilibrium Price (P)

Now, solve the above equation for P (the equilibrium price): \\[8P = 80\\] \\[P = 10\\] The equilibrium price is $10.
03

Find the Equilibrium Quantity (Q)

Substitute the equilibrium price back into either the supply or demand function to find the equilibrium quantity: \\[Q = 20 + 6(10)\\] \\[Q = 80\\] The equilibrium quantity is 80 frisbees. #b. Imposing a Tax on Frisbees#
04

Adjust the Supply Function for the Tax

Since the tax of $4 per frisbee is imposed on the suppliers, we will modify the supply function to account for the tax: \\[Q = 20 + 6(P - 4)\\]
05

Set the New Supply Function Equal to the Demand Function

Now, set the new supply function equal to the demand function: \\[100 - 2P = 20 + 6(P - 4)\\]
06

Solve for the New Equilibrium Price (P')

Solve the above equation for the new equilibrium price P': \\[8P = 100 - 20 + 24\\] \\[8P = 104\\] \\[P' = 13\\] The new equilibrium price that consumers pay is $13.
07

Calculate the Price Firms Receive

Subtract the tax amount from the new equilibrium price to find the price firms receive: \\[P_{firm} = P' - 4\\] \\[P_{firm} = 13 - 4\\] \\[P_{firm} = 9\\] Firms will receive $9 per frisbee.
08

Find the New Equilibrium Quantity (Q')

Substitute the new equilibrium price P' into the demand function to find the new equilibrium quantity Q': \\[Q' = 100 - 2(13)\\] \\[Q' = 74\\] The new equilibrium quantity is 74 frisbees.
09

Determine Tax Burden on Buyers and Sellers

To determine the tax burden on buyers and sellers, we compare the increase in prices for each: Increase for buyers: \(13 - \)10 = $3. Increase for sellers: \(10 - \)9 = $1. Buyers bear \(\$3\) of the \(\$4\) tax, and sellers bear \(\$1\). #c. Analyzing an Alternative Supply Curve#
10

Set the New Supply Function Equal to the Demand Function

Now, set the new supply function equal to the demand function: \\[100 - 2P = 70 + P\\]
11

Solve for the Equilibrium Price

Solve the above equation for P: \\[3P = 30\\] \\[P = 10\\] The equilibrium price remains at $10.
12

Find the Equilibrium Quantity

Substitute the equilibrium price back into the new supply function to find the equilibrium quantity: \\[Q = 70 + 10\\] \\[Q = 80\\] The equilibrium quantity remains at 80 frisbees. By comparing the two supply curves, we can conclude that the impact of the tax depends on the elasticity of the supply function. In this case, the alternative supply function resulted in no change in the equilibrium price and quantity, which means the supply function is more elastic than before. The tax burden still remains the same as in part (b).

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Most popular questions from this chapter

Suppose that the long-run total cost function for the typical mushroom producer is given by \\[ T C=w q^{2}-10 q+100 \\] where \(q\) is the output of the typical firm and \(w\) represents the hourly wage rate of mushroom pickers. Suppose also that the demand for mushrooms is given by \\[ Q=-1,000 \mathrm{P}+40,000 \\] where (Pis total quantity demanded and Pis the market price of mushrooms. a. If the wage rate for mushroom pickers is \(\$ 1,\) what will be the long-run equilibrium out put for the typical mushroom picker? b. Assuming that the mushroom industry exhibits constant costs and that all firms are iden tical, what will be the long-run equilibrium price of mushrooms, and how many mush room firms will there be? c. Suppose the government imposed a tax of \(\$ 3\) for each mushroom picker hired (raising total wage costs, \(w,\) to \(\$ 4\) ). Assuming that the typical firm continues to have costs given by \\[ T C=w q^{2}-10 g+100 \\] how will your answers to parts (a) and (b) change with this new, higher wage rate? d. How would your answers to (a), (b), and (c) change if market demand were instead given by \\[ Q=-1,000 \mathrm{P}+60,000 ? \\]

A perfectly competitive market has 1,000 firms. In the very short run, each 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 noth ing 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 \((

A perfectly 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 \(\left(q_{t}=20\right) .\) 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 (P*)? The total industry output ((?*)? The out put of each firm \((

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost curve of the form \\[ c-3^{\wedge} 09^{3}+0.2 q^{2}+4 q+10 \\]. a Calculate the firm's short-run supply curve with \(q\) as a function of market price (P). b. On the assumption that there are no interaction effects among costs of the firms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by \(Q=-200 \mathrm{P}+8,000\). What will be the short-run equilibrium price-quantity combination?

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 \\[ T C=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 \mathrm{a} \\] 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 hence given by \\[ T C=0.5 q^{2}-10 q+w \\]. a. What is the long-run equilibrium quantity of stilts produced? How many stilts are pro duced 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 \\] Answer the questions posed in part (a). c. Because stilt-making entrepreneurs are the cause of the upward sloping 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 identi cal to the change in long-run producer surplus as measured along the stilt supply curve.
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