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Chapter 6 Henderson and Quandt Microeconomics, Questions 6.4-6.8_ Solutions

6-4 The long-run cost function for each firm that supplies Q is C= -4 +8q. Firms will enter

the industry if profits are positive and leave the industry if profits are negative. Describe the

industry's long-run supply function. Assume that the corresponding demand function is D=

2000-100p. Determine equilibrium price, aggregate quantity, and number of firms.

To find the industry’s long-run supply function, we first need to find the equilibrium output level for each firm. In the long run, each firm will produce at the minimum of its long-run average cost (LRAC), which is the long-run cost per unit of output at the optimal input level. The LRAC curve for each firm can be found by taking the derivative of the long-run cost function:

LRAC = (C/q)’ = 3q^2 – 8q + 8

Setting the LRAC equal to the price of Q, we get the equilibrium output level for each firm:

3q^2 – 8q + 8 = p

Solving for q, we get:

Q = (8 + p)/3

Substituting q into the demand function, we get the industry’s total revenue as a function of the price:

TR = D(p)Q = (2000 – 100p)(100q) = (2000 – 100p)(100/3)(8 + p)

Taking the derivative of TR with respect to p, we get the industry’s marginal revenue:

MR = dTR/dp = -2000/3 + 8000/3p – 200p^2/3

Setting MR equal to zero and solving for p, we get the equilibrium price:

8000/3p – 200p^2/3 = 2000/3

Simplifying, we get:

2000p^2 – 8000p + 2000 = 0

Dividing both sides by 2000, we get:

P^2 – 4p + 1 = 0

Solving for p using the quadratic formula, we get:

P = (4 ± sqrt(16 – 4))/2 = 2 ± sqrt(3)

Since p must be positive, we take the positive solution:

P = 2 + sqrt(3) ≈ 3.732

Substituting the equilibrium price into the equilibrium output equation, we get the equilibrium output level for each firm:

Q = (8 + p)/3 ≈ 3.244

The aggregate quantity is the total output of all firms:

Q = 100q ≈ 324.4

To find the number of firms in the industry, we divide the aggregate quantity by the equilibrium output level of each firm:

N = Q/q ≈ 99.98

Since the number of firms must be a whole number, we round up to get:

N = 100

Therefore, in the long-run, the industry will have 100 identical firms producing a total output of approximately 324.4 units of Q at an equilibrium price of approximately $3.732 per unit.

6-5 Consider an industry with n identical firms in which the ith firm's total cost function is

= a + b (i = 1,...,n), where q = q₁ ++…+. Derive the industry's supply function.

To derive the industry’s supply function, we need to find the profit-maximizing output level for each firm and then sum them up to get the total industry output. The profit function for the ith firm is given by:

Πi = pqi – Ci = pqi – aqi^2 – bqiq

Taking the derivative of πi with respect to qi and setting it equal to zero, we get the first-order condition for profit maximization:

P – 2aqi – bq = 0

Solving for qi, we get:

Qi = (p – bq)/(2a)

Substituting this expression for qi into the definition of q, we get:

Q = q1 + q2 + ... + qn = (p – bq)/(2a) + (p – bq)/(2a) + ... + (p – bq)/(2a) = np/(2a) – nbq/(2a)

Therefore, the industry’s supply function is:

Q = (np)/(2a) – (nbq)/(2a)

Simplifying, we get:

Q(1 + nb/(2a)) = (np)/(2a)

Dividing both sides by (1 + nb/(2a)), we get:

Q = (np)/(2a + nb)

Therefore, the industry’s supply function is:

Q = nq = n(np)/(2a + nb)

Where Q is the total industry output. This shows that the industry’s supply curve is positively sloped since the output of the industry increases as the price of the product increases. The slope of the supply curve depends on the parameters n, a, and b.

6-6 Construct an effective supply curve for an industry which has two sources of supply:

domestic production with the supply curve S = 20 + 8p, and (2) an unlimited supply of imports at

a fixed price of 20.

To construct an effective supply curve for the industry, we need to combine the domestic supply and the supply of imports. Since the supply of imports is unlimited, the price of imports will always be $20.

At any given price level, the domestic producers will supply a number of goods according to their supply curve, which is given as:

Sd = 20 + 8p

Therefore, the total supply at any given price level will be:

Stotal = Sd + Simports

Substituting the value of Simports, we get:

Stotal = 20 + 8p + Simports = 20 + 8p + Qimports(20)

Where Qimports is the number of imports supplied at the fixed price of $20. Since the supply of imports is unlimited, Qimports is effectively infinite, so we can simplify the above equation to:

Stotal = 20 + 8p + ∞

Since we can’t add infinity to a finite number, we can say that the effective supply curve for this industry is simply:

Stotal = ∞ (for p ≥ 0)

This means that as long as the price of the good is greater than or equal to zero, there will always be a supply of the good from either domestic producers or imports, or both. This is because even if the domestic producers decide not to produce anymore at a certain price level, imports will still be available at a fixed price of $20, which will ensure an infinite supply at any price level.

6-7 Determine equilibrium price and quantity for a market with the following demand and supply

functions: D=20-2p and S = 40-6p. Assume that a specific tax of 1 dollar per unit is imposed.

Compute the changes in equilibrium price and quantity.

To determine the equilibrium price and quantity, we need to find the intersection of the demand and supply curves. Setting them equal, we get:

20 – 2p = 40 – 6p

4p = 20

P = 5

Substituting this value of p back into either the demand or supply equation, we get:

Q = 20 – 2p = 20 – 2(5) = 10

So the equilibrium price is $5 and the equilibrium quantity is 10 units.

Now, with a specific tax of $1 per unit, the new supply curve becomes:

S = 40 – 6p – 1 = 39 – 6p

To find the new equilibrium price and quantity, we again set demand equal to supply:

20 – 2p = 39 – 6p

4p = 19

P = 4.75

Substituting this value of p back into either the demand or supply equation, we get:

Q = 20 – 2p = 20 – 2(4.75) = 10.5

So the new equilibrium price is $4.75 and the new equilibrium quantity is 10.5 units.

Therefore, the change in equilibrium price is -$0.25 (a decrease from $5 to $4.75), and the change in equilibrium quantity is +0.5 units (an increase from 10 to 10.5 units).

6-8 Assume fifty firms supply commodity Q at location I and fifty at location II. The cost of

producing output , for the ith firm (in either location) is 0.5. The cost of transporting the

commodity to the market from location I is 6 dollars per unit and from location II, 10 dollars per

unit. Determine the aggregate supply function.

The individual supply function for each firm is:

Si = qi(p – 0.5qi)

Since there are 50 firms in each location, the total supply at each location is:

Qi = 50qi(p – 0.5qi)

To find the aggregate supply function, we need to consider the cost of transportation from each location to the market. Let Q denote the total supply from both locations:

Q = Q1 + Q2 = 50q1(p – 0.5q1) + 50q2(p – 0.5q2)

To account for the transportation costs, we need to adjust the price at each location. Let pi denote the price at each location after accounting for transportation costs:

Pi = p – ci

Where ci is the cost of transportation from location i to the market. Then, we can rewrite the total supply function as:

Q = 50q1(p1 – 0.5q1) + 50q2(p2 – 0.5q2)

Where p1 = p – 6 and p2 = p – 10.

Substituting for p1 and p2, we get:

Q = 50q1(p – 6 – 0.5q1) + 50q2(p – 10 – 0.5q2)

Expanding and simplifying, we get:

Q = 50qp – 800 – 25q1 – 25q2 + 0.5(q1^2 + q2^2)

Thus, the aggregate supply function is:

Q = 50qp – 800 – 25q1 – 25q2 + 0.5(q1^2 + q2^2)

Where q1 and q2 are the quantities supplied by each firm in locations I and II respectively.

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