Question

4. The market for apples is perfectly competitive. The market demand for apples has been estimated as follows: Qd = 54,400 - 900P, where is measured in thousands of cases The market supply is Qs 4000 + 800P Each apple producer has a total cost function given as: TC = 2400 + 54 +0.0625q? Find the market price and quantity, firm output and profits. How many firms are there? Market price $. Market output. Firm output Firm profits $ Number of firms Assuming the cost function given reflects long run costs. Find the long run price, firm output, and market quantity. Now how many firms are there? Number of firms Market output. Market price $_ Firm output

Answer 1

Demand function,

54, 400 – 900P 2p=

Supply function,

Qs -4,000+ 800P

At equilibrium quantity demanded is equal to quantity supply

$\large 54,400 - 900P = -4,000 + 800P$

900P + 800P = 54,400+ 4, 000

$\large \implies 1700P= 58,400$

$\large \implies P= \frac{58,400}{1,700}$

$\large \implies P= \$ \ 34.35$

Q = 54,400 - 900 × 34.35 = 23,485

TC = 2400 +5q + 0.0625q²

MC = 5 + 0.125q

In perfect competition P = MC

5 + 0.125q = 34.35

=> 0.125q = 29.35

=> q = 29.35 / 0.125 = 234.8

Output per firm = 235 (Approximately).

Number of firm = 23,485/234.8 = 100.02

Number of firms = 100

Firms profit = 235×34.35 -(2400 + 5×234.8 + 0.0625 ×234.8^2)

= $ 1045.69

In the long run the P is set equal to minimum ATC

$\large ATC = \frac{2400 + 5q + 0.0625q^2}{q}$

$\large ATC = \frac{2400}{q} + 5 + 0.0625q$

Differentiating ATC wrt q we get

$\large \frac{d}{de}ATC = -\frac{2400}{q^2} + 0.0625$

Now set it equal to zero

$\large -\frac{2400}{q^2} + 0.0625 = 0$

$\large \implies \frac{2400}{q^2} = 0.0625$

$\large \implies q^2 = \frac{2400}{0.0625} = 38,400$

$\large \implies q= \sqrt{38,400}$

$\large \implies q \approx 196$

Firm output = 196 units

Minimum ATC =( 2400/196 )+ 5 + 0.0625 × 196 = $ 29.50

Price = $ 29.50 per unit

Quantity = 54,400 - 900 × 29.50 = 27,850 units

Number of firm = 27,850/196 = 142 firms (approximately)

All the values are calculated please refer the calculations above. Please contact if having any query will be obliged to you for your generous support. Please help me it mean a lot to me. Thank you.