I am wondering where I got lost in this process. Please point out where I missed. Thank you!

A producer of agricultural equipment operating in the U.S. faces demand given by: P = 4,200 - 2Q, where P denotes price in dollars and Q is quantity of units sold per month. The firm's fixed costs are $320,000 per month, and its marginal cost of manufacturing the equipment is $1,400 per unit.

a) Find the firm's profit-maximizing output and price. What is its profit?

b) Suppose that a new market for the firm's product emerges in South America. The firm has begun selling the equipment in several test markets there and has found the elasticity of demand to be E_P= -3.5 for a wide range of prices (between $1,500 and $3,000). The cost of shipping to South America is $200 per unit. One manager argues that the foreign price should be set at $200 above the domestic price (in part a) to cover the transportation cost. Do you agree that this is the optimal foreign price? Justify your answer.

c) Suppose that the firm has produced the optimal level of domestic output as calculated in part a. But before this quantity is sold, demand unexpectedly falls to: P = 2,500 - 2Q, (equivalently Q = 1,250 - .5P). One manager recommends cutting price to sell the entire inventory; another favors maintaining the domestic price as calculated in part a (selling less than the total inventory). Do you agree with either manager? What optimal price would you set?

a) Define the profit-maximizing output and price. And acquire the profit.

P=4,200-2Q

TR=4,200Q-2Q^2

MR=4,200-4Q

According to given number MC=$1,400

4,200-4Q=1,400

4Q=2800

Q=700

Input Q=700 into P=4,200-2Q

P=2,800

Acquire the profit. Profit=TR-TC

TR=4700*2,800=1,960,000

TC=320,000+1,400*700=1,300,000

TR-TC1,960,000-1,300,000=660,000 (profit)

A. Q=700 units, P=$2,800, Profit is $660,000

b) a) The firm is not profit maximizing by setting foreign price simply increasing by $200. The high price leads less demand in quantity.

Based on markup equation,

MR=MC

P* (1+Ep/Ep) = MC

P=Ep/1+Ep*MC

P=(-3.5/1-3.5)*$1,400 = (-3.5/-2.5)*$1,400=$1,960 < $2,800 ($2,800=current profit-maximization price in the U.S. )

For this price elasticity, the optimal price should be lower and consequently the optimal quantity should be higher. It doesn't mean that optimal price should be equal to the product of the optimal markup and MC. If the firm is optimizing then its price ($2,800) should be equal to the product of optimal markup and MC. Since this is not the case, the firm is not profit maximizing by setting its price equal to $1,960.

c) The equation givens are P=2,500-2Q, Q=1,250-0.5P

Acquire quantity intercept

Q=0, P=2,500

P=0, Q=1,250

Input those number into the demand graph and acquire the slope Qd=a-bP

a=quantity intercept

b=change in quantity from a change in the price

the slope is 1,250/2,500=-0.5, Qd=2,500- 0.5P

Caution at intercepts: very high price near the price intercept may cause consumers to buy a lot, on the other hand, very low prices near the quantity intercept may not induce consumers to buy that much.

- Acquire optimal price

P=2,500-2Q

TR=2,500Q-2Q^2

MR=2,500-4Q

MC=1,400

MR=MC, 2,500-4Q1,400, Q=275

P=1,950

A. Optimal Price should be $1,950, with being said, I agree to lower the price however keep in mind the very low price near the quantity intercept may not induce consumers to buy that much.