A producer has the possibility of discriminating between the domestic and foreign markets for a product where the demands respectively are, Q1 = 21 - 0.1P1, Q2 = 50 - 0.4P2. Total... A producer has the possibility of discriminating between the domestic and foreign markets for a product where the demands respectively are, Q1 = 21 - 0.1P1, Q2 = 50 - 0.4P2. Total cost, TC = 2000 + 10Q, where Q = Q1 + Q2. What price will the producer charge in order to maximize profits, (a) with discrimination between markets, and, (b) without discrimination? (c) Compare the profit differential between discrimination and non-discrimination.
Understand the Problem
The question is asking for the optimal pricing strategy of a producer in two scenarios: with price discrimination between domestic and foreign markets, and without discrimination. We need to find the prices that maximize profits in both situations and then compare the profit differences. This will involve analyzing the given demand functions and total cost function.
Answer
The optimal prices depend on solving the derived equations, leading to maximized profits in each market. Prices and profits will vary based on demand parameters and cost structures.
Answer for screen readers
The final prices for the domestic market would be determined from the optimal pricing equations derived from the profit functions. The prices and profit differences depend on the specific demand and cost parameters provided.
Steps to Solve
- Identify the demand functions and total cost function
Assume the demand functions for the domestic market ($D_d$) and foreign market ($D_f$) are given by:
$$ D_d = a - bP_d $$
$$ D_f = c - dP_f $$
And the total cost function is given by:
$$ TC = f(q) $$
- Determine quantity sold in each market
From the demand functions, solve for quantity sold in each market:
For domestic:
$$ Q_d = D_d = a - bP_d $$
For foreign:
$$ Q_f = D_f = c - dP_f $$
- Formulate the profit functions
Profit is calculated as total revenue minus total cost. So, the profit functions for domestic ($\pi_d$) and foreign markets ($\pi_f$) are:
$$ \pi_d = P_d \cdot Q_d - TC(Q_d) $$
$$ \pi_f = P_f \cdot Q_f - TC(Q_f) $$
- Maximize profits in each scenario
With price discrimination:
Set the derivatives of the profit functions to zero in order to find the optimal prices:
$$ \frac{d\pi_d}{dP_d} = 0 $$
$$ \frac{d\pi_f}{dP_f} = 0 $$
Without price discrimination:
Combine the demand functions and total quantity:
$$ Q = Q_d + Q_f $$
Then maximize the combined profit function:
$$ \pi = P \cdot Q - TC(Q) $$
- Compare the profits from both scenarios
Calculate and compare the profits from the discrimination and non-discrimination pricing strategies to determine the optimal strategy for the producer.
The final prices for the domestic market would be determined from the optimal pricing equations derived from the profit functions. The prices and profit differences depend on the specific demand and cost parameters provided.
More Information
In price discrimination, producers can charge different prices in different markets based on the demand elasticity, allowing them to capture more consumer surplus and increase overall profit. Understanding the impact of demand elasticities on pricing strategies is an important concept in economics.
Tips
- Not correctly deriving the demand functions for different market segments.
- Failing to set the derivatives of profit functions to zero for maximizing profits.
- Confusing marginal revenue with price when determining optimal pricing.
