3. Suppose a firm has a constant marginal cost of $10. The current price of the product is $25, and at that price, it is estimated that the price elasticity of demand is -3.0.

a. Is the charging the optimal price for the product? Demonstrate how you know.
b. Should the price be changed? If so, how?

a) it's not charging optimal price because the price exceed MC.

b) it should reduce its price equal to MC in order to maximize profit.

so, the price should be equal to $10

a. To determine if the current price is optimal, we need to calculate the price elasticity of demand (PED) and compare it to the optimal condition. The formula for PED is:

PED = (% change in quantity demanded) / (% change in price)

Given that the estimated price elasticity of demand is -3.0 and the price is currently $25, we can calculate the optimal price. However, we need the specific knowledge of the quantity demanded at the current price and the percentage change required to find the optimal price.

b. Without knowing the specific quantity demanded at the current price or the required percentage change, we cannot determine the exact optimal price. However, we can provide a general guideline for adjusting the price based on the principle of marginal cost pricing.

The principle of marginal cost pricing states that to maximize profit, a firm should set its price equal to its marginal cost. In this case, the constant marginal cost is $10 per unit. So, if the firm wants to maximize its profits, it should consider lowering the price closer to the marginal cost.

Lowering the price would be beneficial because the price elasticity of demand is -3.0, indicating that demand is relatively elastic. With elastic demand, a decrease in price generally leads to a larger percentage increase in quantity demanded.

However, the exact adjustment of the price should be determined based on the market conditions, competition, and other factors that may impact demand and profit.

To determine whether the firm is charging the optimal price for the product, we need to consider the concept of price elasticity of demand and how it relates to marginal cost.

a. To answer whether the firm is charging the optimal price, we first need to calculate the price elasticity of demand. Price elasticity of demand measures how responsive the quantity demanded is to a change in price. In this case, we are given that the price elasticity of demand is -3.0. The negative sign indicates an inverse relationship between price and quantity demanded.

The formula to calculate price elasticity of demand is:
Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)

In this scenario, we are not given specific percentage changes, but since the firm's marginal cost is constant, we can infer that the price is not changing. So we can ignore the denominator and focus on the numerator.

Based on the given information, since the price elasticity of demand is -3.0, it means that a 1% increase in price would result in a 3% decrease in quantity demanded, and vice versa. This indicates that the demand for the product is relatively elastic.

Now, let's analyze whether the firm is charging the optimal price. The optimal price occurs when marginal cost is equal to marginal revenue. In this case, since the marginal cost is constant at $10, the optimal price would be the one at which marginal revenue equals $10.

To calculate the firm's marginal revenue, we need to consider the price elasticity of demand formula:
Price Elasticity of Demand = (Change in Quantity Demanded) / (Change in Price) * (Price / Quantity Demanded)

Since we are assuming no price change, the formula can be simplified as:
Price Elasticity of Demand = (Change in Quantity Demanded) / Quantity Demanded

Rearranging the formula, we find:
Change in Quantity Demanded = Quantity Demanded * Price Elasticity of Demand

Substituting the given values:
Change in Quantity Demanded = Quantity Demanded * (-3.0)

Now, assume the current quantity demanded is "Q" when the price is $25. The change in quantity demanded can be expressed as:
Change in Quantity Demanded = Q * (-3.0)

To determine if the firm is charging the optimal price, we need to compare the marginal revenue to the marginal cost. The marginal revenue can be calculated by multiplying the change in quantity demanded by the price:
Marginal Revenue = (Q * (-3.0)) * $25

If Marginal Revenue equals Marginal Cost ($10 in this case), then the firm is charging the optimal price. If it is greater than $10, the firm is not maximizing its profit and could potentially increase the price. Conversely, if it is lower than $10, the firm could decrease the price to maximize its profit.

b. To decide whether the price should be changed, we compare the calculated marginal revenue to the marginal cost. If the marginal revenue is greater than $10, the firm could consider increasing the price, while if the marginal revenue is lower than $10, the firm could consider decreasing the price.

By following this analysis, you can determine if the firm is charging the optimal price and whether a price change is necessary.