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Marginal revenue (MR) refers to the additional revenue that a firm earns when it sells one more unit of a product. In the context of a monopoly, understanding marginal revenue is crucial as it helps determine the most profitable level of output.

For Country 1 with a demand curve given by \[ p_1 = 110 - Q_1 \]the total revenue function is expressed as \[ TR_1 = 110Q_1 - Q_1^2 \].
When we differentiate this with respect to \( Q_1 \), our marginal revenue becomes \[ MR_1 = 110 - 2Q_1 \].

Similarly, for Country 2 with a demand curve \[ p_2 = 130 - 2Q_2 \], the total revenue function is \[ TR_2 = 130Q_2 - 2Q_2^2 \], giving a marginal revenue of \[ MR_2 = 130 - 4Q_2 \].

Setting marginal revenue equal to marginal cost is how the monopoly finds the output level that maximizes profit.

Total revenue (TR) is the total income a firm receives from selling its goods or services at a given price. It is calculated as the product of the price at which the goods are sold and the quantity sold.

For a monopolist operating in two countries, understanding total revenue helps in devising price discrimination strategies. In Country 1, total revenue can be formulated as \[ TR_1 = (110 - Q_1) \cdot Q_1 = 110Q_1 - Q_1^2 \].

In Country 2, the total revenue equation is \[ TR_2 = (130 - 2Q_2) \cdot Q_2 = 130Q_2 - 2Q_2^2 \].

These functions reflect how changes in the sold quantity impact the revenue under differing demand conditions. Adjusting prices and quantities in each market according to these functions helps maximize profits.

The equilibrium price in a monopoly is where the firm's marginal revenue equals its marginal cost. This is the price that balances production cost with the desired profit margin.

To find this, you substitute the calculated quantity back into the demand equations for each country. In Country 1, given \( Q_1 = 37.5 \), the equilibrium price is calculated as \( p_1 = 110 - 37.5 = 72.5 \).

In Country 2, with \( Q_2 = 23.75 \), it results in \( p_2 = 130 - 47.5 = 82.5 \).

These prices illustrate how the monopoly sets different prices in these separate markets to maximize overall profits, tailoring production and pricing to each market's specific demand characteristics.

Marginal cost (MC) is the additional cost incurred by producing one more unit of a good. In a monopolistic market, the marginal cost plays a pivotal role in determining the profit-maximizing quantity.

In our problem, the marginal cost is consistently \( m = 35 \) for both countries, meaning each additional unit produced costs the firm $35.

To find the equilibrium output, the monopoly equates its marginal revenue with its marginal cost for each market. So in Country 1, \( 110 - 2Q_1 = 35 \), and in Country 2, \( 130 - 4Q_2 = 35 \).

These equations solve for the quantity where the additional revenue from one more unit equals the additional cost of production, ensuring the firm maximizes profit at this output level.

A demand curve represents the relationship between the price of a good and the quantity demanded by consumers at each price level. In a monopoly, understanding the demand curve is essential for setting prices and determining output levels.

For Country 1, the demand curve is linear and expressed as \( p_1 = 110 - Q_1 \). This indicates that the price consumers are willing to pay falls as more units are sold. Conversely, Country 2's demand curve is \( p_2 = 130 - 2Q_2 \), which implies an even steeper decline in price for each additional unit sold compared to Country 1.

Monopolies leverage these demand curves to implement price discrimination strategies, charging different prices in different markets based on their unique demand conditions. This selective pricing helps the firm to capture more consumer surplus and increase overall profits.