A firm's profit function is π(q)=R(q)-C(q) =80 q-(65+40 q+10 q^2) . What is the positive output

A firm's profit function is π(q)=R(q)-C(q) =80 q-(65+40 q+10...

Key Concepts

Profit Maximization

Profit maximization involves choosing the level of production that results in the highest possible profit, where profit is calculated as total revenue minus total cost. This process typically requires analyzing how changes in output affect both revenues and costs to identify the optimal production quantity.

Marginal Analysis

Marginal analysis is a decision-making process that involves comparing marginal revenue (the additional revenue from selling one more unit) with marginal cost (the additional cost of producing one more unit). The profit-maximizing output is found where marginal revenue equals marginal cost, ensuring that producing an extra unit does not decrease overall profit.

Cost Structure

Cost structure refers to the breakdown of a firm’s costs into fixed costs and variable costs. Fixed costs do not vary with output, while variable costs change with the level of production. Understanding the cost structure is essential for making production decisions, particularly in assessing whether losses are due to fixed commitments or variable expenses, which helps in strategic decision-making in both the short and long run.

Shutdown Decision

The shutdown decision in the short run is made by comparing the price to the average variable cost. If the price is lower than the average variable cost, the firm cannot cover its variable expenses and should shut down to minimize losses. Conversely, if the price exceeds the average variable cost, it is advisable to continue operating even if overall profit is negative, as fixed costs are considered sunk in the short run.

Transcript

00:01 Now in this question what has been given a profit function is given and we have to find the output at which the firms maximize its profit or minimize its losses.

00:11 Then we have also to find out the firm's revenue, the variable cost and profit and we have to also make the decision that should the firm operate in the short run or it should shut down.

00:23 First of all, as we know, the profit is total revenue minus total cost and in this case total revenue is 80 q and total cost will be this 65 plus 40 q plus 10 q square this bigger now the profit maximizing firm so the rule is that a firm maximize its profit by producing output at a level where marginal revenue is equal to marginal cost the marginal revenue is the additional revenue which is earned from producing an additional unit of output and marginal cost is the additional cost that is incur due to production of an additional unit of output.

01:00 Output now at the point where mr is equal to mc what happens that the additional revenue on from producing an additional route of output is equal to the additional cost incurred on producing an additional amount of output so as a result the difference between total revenue and total cost is maximized and this results then from making the maximum profit now what we will do to find marginal revenue we will take derivative of 80 q with respect to q and what will get 80 as the marginal revenue.

01:32 Similarly, we'll take derivative of total cost with respect to quantity and what we'll get 0 plus 40 plus 2 into 10 q raised to power 2 minus 1.

01:41 Ultimately what we get mc as marginal cost is 40 plus 20 q is the marginal cost.

01:49 Now we will equate mr is equal to mc that is 80 is equal to 40 plus 20 q and so what we get we solve it at further and what we get as q is 2.

02:00 That is the profit maximizing output is two units.

02:03 So at two units of output, the firm will earn maximum profit or it will minimize its losses.

02:09 Now we will find total revenue.

02:11 Total revenue is what price into quantity.

02:14 That is 80 multiply by 2 and it is equal to 160.

02:18 Similarly total cost, we will put q is equal to 2 in this cost function.

02:22 What we get 65 plus 40 multiply by 2 plus 10 into 2 into 2...