Using the cost and revenue graphs in Figure 4.50, sketch the following functions. Label the poin

Using the cost and revenue graphs in Figure 4.50, sketch the...

Key Concepts

Marginal Revenue

Marginal revenue is the extra revenue generated by selling one additional unit of a product, defined as the derivative of the total revenue function with respect to quantity. It is depicted graphically as the slope of the revenue curve, and is essential for understanding how revenue changes with incremental increases in output, guiding decisions about maximizing profit.

Total Profit Function

This concept represents the overall earnings of a firm calculated as the difference between total revenue and total cost at each production level. Graphically, it is obtained by subtracting the cost curve from the revenue curve, which helps in visualizing profit margins, break?even points, and potential losses or gains as production increases.

Marginal Cost

Marginal cost is the additional cost incurred when producing one extra unit of output, represented mathematically as the derivative of the total cost function with respect to quantity. On a graph, it corresponds to the slope or rate of change of the total cost curve, offering insight into cost efficiency and optimal production levels.

Transcript

00:02 Hello, we are doing problem 4 in which you are given a graph similar to the one that i've drawn here.

00:12 On the graph, this is the revenue function, the x -axis is a q function and this is a dollar amount.

00:27 Cost function is a brown curve here.

00:35 Alright, and this value.

00:40 Q1 and this value is given also.

00:51 Now what happens at q1 and q2? first, let's see what we need to do.

00:57 We need to, based on these on the given graph, we need to produce a graph of total profit.

01:08 What do we know about profit is the function that is q equals revenue of q minus.

01:21 These points are the points in which the graph of c and and r have the same y value so the difference between two values is going to be zero so at this point here we have that pi equal zero and at this point here we have pi we have pi in the interval from zero to q1 the cost function is above this is the cost function and here in this interval here the cost function is above the revenue function so we are subtracting a larger number from a smaller number so we will have that the total profit is going to be less than zero negative here the revenue is greater than in the interval from q1 to q2 the revenue is greater than the cost so here in this interval we're going to have that pi is greater than zero and for q greater than q2 phi is going to be negative again because the cost is going to be greater than the revenue.

02:38 Now these two lines that i've added on to the given graph our lines are parallel to the revenue function.

02:48 Why is that? these are the points, this point here, and this point here, are the points where the marginal revenue equals the marginal cost function.

03:02 Why is that? what is c? the derivative of c.

03:09 The derivative of c in q is the slope of the tangent.

03:21 Now, when we observe r, this is a linear function, and r of q is some mq plus b, let's say, where the slope we can...

03:35 Now, marginal revenue is going to be the derivative of this side is m.

03:41 So the slope of the marginal revenue function is going to be m, which is the same as the slope of this graph here.

03:53 We search where are the points on c of q that have tangents in the same graph, tangents as the same as the tangents to the graph are.

04:07 And these points are here.

04:09 We can label them q3.

04:12 And now we have the necessary ingredients to a graph of the turtle profit function my graph has escaped a little bit from up here but i think we can follow it from this point on now let's see what happens at the beginning at the beginning we have that said that the total profit is negative and up to this point let's say this is a point of q3 is negative and this is at the most negative.

05:05 From there on it climbs back up to zero at around q1.

05:13 Not around q1 but at q1 it will rise up and cross the x -axis.

05:22 What happens in the next? we've reached the point where 5 -0.

05:26 Now in the next interval in this interval here from q1 to q2.

05:33 We have value q4 where the profit is at its greatest.

05:39 We found out this was our point of the critical point...

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