## Saturday, 1 June 2013

### Why profit is the highest when marginal cost curve crosses the marginal revenue curve ( Competitive firm model )

The definitions and the characteristics for the various terms need to be referred to a standard text book.

This blog will discuss only the reason for why the profit is maximized when Marginal cost curve crosses the Marginal revenue curve.

Abbrevations

MC=    Marginal Cost                                                     MR=   Marginal Revenue
ATC=  Average Total Costs                                           TC=     Total Costs
AVC=  Average Variable Costs                                     TR=     Total Revenue
AFC=  Average Fixed Costs                                          P=        Price; Q=    Quantity

Consider the following example

The table below gives the details of a factory producing a certain unit of a product.

Table 1

It is mentioned in a previous blog, why marginal cost crosses the Average cost curves at their lowest points,  and how the marginal cost, which is a change in cost, actually affects all the average costs.

EXPLANATION

Consider the following plots, plot 1 and plot 2 derived from the factory data in table 1.

Plot 1

Plot 2

The Total Cost is equal to Average Total Cost  multiplied by the quantity, and Total Revenue is equal to Average Total Revenue ( equal to price or MR for a competitive firm) multiplied by the quantity. Total Profit is the difference between the two.

In Plot 2, Total cost is drawn as an area which is equal to ATC multiplied by Quantity. And Total Revenue is drawn as an area which is equal to ATR multiplied by Quantity. Total profit is the Total Revenue area minus the Total cost area.

For example.

For a Quantity H, the TC=     area EYHF
TR=      area DIHF
Profit=  area DIYE.

For a Quantity G, the TC=     area ABGF
TR=     area DCGF
Loss=  area ABCD

CONFUSION

1.       In theory the profit has to be highest at a production output where the marginal cost crosses the marginal revenue, which is at point Z. whereas, it can be clearly seen that the Average total cost is lowest at point Y. And also, marginal cost is lower at point Y, which is lower than at point Z.

2.       At point W, the marginal cost is the lowest, much lower than the marginal revenue. But the production is at a significant loss at the same point.

CLEARING CONFUSION 1

Consider the following plots 3 and 4. Plot 3 plots the Total Revenue and Total costs. And plot 4, plots the difference of Total cost and Total Revenue ie the profit.

Plot 3

Plot 4

From plot 4 it is quite obvious that the maximum profit occurs at point Z ( where the MC crosses the MR). It is also observable from plot 3.

REASON

The reason for this confusion is that, MC as defined previously is a change in cost. The difference between AVR( the straight line of \$2) and the ATC curve gives us Average total profit(AVP), and it is quite clear that AVP is the highest at the point Y where ATC is the lowest. We get confused because we think that the profit should be highest when the AVP is the highest. But AVP is only an average. Increasing the production from point Y to point Z definitely reduces the AVP. But it increases the total profits. AVP is reduced because, each quantity  produced after point Y yeilds lesser profits, bringing down the average. But  the total profit gets increased albeit with decreased rate. Even when the Average Total Cost is at its lowest when the production reaches point Y (370 units), we can still squeeze in more profit per unit, by producing until the production reaches point Z (440 units) . Until then, each change in cost/ unit (MC) is still less than the price for which those units can be sold. The increase in total profits becomes zero only when the production reaches point Z. This is quite clear in the Plot 5.

The difference between MR and MC is called Marginal profit, whereas, the difference between ATR and ATC is Average profit. In order to maximize the Total profit, we have to go on producing until the marginal profit is diminished to zero. Since Average profit is just an average, it might be still be positive for many more units of production even after the production crosses point Z. For more clarity refer to plot 5.

Plot 5

In Plot 5, we can see that the average profit is maximum at point Y, which is self explanatory.

CLEARING CONFUSION 2

Marginal Cost, Marginal Revenue and Marginal Profit  are also the slopes of the Total Cost, Total Revenue and the Total profit curves respectively.

So in this case, when Marginal Cost is at the lowest, the Marginal Profit is at the maximum. But the factory is in loss. It just says that the slope of the Total cost curve is at the lowest and is about to change, and the slope of the Total profit curve is at the maximum and is about to change. At this point the Total cost is not the lowest, neither is the Total profit the highest. This is amply demonstrated in the plots 3,4 and 5.