# How to Find the Maximum Profit for a Perfectly Competitive Firm

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## Introduction: How to Find the Maximum Profit for a Perfectly Competitive Firm

Target Audience:
This is aimed toward those who have taken or are currently taking Intermediate Microeconomics.
Need to understand how to plot the Total Product of Labor Curve, Average Product of Labor Curve, and the Marginal Product of Labor Curve.

Background:
Characteristics of Perfect Competition:
1. Many producers
2. Homogenous product (perfect substitutes)
3. Free entry and free exit.

Necessary Conditions:
Profit = Total Revenue – Total Cost
π = TR - TC

We want to look at how profit changes with respect to quantity, meaning we want to look at the slope. We want to change the equation above to look at the change in profit divided by the change in quantity. (π = Profit) These slopes are referred to as marginals.
Quantity = Q
Δ = the change in

Δπ/ΔQ=ΔTR/ΔQ-  ΔTC/ΔQ
MNR = MR – MC

MNR – Marginal Net Revenue
MC – Marginal Cost
MR – Marginal Revenue

The firm will continue to produce if Marginal Revenue is greater then the Marginal Cost. This means that we have a positive marginal profit.  For perfect competition in order to maximize profit the MNR must equal zero.

MNR = MR – MC = 0
MR = MC

MR = MC is a necessary condition for perfect competition.

We want to begin by starting with revenue. Total Revenue (TR) is equal to the Price (P) multiplied by the Quantity (Q).

TR = P*Q

Next we want to observe the average value of the revenue and to do this we must divide the total revenue by the quantity. This will give us our Average Revenue (AR)

AR=  TR/Q=(P*Q)/Q=P

We substitute P*Q into the equation and we come to see that AR = P because the Q cancels in the numerator and denominator.

Next we want to look at the change in Revenue, which is the slope and also known as the Marginal Revenue (MR.) We must divide the change in Total Revenue by the change in Quantity.

MR=  ΔTR/ΔQ=  (Δ(P*Q))/ΔQ=(P* ΔQ)/ΔQ=P

We substitute P*Q again into the equation and can pull out the P because it is constant. From this the ΔQ’s cancel leaving only P. From this we see MR = P

Next we combine all of the information we just found.

***AR = MR = P

***This equation only holds for perfect competition

This last equation is incredibly important to understand. It means that at some price you will have a horizontal AR and MR curve and this coincides with the demand curve. For a perfectly competitive market to maximize profits MR must equal Marginal cost and in the long run this profit will be equal to zero.

## Step 1: Begin With Previous Knowledge of Production Theory

*Begin with previous knowledge of the Production Theory.

The Total Product of a variable factor of production identifies which outputs are possible using various levels of the variable input.
The Total Product Curve is shown in the first graph.

AXES
Q = Quantity
L = Labor

The first graph is the Total Product of Labor Curve (TPL)

There are three characteristic points that have been pointed out:
A = Inflection Point
B = Point of Maximum Slope
C = Slope of zero

Previously known information:
TPL = Total Product of Labor
APL = Average Product of Labor
MPL = Marginal Product of Labor
TC = Total Cost
w*L =wage rate* Labor
r*K = wage rate * Capital
APL=  TPL/Q=  Q/L
MPL=  ΔTPL/ΔL=  ΔQ/ΔL

TC=w*L+r*K

The average product is the TPL/Q and the MPL is the slope of the TPL curve.
At point B the slope reaches its maximum and this is where the Average will reach its maximum as well. At the inflection point (A) the MPL reaches its maximum and continues to decline from that point and intersects the maximum of the APL. At point C the slope is zero meaning that the MPL is as well zero. From this point MPL declines and has a negative slope meaning that the MPL will be negative. This is shown in the  second graph.

***It is important to note that between point B and C the MPL is positive and declining. In the firm this in the only range in which it will produce output.

## Step 2: Derive the Cost Curve From the APL/MPL Curves

Total Cost = Variable Cost + Fixed Cost
TC = VC + FC

To find the average you must divide by the quantity.

TC/Q=TVC/Q+TFC/Q
AC=AVC+AFC

To find the Average of the variable cost we must divide by Q. From previous knowledge we know that TVC =wL. We can now manipulate the equation and we know that Q/L = APL from above. This is also previously known.

AVC=  TVC/Q=  wL/Q=w/(Q/L)=  w/APL

As average product of labor (APL) increases the AVC decreases and as the APL decreases the AVC increases.

Next we find the slope of the cost curve. We divide the change in Total Cost by the change in Quantity

MC=  ΔTC/ΔQ=  ΔTVC/ΔQ=  Δ(w*L)/ΔQ=  wΔL/ΔQ=  w/(ΔQ/ΔL)=  w/MPL

The change in Total Cost is equal to the change in total variable cost because the fixed cost is not changing.
As the MPL increases the MC decreased and as the MPL decreases the MC increases. This is shown in the graph.

As the marginal product of labor increases the MC decreases and when the marginal product of labor decreases the MC increases. This is how we will derive the MC and AVC curve. The AC curve will be above the AVC curve and the MC will intersect at the minimum of the AVC and AC curve.

## Step 3: Profit, Average Revenue, Marginal Revenue Curve

We will begin with the definition of profit.
These equations were defined and explained in the Background.

Profit = Total Revenue – Total Cost
π=TR-TC

Δπ/ΔQ=ΔTR/ΔQ-  ΔTC/ΔQ

MNR = MR – MC = 0

The firm will continue to produce if Marginal Revenue is greater then the Marginal Cost. This means that we have a positive marginal profit.  We want for our marginal net revenue to equal 0.

MR = MC is a necessary condition for perfect competition

Revenue = Price * Quantity
AR=  TR/Q=(P*Q)/Q=P
MR=  ΔTR/ΔQ=  (Δ(P*Q))/ΔQ=(P* ΔQ)/ΔQ=P

AR = MR =P
This means we will have a horizontal line at the chosen price which is shown on the graph.

## Step 4: Combine Graphs: P Is Greater Than AC

First we will look at when Price is greater then the Average Cost.

P>AC
We want to first identify where our TR is on our graph. TR = P*Q So we must find where MC =MR and draw a vertical line down to the Quantity axis and find the Quantity which correlates to the Price chosen. As you can see this forms a rectangle and the area of the rectangle is the TR.  The shaded box represents the TR.

TR = Q*P
= P0Q0

The Second Graph

Next we have to find the TC. We have our necessary quantity marked and now we must look at the area under the AC curve. We draw a straight line from the price axis to where the price lays tangent to the AC curve where the Q = AC and use this new price to find the Area under the curve. It should be noticeable from the graphs that the TR area is larger than the TC area.

TC = P1Q

Now we can find the profit. The TR –TC will be the shaded region below.
TR was greater than TC and therefor the profit was positive.

The Third Graph

π=TR-TC
π=ABCD=positive profit

## Step 5: P Is Greater Then Average Variable Cost and Less Than Average Cost

AVC<P<AC

The First Graph
We want to first identify where our TR is on our graph. TR = P*Q So we must find where MC =MR and draw a vertical line down to the Quantity axis and find the Quantity which correlates to the Price chosen. As you can see this forms a rectangle and the Area of the rectangle is the TR.  The shaded box represents the TR.

TR = PQ
Next we have to find the TC. We have our necessary quantity marked and now we must look at the area under the AC curve. We draw a straight line from the price axis to where the price lays tangent to the AC curve where the Q =AC and use this new price to find the Area under the curve. It should be noticeable from the graphs that the TC area is larger than the TR area.

Second Graph

TC = P0Q

Third Graph
Profit is negative. The firm will continue to operate as long as it covers its variable cost, which is does.  When AVC<P<AC the profit is negative but the firm will continue to produce because it will lose more if it shuts down.

## Step 6: AVC Is Greater Than P

The First Graph

We want to first identify where our TR is on our graph. TR = P*Q So we must find where MC =MR and draw a vertical line down to the Quantity axis and find the Quantity which correlates to the Price chosen. As you can see this forms a rectangle and the Area of the rectangle is the TR.  The shaded box represents the TR.

TR = PQ

Next we have to find the TC. We have our necessary quantity marked and now we must look at the area under the AC curve. We draw a straight line from the price axis to where the price lays tangent to the AC curve where the Q = AC and use this new price to find the Area under the curve. It should be noticeable from the graphs that the TC area is larger than the TR area.

The Second Graph

TC = P0Q

The Third Graph

Loss is greater then the variable cost therefor the firm will shut down.

## Step 7: The Shutdown Point: P = AVC

P=AVC

P = AVC which is the point at which the firm is not sure whether is should shut down or continue producing. As we have seen when P>AVC the firm continues to produce and when P<AVC the firm will shutdown. At this point P =AVC the firm must make decisions as to whether it should continue to produce or shut down.

## Step 8: Look at Profit From TC and TR Curves

First Graph
From the TR and TC curves we will now find the maximum profit.

TC is always above TVC. Between TC and TVC the distance is TFC.

TC = Total Cost
TVC = Total Variable Cost
TFC = Totao Fixed Cost

The TC curve from above is incorporated in the graph below. The TC and TR are combined. TR is P*Q which is a linear relationship and increases as Price and Quantity increase.

Second Graph

As we can see from the graph above we can observe profit by looking at the change in TR and TC.
A) TC >TR : profit is negative
B) TR = TC : profit is zero
C) TR >TC : profit is positive
D) TR > TC : profit is maximized

## Step 9: Combine Graphs to Find Max Profit

When Profit is maximized and minimized the MC = MR. When the TC = TR the AC = MR. As we stated above when the total revenue is greater then the total cost we have positive profit and when the TC is greater then the TR the profit is negative. From this we can Combine the TR,TC curve with the MC, AC, and the Profit graphs to find the point at which the firm maximizes profit.

As we can see the firm maximizes profits when the profit graph reaches its maximum. This is when on the TC, TR curve the TR is greater and the vertical distance between the TC is at is maximum. This is also the point where our MC = MR.