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How to find the Maximum Profit for a Perfectly Competitive Firm

Picture of 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.
 
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