# Break-even Analysis Form BREAK-EVEN ANALYSIS - 6
3
Proof: Using the definition of Total Profit (
A
) we can take the partial derivative of it with
respect to output:
M
A
/
M
Q =
M
(TR
!
TVC
!
TFC)/
M
Q. Since TFC is a constant (its
derivative is thus zero), we obtain
M
A
/
M
Q =
M
(TR
!
TVC)/
M
Q. Now, if P and Q do not
change with Q, this would become M
A
/MQ = (P ! AVC)(MQ/MQ) = P ! AVC. This simply
says that for an unit increase in output, profit will increase by an amount equal to the
difference between price and average variable cost. Substituting
M
A
/
M
Q = P
!
AVC back
into the definition of DOL (= (Q/
A
)(
M
A
/
M
Q)) and simplifying, we get the boxed property
given above.
4
Proof: Using, say, the definition DOL = (TR
!
TVC)/(TR
!
TC)] above, we take the
partial derivative of DOL with respect to AVC, obtaining: M(DOL)/M(TVC) = ! 1/(TR !
TC) which is clearly negative if the firm is making profit. Indeed, since Q is a constant, we
can rewrite this as M(DOL)/M(AVC) = ! Q/(P ! ATC) < 0.
DOL
TFC
Profit
TR TVC
TR TC
TR TVC
TR TVC TFC
= =
=
are constant), it can be readily derived that:
3
(3)
Several conclusions can be drawn:
(1) The DOL generally depends on the output level, and is equal to zero at the profit-
maximizing output (because M
A
/MQ = 0 at that point)
(2) The DOL is negative (positive) below (above) the break-even level. Equation (3)
also yields an interesting result:
! Rule: For the same total cost, the Degree of Operating Leverage increases with fixed
costs and decreases with variable costs.
4
(3) Consider 2 (almost identical) plants, except that
Plant A (capital intensive): low fixed costs, high variable costs
Plant B (labor intensive) high fixed costs, low variable costs
Since fixed costs are outlays already made, if the firm chooses to built Plant A
instead of B, it can be said that the firm “gets more leverage” out of the resources
(4) A plant with high fixed costs and low variable costs will also have a higher break-
even point than a plant with low fixed costs and high variable costs.
The significance of this relationship is that a firm with large fixed costs usually
breaks even at a higher output level. However, this firm’s DOL is also higher, its
profits rises at a relatively high rate when production rises above break-even.
Likewise, its profits declines more quickly during economic downturns, and the firm
would become unprofitable at a relatively large output quantity (since the break-even
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