# Synopsis Profit maximization

```Profit maximization
This chapter describes solutions to proﬁt-maximization problem of a ﬁrm in competitive markets for the factors of
production it uses and the output goods it produces.
Proﬁts = revenues- cost.
Suppose that the ﬁrm produces n outputs (y1,...,yn) and uses m inputs (x1,...,xm). Let the prices of the output
goods be (p1,...,pn) and the prices of the inputs be (w1,...,wm).
The proﬁts the ﬁrm receives, π, can be expressed as
Cost includes all factors of production used by the ﬁrm, valued at their market price. If an individual works in his
own ﬁrm, then his labor is an input it should also be counted as part of the costs. Similarly, if a farmer owns some
land and uses it in his production, that land should be valued at its market value for purposes of computing the
economic costs. Costs like these are referred to as opportunity costs.
Objective of profit maximization of a firm is to generate maximum present value of the stream of profit that a firm
can generate.
Fixed and variable costs: Cost associated with fixed factor of production which does not change with the volume of
production is called fixed cost., and the cost associated with the factors which can be used with different amount
is called variable cost. However, in the long run all costs are variable because in long-run the firm is free to vary all
factors of production.
If a firm chooses to maximize its profits, it must choose that level of output where Marginal Cost (MC) is equal
to Marginal Revenue (MR) and the Marginal Cost curve is rising. In other
words, it must produce at a level where MC = MR.
At A, Marginal Cost < Marginal Revenue, then for each extra unit
produced, revenue will be higher than the cost so that you will generate
more. At B, Marginal Cost > Marginal Revenue, then for each extra unit
produced, the cost will be higher than revenue so that you will create
less.
Thus, optimal quantity produced should be at MC = MR
Shor run-profit maximization:
In short run, if we fixed factor of production, the production function will be f(x1, x2), then the profit maximization
function
If x1* is the profit-maximizing choice of factor 1, then the output price times the marginal product of factor 1
should equal the price of factor 1.
ie Value marginal product of factor 1 would be equal to price of factor 1
Profit maximization in long-run:
In the long run the all cost can vary as firm is free to choose the level of all inputs. Thus the long-run profitmaximization problem will be
The equation for long-run profit is
y
w1
  w2 x2
x1 
p
p
Change in x2 causes no change to the slope, and an increase in
the vertical intercept.
The optimal choice of the factor of production will be
In the long run, output can be increased by increasing all factors in the same proportion. Generally, laws of returns
to scale refer to an increase in output due to increase in all factors in the same proportion (constant return to
scale), output increases more than proportionate to increase in factor of production (increasing return to scale)
and output increases less than proportionate to increase in factor of production (diminishing return to scale)
If a competitive firm’s technology exhibits decreasing returns-to-scale then the firm has a single long-run profitmaximizing production plan.
So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive
Revealed profitability:
When a profit-maximizing firm makes its choice of inputs and outputs it reveals two things: first, that the inputs
and outputs used represent a feasible production plan, and second, that these choices are more profitable than
other feasible choices that the firm could have made. That is, the profits that the firm achieved facing the t period
prices must be larger than if they used the s period plan and vice versa. If either of these inequalities were
violated, the firm could not have been a profit-maximizing firm (with an unchanging technology).
Limitation of profit maximization rules:



In the real world, it is not so easy to know exactly your Marginal Revenue and Marginal Cost of the last
products sold. For example, it is difficult for firms to know the price elasticity of demand for their good –
which determines the MR.
The use of the profit maximization rule also depends on how other firms react. If you increase your price,
and other firms may follow, demand may be inelastic. But, if you are the only firm to increase the price,
demand will be elastic.
It is difficult to isolate the effect of changing the price on demand. Demand may change due to many
other factors apart from price.

Increasing price to maximize profits in the short run could encourage more firms to enter the market.
Therefore firms may decide to make less than maximum profits and pursue a higher market share.
Chapert 21:
Cost curves
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