# conditional factor demand curve ```Intermediate Microeconomic Theory
Factor Demand/Firm Behavior
1
Firm Behavior

Given its technology, we now want to develop a
model of firm behavior.

What is our basic assumption about firm behavior?
2
Firm Behavior

Standard assumption: firms make decisions
to maximize profits, or maximize total
revenue minus total costs.
  pq  ( w1 x1  w 2 x 2  ...  w n x n )
where q = f(x1,…,xm)

Decision process can be broken up into two
parts:

For any given level of output, what combination
of inputs should firm use? (Optimal Production)

Given optimal production, how much should it
produce/supply? (Supply)
3
Production

We will first consider the Production decision.

Key idea:
 Consider any level of output for firm: q
 If firm is producing q in the profit maximizing
way, it must be producing q using the cost
minimizing process (why must this be the
case?)

So, key to modeling production behavior is to
think about how firms can minimize cost of
producing any given level of output.
4
Costs

When economists think about costs, they
think much more broadly than do
accountants.

Costs include not only direct costs that must
be paid for, but indirect costs or
“opportunity costs”

Opportunity cost - Lost revenue from
failing to use an input factor for next best
use.
5
Costs

Suppose Bill is currently working at a coffee shop on Saturday
nights 8-12 for \$10/hr (for which his is paid at the end of every
month). He also has \$1500 in a money market account earning 1%
per month.

A friend suggests Bill should DJ parties on Sat nights for the next
month. To do so, Bill would have to rent a sound and light system
for \$1000 (which would have to be paid at the beginning of the
month).

From the perspective at the end of the month, what would be the
economic cost for Bill to be a DJ?
6
Iso-Cost Curves

From now on, when we consider costs, we consider economic costs
(meaning opportunity costs are implicitly included)

Now we want to consider the costs of different input bundles.

If we again consider the simple two-input case we can think of the cost of
different input bundles via Iso-cost curves.

What is an Iso-cost curve?
7
Iso-Cost Curves


Iso-cost curves - all combinations of inputs that cost the
same amount.
Example: Suppose input prices are w1 = \$10 w2 = \$20.

What would \$100 Iso-cost curve look like?

How about for the \$200 Iso-cost curve?

What happens to Iso-cost curves when input prices
change to w1 = \$20 w2 = \$20?

So what is slope and interpretation of slope of an Iso-cost
curve?
8
Production Decision

Consider a firm with a technology given by a production function
F(x1, x2) = x10.5 x20.5, and price of the inputs are w1 = 1 and w2 = 2.





What would q = 100 iso-quant look like? What would iso-cost curves
look like?
So how would we graphically characterize the “optimal” input bundle for
this firm to use to produce 100 units of output?
How do you interpret this?
What about a different level of output, say q = 200?
What if prices were w1 = 2 and w2 = 2?
9
Production Decision

So firm’s decision regarding which input bundle to use to produce a
given level of output is similar (but not identical) to individual’s
decision regarding which bundle good he should consume to
produce utility.


Individual - Chooses bundle that is on highest Indifference Curve, but
is still on budget constraint for \$m.
(“Utility Maximization”)
Firm – To produce q units of output, choose input bundle that is on
lowest Iso-cost curve, but still is on Isoquant curve for q.
(“Cost Minimization”)
10
Conditional Factor Demands

In consumer theory, choice problem over
good given different prices gives demand
curve for a good.

In producer theory, choice problem over
different input factors given prices gives
conditional factor demand curve



Denoted x1(w1, w2, q)
Still the relationship between quantity
demanded (in this case of an input) and its
own price.
However, it is “conditional” because it gives
demand for inputs conditional on producing
a given amount q.
11
Conditional Factor Demand Curve

We can derive the Conditional Factor Demand Curve for a given
input factor by choosing a given output level, then varying the
price of one input while the holding prices of other inputs constant.

Graphically?

How will this conditional factor demand curve change if we consider a
higher level of output?

How about higher price for the other input?
12
Thinking about Conditional Factor Demand Curves

We can still think about things like factor demand elasticity, or
the percentage change in quantity demanded due to a
percentage change in its own price.

What will factor demand elasticity generally depend on?
13
Conditional Factor Demands Analytically

How would we derive conditional factor demand function for
quantity q analytically, say for a Cobb-Douglas production function:
F(x1, x2) = x1ax2b and input prices w1 and w2?

What two conditions hold at optimal input bundle in example from
above?

How do we use these to derive optimal input bundle for any given q ?
14
Conditional Factor Demands Analytically

So for Cobb-Douglas Technology, conditional factor demands for the two
factors, conditional on producing q units, are given by the following two
equations:
 a w2 

x1 ( w1 , w 2 , q 0 )  

b
w
1 

b /( a  b )
 b w1 

x 2 ( w1 , w 2 , q 0 )  

a
w
2 

q
1 /( a  b )
a /( a  b )
q
1 /( a  b )
15
Conditional Factor Demand Curve Analytically

So what will be conditional factor demand curve for input x1 given
production function F(x1, x2) = x10.5x20.5, w1 = 2 and w2 = 8?


What happens to conditional demand for x1 as its own price rises? The
price of the other input falls?

So if x1 and x2 are only inputs (i.e. no other direct or opportunity costs) and
firm is a cost minimizer, how much will it cost to produce 100 units of
16
Is there (Middle Class) Life After Maytag?

Describe the “economics” of the New York
Times article on the changes in the factor
demand for labor by Maytag.
17
Is there (Middle Class) Life After Maytag?
K
K
\$1 million isocost
\$1 million isocost
q=1500
q=1500
900
Newton Factory
LNewton
900
LX
Factory in City X
So if labor in Newton and City X cost the same, the initial situation for
producing 3000 units of output would be like above, and would cost \$2 million.
But cost of Labor is not the same in Newton and Monterey, so what happens?
18
Is there (Middle Class) Life After Maytag?
K
K
\$1 million isocost (Monterey)
\$1 million isocost
\$1.2 million isocost
\$700,000 isocost
q=2000
q=2000
900
Newton Factory
LNewton
900 1000 1600
q=4000
LMonterey
Monterey Factory
So in example above, what happens to labor input? How much money
is saved?
Besides the “economics,” what else does the article emphasize?
19
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