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Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 15 Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets
IV. Market failure Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets Partial equilibrium (Ch. 16)
Exchange economies (Ch. 31)
Production (Ch. 18–23)
IV. Market failure Production
Technological constraints of a ﬁrm represented by a production set in
inputoutput space.
Consider the one input one output case. Boundary of production set is y D f .x/;
where the function f is called the production function. Production
Now consider the two input one output case.
Production function: y D f .x1; x2/;
where y is the output level, and x1 and x2 are the input levels.
Inputs 1 and 2 are sometimes called production factors. Production
N
Isoquants: Set of input combinations .x1; x2/ with f .x1; x2/ D y , for given
y.
N Three leading examples
ab
CobbDouglas: f .x1; x2 / D Ax1 x2 . Perfect complements: f .x1; x2/ D minfx1; x2g.
Perfect substitutes: f .x1; x2/ D ax1 C bx2. Standard assumptions on technology
Monotonicity: production function f .x1; x2/ is increasing in x1 and x2
Convexity: isoquants are convex Marginal product. Technical Rate of Substitution.
Consider two input model.
Marginal product of factor 1: MP1.x1 ; x2 / D @f
.x1 ; x2 /;
@x1 and analogously for factor 2.
Technical rate of substitution: TRS.x1; x2/
D slope of the isoquant that passes through .x1; x2/ MP1.x1 ; x2/
MP2.x1 ; x2/ Diminishing MP and TRS
Two standard properties of technologies: Law of Diminishing MP: MP1 decreases in x1 for ﬁxed x2, and likewise
for MP2.
Diminishing TRS: jTRS j decreases in x1 along isoquant. Longrun vs. shortrun
Within a certain ﬁxed period of time it might be infeasible to change some
production factors. y D f .x1; x2/
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with x2 ﬁxed. This is the short run.
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Eventually, all factors can change. This is the long run. Returns to scale
When we talked about the law of diminishing MP, we consider the change in
MP1 due to a change in x1 holding x2 ﬁxed.
Now we are going to consider joint changes in x1 and x2.
If we double all inputs, does output double, less than double or more than
double? Returns to scale
Constant returns to scale: for all t > 1,
f .tx1; tx2 / D tf .x1; x2/
Increasing returns to scale: for all t > 1,
f .tx1; tx2 / > tf .x1; x2/
Decreasing returns to scale: for all t > 1,
f .tx1; tx2 / < tf .x1; x2/ Proﬁt maximization
One output y , two inputs x1 and x2.
A competitive ﬁrm takes prices p , w1 and w2 as given and solves: max .y;x1 ;x2 / py w1x1 w2x2 subj. to y D f .x1; x2/
or, equivalently, max pf .x1; x2/ .x1 ;x2 / w1x1 w2x2: Proﬁt maximization
Firstorder conditions: MP1.x1; x2/ D w1=p
MP2.x1; x2/ D w2=p;
which is TRS.x1; x2/ D w1
:
w2 (Note the analogy with consumer theory.)
This characterization is valid whenever isoquants are smooth and convex
and solution is interior (i.e., both x1 and x2 are positive). Constant returns implies zero proﬁts
If proﬁts were not zero, by doubling input one could double the proﬁt. If
initial proﬁt were positive, ﬁrm could not be choosing an input combination
that maximizes its proﬁts!
Let us see this more formally....
At the optimum, proﬁt is
… D pf .x1 ; x2 /
w1x1
w2x2 ;
* where .x1 ; x2 / is the proﬁtmaximizing combination of inputs. Using the constant returns to scale assumption, doubling the inputs yields a
proﬁt of
pf .2x1 ; 2x2 /
2w1x1
2w2x2 D 2…; which is strictly greater than …, unless … D 0. Hence, … must be zero,
otherwise .x1 ; x2 / would not be proﬁtmaximizing. Cobbdouglas example
ab
Technology: y D f .x1; x2/ D x1 x2 . FOC:
a
b
ax1 1x2
ab
bx1 x2 1 w1
D MP1.x1; x2/ D
p
w2
D MP2.x1; x2/ D
p A bit of algebra (see appendix of Chapter 19) yields yDp aCb
1ab a
w1 a
1ab b
w2 b
1ab Notice that this is not well deﬁned for a C b D 1 (constant returns). Cost minimization
For each output level y , a competitive ﬁrm must choose a combination of
inputs to minimize cost subject to the constraint that the combination of
inputs yields output y .
Cost minimization: min .x1 ;x2 / w1x1 C w2x2 subj. to f .x1; x2/ D y The value of w1x1 C w2x2 at the minimum is denoted c.y/ and it is called
the cost function. Cost minimization
Proﬁt maximization then becomes: max py
y 0 c .y/ with ﬁrstorder condition: p D c 0.y/ M C.y/;
for interior solutions y > 0. Cost minimization
Now let c.y/ D cv .y/ C F , where F > 0 is the ﬁxed cost and cv is the
variable cost, i.e., cv .0/ D 0.
The average cost is deﬁned as AC.y/ D c.y/=y , and the average variable
cost is AV C.y/ D cv .y/=y .
The marginal cost curve intersects the average cost and the average
variable cost curves at their minima.
In fact, if y minimizes AC then M C.y /y c .y /
0 D AC .y / D
y 2
0 and analogously for AV C . ) M C.y / D c.y /=y D AC.y /; Supply curve
The (inverse) supply curve is obtained from the ﬁrstorder condition p D M C.y/;
with two exceptions.
First, the ﬁrm would never choose an output level y with p D M C.y/ when
M C is decreasing around y . Otherwise, by increasing y a little bit it would
increase proﬁts.
Second, the ﬁrm will choose a positive output level y with p D M C.y/ only
if this yields higher proﬁts than producing zero: py cv .y/ F F ) p AV C.y/: In sum, the inverse supply curve is the increasing portion of the marginal
cost curve that lies above the average cost curve. ...
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.
 Spring '09
 SAMUELSON
 Microeconomics

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