Isoquants,%20Isocosts%20and%20Cost%20Minimization

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Isoquants, Isocosts and Cost Minimization Overhead
s
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We define the production function as
y represents output
f represents the relationship between y and x
xj is the quantity used of the jth input
(x1, x2, x3, . . . xn) is the input bundle
n is the number of inputs used by the firm
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Holding other inputs fixed, the production
function looks like this
y f (x1, x2, x3, . . . xn )
350
300
Output -y
250
200
150
100
50
0
0
2
4
6
8
10
12
Input -x
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Marginal physical product
Marginal physical product is defined as the
increment in production that occurs when an
additional unit of a particular input is employed
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Mathematically we define MPP as
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Graphically marginal product looks like this
60
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Output -y
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
-10
-20
-30
-40
Input -x
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The Cost Minimization Problem
Pick y observe w1, w2, etc choose the least
cost xs
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Isoquants
An isoquant curve in two dimensions
represents all combinations of two inputs that
produce the same quantity of output
The word isomeans same
while quant stand for quantity
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Isoquants are contour lines of the production
function
If we plot in x1 - x2 space all combinations of
x1 and x2 that lead to the same (level) height
for the production function, we get contour
lines similar to those you see on a contour map
Isoquants are analogous to indifference curves
Indifference curves represent combinations of
goods that yield the same utility
Isoquants represent combinations of inputs that
yield the same level of production
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Production function for the hay example
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Another view
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Yet another view (low xs)
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With a horizontal plane at y 250,000
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With a horizontal plane at 100,000
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Contour plot
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Another contour plot
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There are many ways to produce 2,000 bales of hay
per hour
Workers Tractor-Wagons Total Cost AC
10 1 80 0.04 6.45 1.66 71.94 .03597 5.
48 2 72.8658 0.0364 3.667 3 82.0015 0.041
2.636 4 95.8167 0.0479 1.9786 5 111.872 .05
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Plotting these points in x1 - x2 space we obtain
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Or
Isoquant y 2,000
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X
10
1
8
6
4
2
0
0
2
4
6
8
10
12
14
X
2
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Cutting Plane for y 10,000
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Isoquant for y 10,000
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Only the negatively sloped portions of the
isoquant are efficient
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Isoquant for y 10,000
x1 x2 output y -- 1 10,000 -- 2 10,000 -- 3
10,000 12.469 4 10,000 9.725 5
10,000 8.063 6 10,000 6.883 7 10,000 5.990 8 1
0,000 5.290 9 10,000
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y 10,000
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y 2, 000 y 10,000
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Graphical representation
Isoquants y 2,000, y 10,000
14
12
x1
10
8
6
4
2
0
0
2
4
6
8
10
12
14
x
2
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y 2,000, y 5,000, y 10,000
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More levels
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And even more
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Comparison to full map
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Slope of isoquants
An increase in one input (factor) requires a
decrease in the other input to keep total
production unchanged
Therefore, isoquants slope down (have a negative
slope)
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Properties of Isoquants
Higher isoquants represent greater levels of
production
Isoquants are convex to the origin
This means that as we use more and more of an
input, its marginal value in terms of the
substituting for the other input becomes less and
less
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Slope of isoquants
The slope of an isoquant is called the marginal
rate of (technical) substitution MR(T)S
between input 1 and input 2
The MRS tells us the decrease in the quantity of
input 1 (x1) that is needed to accompany a one
unit increase in the quantity of input 2 (x2),
in order to keep the production the same
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The Marginal Rate of Substitution (MRTS)
14
12
x1
10
8
?
6
?
4
2
?
0
0
2
4
6
8
10
12
14
x
2
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Algebraic formula for the MRS
The marginal rate of (technical) substitution
of input 1 for input 2 is
We use the symbol - y constant - to
remind us that the measurement is along a
constant production isoquant
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Example calculations y 2,000
Workers Tractor-Wagons x1 x2 10 1 5.48 2
3.667 3 2.636 4 1.9786 5
Change x2 from 1 to 2
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Example calculations y 2,000
Change x2 from 2 to 3
Workers Tractor-Wagons x1 x2 10 1 5.48 2
3.667 3 2.636 4 1.9786 5
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More example calculations y 10,000
x1 x2 12.469 4 9.725 5 8.063 6 6.883 7 5.990 8
Change x2 from 5 to 6
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A declining marginal rate of substitution
The marginal rate of substitution becomes larger
in absolute value as we have more of an input.
The amount of an input we can to give up and
keep production the same is greater, when we
already have a lot of it.
When the firm is using 10 units of x1, it can
give up 4.52 units with an increase of only 1
unit of input 2, and keep production the same
But when the firm is using only 5.48 units of x1,
it can only give up 1.813 units with a one unit
increase in input 2 and keep production the same
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Slope of isoquants and marginal physical product
Marginal physical product is defined as the
increment in production that occurs when an
additional unit of a particular input is employed
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Marginal physical product and isoquants
All points on an isoquant are associated with
the same amount of production
Hence the loss in production associated with
?x1 must equal the gain in production from ?x2 ,
as we increase the level of x2 and decrease the
level of x1
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Rearrange this expression by subtracting MPPx2 ?
x2 from both sides,
Then divide both sides by MPPx1
Then divide both sides by ? x2
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The left hand side of this expression is the
marginal rate of substitution of x1 for x2, so
we can write
So the slope of an isoquant is equal to the
negative of the ratio of the marginal
physical products of the two inputs at a given
point
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The isoquant becomes flatter as we move to the
right, as we use more x2 (and its MPP declines)
and we use less x1 ( and its MPP increases)
So not only is the slope negative, but the
isoquant is convex to the origin
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The Marginal Rate of Substitution (MRTS)
14
12
x1
10
8
?
6
4
2
?
0
0
2
4
6
8
10
12
14
x
2
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Approx x1 x2 MRS MPP1 MPP2 12.4687 4.00
00 --- 664.6851 2585.7400 11.8528 4.1713 -3
.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1
010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.13
21 1975.4051 8.0629 6.0000 -1.5941 1254.9695 172
4.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951
6.8827 7.0000 -1.0291 1466.3867 1489.5380
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Approx x1 x2 MRS MPP1 MPP2 12.4687 4.00
00 --- 664.6851 2585.7400 11.8528 4.1713 -3
.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1
010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.13
21 1975.4051 8.0629 6.0000 -1.5941 1254.9695 172
4.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951
6.8827 7.0000 -1.0291 1466.3867 1489.5380
x2 rises and MRS falls
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Isocost lines
An isocost line identifies which combinations of
inputs the firm can afford to buy with a given
expenditure or cost (C), at given input prices.
Quantities of inputs - x1, x2, x3, . . .
Prices of inputs - w1, w2, w3, . . .
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Graphical representation
Cost 120 w1 6 w2 20
22
20
x
1
18
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
x
2
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Slope of the isocost line
So the slope is -w2 / w1
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Example
C 120, w1 6.00, w2 20.00
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Intercept of the isocost line

So the intercept is C / w1
With higher cost, the isocost line moves out
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Isoquants and isocost lines
We can combine isoquants and isocost lines to
help us determine the least cost input combination
The idea is to be on the lowest isocost line that
allows production on a given isoquant
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Combine an isoquant with several isocost lines
Isocost lines for 20, 60, 120, 180, 240, 360
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Consider C 120 and C 180
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X
20
1
16
12
8
4
0
3
4
5
6
7
8
9
10
11
12
13
X
2
At intersection there are opportunities for trade
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Add C 160
24
X
20
1
16
12
8
4
0
4
5
6
7
8
9
10
11
12
13
X
2
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Add C 154.6
24
X
20
1
16
12
8
4
0
4
5
6
7
8
9
10
11
12
13
X
2
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In review
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X
20
1
16
12
8
4
0
4
5
6
7
8
9
10
11
12
13
X
2
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The least cost combination of inputs
The optimal input combination occurs where the
isoquant and the isocost line are tangent
Tangency implies that the slopes are equal
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Slope of the isocost line
-w2 / w1
Slope of the isoquant
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Optimality conditions
Slope of the isocost line Slope of the
isoquant
Substituting we obtain
The price ratio equals the ratio of marginal
products
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We can write this in a more interesting form
Multiply both sides by MPPx1
and then divide by w2
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Graphical representation
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Statement of optimality conditions
a. The optimum point is on the isocost line
b. The optimum point is on the isoquant
c. The isoquant and the isocost line are
tangent at the optimum combination of x1 and x2
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(No Transcript)
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f. The marginal product of each input divided
by its price is equal to the marginal product
of every other input divided by its price
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Example Table w1 6, w2 20
To get an x1, I can give up 3.33 x2 in terms of
cost
Approx x1 x2 MRS MPP1 MPP2 MPP1/w1 MPP2/w2 MR
S -w2 / w1 1.0000 -3.3333 2.0000
-3.3333 3.0000 -3.3333 12.4687 4.0000 66
4.6851 2585.7400 110.7809 129.2870 -3.8902 -3.3333
11.8528 4.1713 -3.5946 739.5588 2465.2134 123.259
8 123.2607 -3.3334 -3.3333 9.7255 5.0000 -2.5672 1
010.5290 2050.0940 168.4215 102.5047 -2.0287 -3.33
33 9.3428 5.1972 -1.9411 1063.1321 1975.4051 177.1
887 98.7703 -1.8581 -3.3333 8.0629 6.0000 -1.5941
1254.9695 1724.5840 209.1616 86.2292 -1.3742 -3.33
33 6.9792 6.9063 -1.1959 1447.3307 1508.9951 241.2
218 75.4498 -1.0426 -3.3333 6.8827 7.0000 -1.0291
1466.3867 1489.5380 244.3978 74.4769 -1.0158 -3.33
33 5.9898 8.0000 -0.8929 1663.9176 1305.9560 277.3
196 65.2978 -0.7849 -3.3333 5.2904 9.0000 -0.6994
1855.3017 1155.0760 309.2169 57.7538 -0.6226 -3.33
33 4.7309 10.0000 -0.5595 2044.1823 1026.1700 340.
6971 51.3085 -0.5020 -3.3333 4.2773 11.0000 -0.453
5 2232.4134 912.4680 372.0689 45.6234 -0.4087 -3.3
333 3.9071 12.0000 -0.3702 2420.9761 809.4240 403.
4960 40.4712 -0.3343 -3.3333 3.6042 13.0000 -0.302
9 2610.3937 713.8360 435.0656 35.6918 -0.2735 -3.3
333
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Intuition for the conditions
The isocost line tells us the rate at which the
firm is able to trade one input for the
other, given their relative prices and total
expenditure
For example in this case the firm must give up 3
1/3 units of input 1 in order to buy a unit of
input 2
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w1 6 w2 20 C 180
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X
20
1
16
12
10
8
4
0
4
5
6
7
8
9
10
11
12
13
3
X
2
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The isoquant tells us the rate at which the
firm can trade one input for the other and
remain at the same production level
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x
20
1
16
12
8
4
0
4
5
6
7
8
9
10
11
12
13
x2
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If there is any difference between the rate
at which the firm can trade one input for
another with no change in production and the
rate at which it is able to trade given relative
prices, the firm can always make itself better
off by moving up or down the isocost line
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The isoquant tells us the rate at which the
firm can trade one input for the other and
remain at the same production level
24
x
20
1
16
12
8
4
0
4
5
6
7
8
9
10
11
12
13
x2
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When the slope of the isoquant is steeper than
the isocost line, the firm will move down the line
When the slope of the isoquant is less steep than
the isocost line, the firm will move up the line
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When the slope of the isoquant is steeper than
the isocost line, the firm will move down the line
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x
20
1
16
12
8
4
0
4
5
6
7
8
9
10
11
12
13
x2
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The End