Transcript Chapter 10 - Pegasus @ UCF

Chapter 10 –Theory of Production
and Cost in the Long Run(LR)
 The theory of production in the LR provides the
theoretical basis for firm decision-making and LR
costs and supply.
 In essence, we will assume that the firm’s goal is
to maximize output subject to a cost constraint.
We will see that this is the same as minimizing the
cost of producing a given level of output.
 Keep in mind that all inputs are variable in the LR
– plant size can be changed,
– new locations can be chosen
10.1
Production Isoquants
 An isoquant is a locus of points indicating
different combinations of 2 inputs each of
which yields the same level of output.
 Note 2 inputs are assumed since we desire
to present model graphically.
Q  f ( L, K )
10.2
Characteristics of Isoquants
 Negative slope – tradeoffs, if more of L then
less of K if output is held constant
 Convex to the origin – diminishing MRTS,
the more of L you have relative to K the
more able you are to trade L for K and hold
output constant.
 Isoquants cannot intersect
10.3
Marginal Rate of Technical
Substitution
 The MRTS is the (negative of the) slope of
the isoquant. Therefore it reflects
K
MRTS 
L
It is a measure of the number of units of K that must be
given up if L is increased by a single unit, holding output
constant. Note it will diminish as we move down an
isoquant.
10.4
Concept of an Isoquant Map
 Graph of several isoquants each
representing different levels of output.
 The higher (further from the origin) an
isoquant, the greater the level of output.
10.5
Marginal Product and MRTS
 Marginal product of an input is the change
in total product in response to increasing the
variable input by a single unit.
 The change in total product is given by the
following equation
Q  (MPL  L)  (MPK  K )
10.6
Marginal Product and MRTS
Q  (MPL  L)  (MPK  K )
Along an isoquant the change in output is equal to zero
and
0  ( MPL  L)  ( MPK  K )
 ( MPK  K )  ( MPL  L)
 K
L
MP
L

MPK
 MRTS
10.7
The Cost Constraint – Isocost
Lines
 Suppose you have $100, C, to spend on two inputs
, L & K, and the prices of each are $10, PL, and
$20, PK, respectively. Determine the equation
relating K to L reflecting your budget constraint.
 100 = 10L+20K or
 K=5-0.5L
 In general, the cost constraint is
 K = C/PK-(PL/PK)L
 Note linear and slope is ratio of prices
10.8
Changes in Isocost
 What happens to the isocost if cost, C,
changes?
 What happens to budget line if one of the
prices change?
 K = C/PK-(PL/PK)L, w=PL, r=PK,C-bar =
cost level then isocost is
C w
K
 L
r
r
10.9
Change in Cost
Budget line I – C=100, PX=10, PY=20
Y Budget Line II – C=140, Prices same
7
5
I
10
14
X
10.10
Change in Price
K
Isocost I – C=100, PL=10, PK=20
Isocost II – C=100, PL=20, PK=20
7
5
II
5
I
10
14
L
10.11
Determining the Optimal
Combination of Inputs
 Producer’s goal is to maximize profits:
– Minimize cost of producing a constant level of
output
– Maximize output subject to a cost constraint
 The isocost line shows what combinations
of L and K that the producer is able to
purchase with a fixed cost level.
 The isoquant map shows the producer’s
preferences for X and Y.
10.12
Minimizing Cost of Producing a
Given Level of Output
 The Optimal Solution, where the producer
minimizes cost subject to an output
constraint, is found where the isocost line is
tangent to an isoquant. Since isoquants
cannot intersect this will be the highest
possible level of utility given the constraint.
 See Figure 10.4 page 366.
10.13
Cost Minimization
 At any tangency point the slopes of the two
relationships must be equal.
 Slope of isoquant is the MRTS – the rate the
producer is willing to substitute K for L, holding
output constant.
 Slope of isocost line is the ratio of prices, PL/PK,
which reflects the rate the producer is able to
substitute K for L and maintain constant cost.
10.14
Cost Minimization
PL
MRTS 
PK
Rate willing to sub = Rate able to sub
10.15
Cost Minimization
 Recall the Marginal Product interpretation
of the MRTS or slope of the isoquant. Note
PL = w and PK = r in text.
MPL
PL
MRTS 

MPK
PK
MPL
MPK

PL
PK
10.16
Equilibrium for the Firm
 A producer is hiring 20 units of labor and 6
units of capital (bundle A). The price of labor
is $10, the price of capital is $2, and at A, the
marginal products of labor and capital are
both equal to 20.
 Is the firm in equilibrium?
 No, MP to price ratios are not equal, should
use more capital and less labor.
 Beginning at A, what happens to output and
cost if the producer increases labor by one
unit and decreases capital by 1 unit?
10.17
 Output remains constant and cost increases
Equilibrium for the Firm
 A producer is hiring 20 units of labor and 6
units of capital (bundle A). The price of labor
is $10, the price of capital is $2, and at A, the
marginal products of labor and capital are
both equal to 20.
 In equilibrium, which of the following will be
true?
– MPL will be less than 20.
– MPK will be more than 20.
– MPL will be 5 times MPK.
10.18
Expansion Path
 An expansion path is a curve that shows the
least costly combination of two inputs
required to produce each level of output,
holding the input price ratio constant.
 See Figure 10.6, page 373.
 Along an expansion path,
MPL
MRTS 
MPK
MPL
MPK

PL
PK
PL

PK
10.19
Expansion Path
The following is always true along an expansion
path.
MPL
PL
MRTS 

MPK
PK
MPL
MPK

PL
PK
10.20
Cost Curve Derived from
Expansion Path
 Since the Expansion Path plots points the optimal
combination of inputs required to produce each
level of output, total cost for each level of output
can be determined since it is assumed that the
prices of inputs are fixed.
 Thus, if the optimal quantity of labor and capital
to produce 100 units of output are 10 and 5
respectively, and the wage rate is $20 and price of
capital, $50 then the total cost is
$20(10) + $50(5) = $450
10.21
Returns to Scale
 Returns to Scale deals with the impact on output
of a change in the scale(proportional changes in all
inputs) of a firm’s operations.
 Returns to scale can be classified as
– Constant: output changes proportionately to the change
in the inputs
– Increasing: output changes more than proportionate to
the change in the inputs
– Decreasing: output changes less than proportionate to
the change in the inputs
10.22
Returns to Scale
 Recall the general form of our production
function is Q = f(L,K). Now, suppose we
increase all inputs by the factor c as
represented in the following production
function,
F(cL, cK) = zQ
 What are the returns to scale if
• z=c?
• z>c?
• z<c?
10.23
Long Run Costs
 The long run average, LAC, and marginal,
LMC, cost curves have the same basic
shape that the equivalent short run cost
curves.
 However, the reason why each is U-shaped
is for different reasons, which are
– Short run – the Law of Diminishing Marginal
returns
– Long run – economies/diseconomies of scale
10.24
Economies of Scale
 Economies of Scale exist when LAC
decreases as output increases.
 Diseconomies of Scale exist when LAC
increases as output increases.
LAC
economies
diseconomies
Q
10.25
Economies of Scale
 Reasons for economies of scale are
– Specialization and division of labor
– Better meshing of equipment
– Economies on capital purchases – machines
that are 10 times as productive may not cost 10
times as much
– More capital intensive
 Reason for diseconomies of scale
– Inefficiency in management
10.26
Economies of Scope
 Scope economies exist if the joint costs of
producing two or more products is less than
the separate costs of producing each
individually.
 An example might be an auto air
conditioning repair shop that adds
radiator/cooling system repairs
10.27
Relationship between SR and LR
Cost Curves
 The LAC curve is a locus of points on SAC
curves, which represent the most efficient
(cost effective) way of producing each level
of output given that the firm has the
opportunity and ability to change the
quantity of any and all inputs.
 See Figure 10.14 page 391.
10.28