Microeconomics and Policy Analysis PUAF 640

1
Microeconomics and Policy Analysis PUAF 640

• Professor Randi Hjalmarsson
• Fall 2008
• Lecture 8

2
Class Outline (Chapters 8, 9)

• Production functions (contd)
• Optimal Input Choices (finally!) the third
  question
• More on cost curves
• Short run versus Long run

3
Brief Review of Firm Concepts

• Question 1 If the firm stays open, how does it
  choose Q?
• It chooses Q to maximize profit TR(Q) TC(Q)
• Marginal Output Rule Q such that MR(Q) MC(Q)!

TC
MR, MC ()
TR, TC ()
TR(Q) p(Q)Q
MC
p
MR
Q
Q
Q
Q
4
Brief Review of Firm Concepts

• Second Question When should the firm stay open
  or shut down?
• Shut down rule. Shut if
• p lt 0, or
• Average p lt 0 for all Q, or
• AR lt AC for all Q

()
Price taking firm
Non-Price taking firm
()
AC
AC
ARp
ARD
Q
Q
5
Brief Review of Firm Concepts

• Third Question What combination of inputs should
  the firm use to produce Q?
• Responses of production to changes in inputs
• Production functions
• Marginal physical product (MPP) of an input
• Changes in 1 input, holding other input constant
• Returns to scale
• Changes in output when both inputs change
  proportionally

6
Representation of the firms production
technology (Q as a function of L and K)

• Isoquant curve showing all the combinations of
  K and L that yield the same level of output.

K
Why does it slope downward?
b
Q30
Q20
a
Q10
Process can be labor intensive a
L
Process can be capital intensive b
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Marginal rate of technical substitution

• MRTSKL -(Slope of isoquant) -?K/?L
• MRTS of K for L
• It measures the tradeoff firms can make between K
  and L, holding output fixed.
• At low L and high K (b), MRTS is high.
• Can cut a lot of capital but keep Q constant with
  just a little more labor.
• MRTS decreases along isoquant.
• MRTSKL MPPL/MPPK.
• If change K and L, then change in Q is
• ?Q ?LMPPL ?KMPPK
• But, if moving along an isoquant, ?Q 0
• ?LMPPL ?KMPPK ? ?K/?LMPPL/MPPK

K
b
a
Q0
L
8
Special Cases of Isoquants

• A person (L) produces 10 units/hr. A robot (K)
  produces 20 units/hr. What is the production fcn?
• f(K,L) 10L 20K
• Draw isoquants Q0 100, Q1 200
• MRTS ?
• -?K/?L -(-5/10) .5
• Every isoquant has same MRTS.
• And MRTS is constant along isoquant.
• Can give up ½ robot for 1 person,and produce
  same amount.
• Perfect Substitutes
• L and K can always be substituted at the same
  rate.

K
10
5
200
100
10
20
L
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Perfect Complements

• 2 inputs used together in constant proportion
  no substitution of factors is possible.
• Extra L doesnt yield more Q unless K is also
  increased in a fixed proportion.

K
Q2
Q1
Q0
L
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Total Cost (as a function of K and L)

• What is the total cost of production?
• w cost of unit of labor (wage)
• v cost of unit of capital (rental rate)
• TC wL vK
• For a given total cost, there are multiple
  combinations of capital and labor.

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Isocost Lines

• TC0 wL vK
• All combos of L and K that give same cost, TC0.
• K (TC0/v) (w/v)L
• Slope of isocost line -w/v
• Ratio of input prices
• Rate at which firm can substitute purchase of one
  input for another without changing expenditure.
• Rate at which can substitute in the market, not
  in production.

K
TC2
TC0/v
TC1
TC0
TC0/w
L
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Question 3 Input choice, finally!

• A firm decides to produce Q. How much K and L
  should it use?

Firm wants to maximize profits ? minimize costs.
K
Choose K, L isocost tangent to isoquant
Optimality Condition Slope isocost Slope
isoquant w/v MRTS MPPL/MPPK
K
Q
TC2
TC1
TC0
MPPL/w MPPK/v
L
L
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• L and K will be a function of
• Price of labor (w)
• Price of capital (v)
• Output level, Q
• Demands for L and K that we get from cost
  minimization are called derived demands, because
  they are demands given output level, Q.

14
Lagrangians Again

• Now have basis for a Lagrangian.
• Choose K,L to minimize TC subject to Q Qwhere
  TC wL vK and Q f(K,L)
• Note that Q is already decided.

L wL vK ?(Q - f(K,L))
Constraint, rewritten so that it is equal to zero.
Function to be minimized
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Lagrangian Example

• TC wL vK and Q L½K½
• Firms problem Find L and K
• Min TC wL vK s.t. Q L½K½
• Step 1 What is the Lagrangian?
• L wL vK ?(Q - L½K½)

K
K
Q
TC1
L
L
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Step 2 Take partial derivatives for K, L, and ?.
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Step 3 Solve system of equations.
Set (i) and (ii) equal
Plug into (iii)
These are derived demands for L and K.
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Can also find total cost
19
Practice Question

• Production Function f(K,L) 2KL
• Profit maximizing output Q
• TC wL vK
• Solve for optimal input levels L and K
• (Note these are in terms of w, v, and Q)
• If the price of labor 10 and the price of
  capital 20, what quantity of labor and capital
  maximizes profits?

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Comparative Statics Response of optimal inputs
to changes in input prices

• Consider wage increase.
• Affect on isocost line?
• Rotate in (afford less L for same ).
• This isocost line is still for TC0.
• Can no longer afford to produce Q.
• A new family of isocost lines. (A family of
  isocost lines always has the same slope.)
• Find new isocost line that is tangent to Q.
• Substitute away L to K (from input that became
  relatively more expensive).

K
New equilibrium
TC0/v0
Initial equilibrium
K1
K0
Q
L0
L1
L
TC0/w1
TC0/w0
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Comparative Statics Response of optimal inputs
to changes in Q

• Suppose Q increases.
• Clearly cant produce Q1 with TC0.
• Total cost must increase andat least one input
  must increase.
• Draw expansion path that traces out optimal input
  combinations for varying levels of output.

K
TC0/v0
Q2
K0
Q1
Q0
L0
L
TC1/w0
TC2/w0
TC0/w0
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Cost Functions in More Detail

• Total Cost Functions found by putting derived
  input demands back into the total cost function.
• TC(Q,w,v) wL(Q,w,v) vK(Q,w,v)
• Can also include a Fixed Cost component (as well
  soon see).
• Should now better understand the shape of TC
  curves.
• Expansion path indicates that it slopes up (costs
  more to produce more).
• Shape also depends on whether the production
  function displays CRTS, DRTS, or IRTS.

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• CRTS ? ?ing inputs by a yields ? in Q by a
• TC is linear
• To double output, you double input. Also double
  costs.

• IRTS ? ?ing inputs by a yields ? in Q by gt a
• TC increases at decreasing rate.
• To double output, you need less than double
  inputs, and less than double costs.

TC
TC
Q
Q
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• DRTS ? ?ing inputs by a yields ? in Q by lt a
• TC increases at increasing rate.
• To double output, you need more than double input
  and more than double costs.

• Typical firm
• Experiences portion of each of these curves over
  all Q.

TC
TC
Q
Q
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Review Marginal and Average Cost

• MC slope of TC curve
• MC ?TC/?Q dTC/dQ
• AC ratio of TC to output TC/Q
• Slope of ray from the origin to point on curve.

MC, AC
TC
MC
AC
AC initially gt MCCross at min ACAC then lt MC
MC AC(min AC)
Q
Q
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Short Run (SR) versus Long Run (LR)

• So far, weve assumed that firms can freely
  choose how much L and K to use.
• Is this realistic?
• In the short run, inputs may not be variable.
• Union contracts may limit size of L.
• Factory size cannot be changed.
• So, weve been looking at long run cost curves,
  since all inputs are variable.
• In the short run, hold one input fixed.

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Short Run Cost Functions

• If K is variable, a, b, and c are cost minimizing
  bundles for each Q (long run).
• K is fixed K K
• Optimizing bundles changes if firm still wants to
  produce same Q a, b, c
• b is same as b
• Dashed isocost lines SR TC!
• Solid isocost lines LR TC

K
c
b
c
a
K
Q2
a
Q1
Q0
L
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Short Run Cost Functions

• At a and c, MRTS ? w/v !
• At a, firm would like to hire more L and less K,
  but it cant change K.
• What about at c?
• At a and c, TCSR gt TCLR.
• Isocost lines are higher.
• Not being able to adjust an input tends to
  increase production costs.
• Derived demand for L
• LL(Q, K) w is irrelevant.

K
c
b
c
a
K
Q2
a
Q1
Q0
L
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What does SRTC function look like?

• TC wL vK
• Find SRTC by substituting in derived demands
• TCSR wL(Q,K) vK.
• TCSR VCSR FCSR

Variable Costs
Fixed Costs
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Graph of TCSR TCSR VCSR FCSR
TCSRwL(Q,K) vK
TC ()
VCSRwL(Q,K)
Can find MC, SRATC, and SRAVC in same way as for
long run curves. Again, MC intersects at min of
AC.
FCSRvK
Q
Why does VCSR look like this?
MPPL If ? at low Q, then labor costs ? at ?ing
rate at low Q.
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MCSR, ATCSR, AVCSR
()
MCSR
ATCSR
AVCSR
Q
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Relationship between SR and LR total cost curves
TC is based on optimal choice of inputs for each
Q.
TC ()
TCSR
In LR, there is some Q, Q1, for which K is
optimal.
TCLR
At this output level, TCSR TCLR.
Q1
Q