Economics 202: Intermediate Microeconomic Theory

1
Economics 202 Intermediate Microeconomic Theory

• Any questions?
• Read Ch 7 (if you havent already) and Ch 8
• HW 7 on website due Thursday.

2
Production Technology

• So far the supply of goods services was given
• Tom Hank picked coconuts caught fish and then
  traded them with one another
• What determines the quantity of goods a firm
  produces?
• How productive are our inputs (K, L, land, raw
  materials)?
• What is our technology in the production process?
• How much do they cost? We want to minimize
  costs.
• What is the profit-maximizing level of output?
• Begin with a production function which represents
  the relationship between inputs and output
• Q f (L, K, M, )
• Tells us the maximum output level, Q, for given
  inputs (L, K, M, etc.) using technology f

3
Short-term Production

• Of course there are dozens (hundreds?) of inputs
• Short-term is a period of time over which the use
  of a given input is very costly to change.
  (land, building, etc.)
• In the long-run, all inputs are variable.
• Assume that K is fixed in short-term and L is the
  only input that a firm can change. (two input
  model)
• Total Product total output produced by a firm
• Average Product of Input i total product
  divided by amount of input used to make that
  total product
• Marginal Product of Input i change in total
  output that results from a one-unit change in the
  amount of input, holding other inputs constant

4
Geometry of TP, APL, MPL
Q

• APL Q/L
• APL at a given level of L is the slope of ray
  from origin to TP curve
• MPL ?Q/?L
• MPL measures how much Q changes for another unit
  of L, holding K constant
• MPL at a given level of L is the slope of TP
  curve at that point
• MPL rises until the inflection point and then
  falls
• When MPL 0, TP is at maximum

TP
L
Q per worker
MPL
APL
L
5
Total, Average, and Marginal Product
Q

• Key characteristics
• If MPL gt 0, TP rises
• If MPL lt 0, TP falls
• If MPL 0, TP at maximum
• If MPL gt APL, APL rising
• If MPL lt APL, APL falling
• ?MPL APL at APLs max
• Examples avg height in room
• Batting averages

49
TP
40
18
4
2
8
L
Q per worker
MPL
13

• Q f(L, K) 600K2L2 K3L3
• MPL ? (let K 10)
• maximum Q ?
• APL ? Max APL ?
• General result ?(Q/L) 0 implies

10
9
APL
6.125
L
2
4
8
6
Diminishing Marginal Returns

• Diminishing Marginal Productivity (Returns)
• Empirical observation
• Key points
• 1. It says the MPL begins to decrease after some
  point, not that it becomes negative.
• Malthus prediction With fixed land and
  diminishing MPL, food supplies will become
  insufficient. What did he miss?
• 2. Ceteris paribus. i.e., other inputs are held
  constant K, energy, raw materials, technology.
  If you find a better way to produce, that shifts
  your entire TP curve!

7
Long-Run Production

• In the long-run, all inputs are variable.
• The long-run and short-run will be different time
  spans for different industries and firms.

K (units of capital)

• Isoquant depicts the possible combinations of K
  and L which produce the same level of max
  output.
• f(K,L) Q0 (Q0 is efficient or max level)

A
Q3 500
Q2 200
B
Q1 100

• Properties of Isoquants
• (1) isoquants are downward-sloping
• MPL gt 0 and MPK gt 0

L (units of labor)
(4) isoquants are strictly convex to the
origin. The absolute value of the slope gets
smaller from left to right.

• (2) isoquants do not intersect

(3) higher isoquants are associated with
higher levels of Q (MPL gt 0, MPK gt 0)
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Slope of an Isoquant

• Slope measures the amount by which a firm can
  reduce K and use 1 more unit of L, maintaining
  the same Q
• Solve for dK/dL
• We call this the Marginal Rate of Technical
  Substitution, the rate at which L can be
  substituted for K, holding Q constant
• Slope is flat when lots of L, little K (easier
  for K to replace L), but steeper when little L
  lots of K (harder to replace worker with machines)

K
Q 500
L
9
Production Functions

• Linear Production Function (? ?)
• Q f(K,L) aK bL
• IRS, CRS, or DRS?
• Realistic?
• Fixed-Proportions (Leontief) Production Function
  (? 0)
• Q min(aK, bL) a, b gt 0
• Such firms will always operate along the ray
  where (K/L) is constant
• IRS, CRS, or DRS?
• Realistic?
• Cobb-Douglas Production Function (? 1)
• Q f(K,L) AKaLb
• IRS, CRS, or DRS? C-D can exhibit any of these
• For CRS case (a b) 1, ? 1
• Homothetic?
• Constant Elasticity of Substitution Production
  Function (? 1/(1- ?) )
• Incorporates the first 3 as special cases
• Q f(K,L) K? L? ?/? for ?? 1, ? ? 0, ?
  gt 0
• ? 1 ? CRS, ? gt 1 ? IRS, ? lt 1 ? DRS

10
Returns to Scale

• What relationship exists between inputs and
  outputs in the Long-Run?
• All Inputs are Output Production exhibits
• Doubled Doubles Constant Returns to Scale
• Doubled lt doubles Decreasing Returns to Scale
• Doubled gt doubles Increasing Returns to Scale
• However, different factors lead to different
  outcomes.
• Increasing Returns to Scale
• Decreasing Returns to Scale (the Dilbert
  effect)
• Constant Returns to Scale
• Graphical representation of IRS, CRS, DRS
• CRS production functions are homogeneous of
  degree 1
• Note that ALL inputs must be increased by the
  same proportion to call it returns to scale.

11
Elasticity of Substitution

• How easy is it to substitute K for L? (look at
  one isoquant, not the the map)
• Typically, MRTS falls as K/L ratio falls
• What if MRTS stays the same as K/L falls?
• Substitution is easy
• If MRTS changes a lot as K/L falls?
• Substitution is difficult
• Elasticity of Substitution ? ?(K/L) /
  ?(MRTS)
• High ? implies MRTS does not change much relative
  to (K/L) flatter isoquant
• Low ? implies MRTS changes a lot relative to
  (K/L) steeper isoquant
• Typical assumption is constant ? along an isoquant

12
Long-Run Costs of Production

• All inputs are variable
• Firms costs can be represented by an Iso-Cost
  line, which identifies all the combinations of
  (L,K) that can be purchased for a given total
  cost.
• TC wL rK
• Rewrite to get K (-w/r)L (TC/r)
• Y-intercept is TC/r
• X-intercept is TC/w
• Slope indicates the relative prices of the inputs
  (slope -2 says hiring 1 more L, means must buy
  2 less K)
• Analogy with consumers problem
• Exception?
• Consumers are stuck with feasible set
• Firms can increase TC by hiring more inputs and
  paying for them by selling more output

• Assumptions
• Homogeneous labor and capital
• Perfectly competitive input markets

13
Least Cost/Max Output

• At the tangency point, slope of the isoquant
  slope of the isocost line
• MPL/ MPK w/r
• Two ways to interpret
• 1. Least-cost way to produce a given Q
• If firm decides to produce Q2, the
  cost-minimizing way is TC2.
• 2. Maximum output possible for a given TC
• If firm decides to spend TC1, Q1 is the most
  they can produce.
• This is the Dual problem to the firms Primal
  cost-min problem

Capital (K)
TC3/r
TC2/r
slope -w/r
TC1/r
Q39
A
Q26
Q13
TC1/w
TC3/w
Labor
TC2/w
14
Output Maximization

• Lets rearrange the equation MPL/ MPK w/r as
  follows
• MPL MPK
• w r
• This says that the firm should use K L in such
  a way that the additional output per dollar spent
  on L additional output per dollar spent on K
• Firm decides to spend TC2. Whats the most Q
  they can make?
• At point A
• MPL 100 widgets, w 20
• MPK 25 widgets, r 25
• MPL/w 5 widgets/dollar
• MPK/r 1 widget/dollar

Capital (K)
TC2/r
A
M
Q2
Q1
Labor
TC2/w

• Firm can increase Q and keep the same total
  cost A ? M
• Spend 1 less on K ? lose 1 widget
• Spend 1 more on L ? gain 5 widgets

15
Cost Minimization

• Interpretation 2, rearrange another way w
  r
• MPL MPK
• This says that the last widget made using L
  should cost the same as the last widget made
  using K.
• Firm decides to make Q1 widgets. Whats the
  least-cost way to do it?
• MPL 10 widgets, w 20 MPK 8 widgets, r
  10
• w/MPL 2/ widget
• r/MPK 1.25/ widget
• Firm can decrease TC and still produce Q1
  widgets B ? N

Capital (K)
TC2/r
TC1/r
N
B
Q1
Labor
TC2/w
TC1/w

• Produce 1 less widget using L ? save 2
• Produce 1 more widget using K ? costs only
  1.25 more

16
Expansion Path

• The long-run Expansion path traces out the
  least-cost way to produce different levels of Q
• As drawn, weve assumed that input prices are
  constant (since the slope of isocost lines do
  not change)
• Cost-minimization is not the same as
  ?-maximization
• ? TR - TC so cost-minimization is necessary for
  profit maximization
• Cost-minimization occurs at all points on the
  expansion path, but ?-max involves choosing the
  point on the path that yields the most profit.

Capital (K)
TC3/r
TC2/r
TC1/r
Expansion Path
C
B
Q3
A
Q2
Q1
TC1/w
TC3/w
Labor
TC2/w
17
Long-Run Cost Curves

• Q L2/3K2/3
• IRS, CRS, or DRS?
• Find LRTC function.
• Decreasing, constant, or increasing LRTC? That
  is, as Q goes up, LRTC does what?

LRTC
100
20
Q
/unit
MC
LRAC
Q
18
Shape of Long-run AC curves
LAC ()
IRS
DRS
LAC
LAC
CRS
Output

• Many are U-shaped, but some are L-shaped
• L-shape ? IRS/economies of scale are quickly
  exhausted, CRS exist over a wide range of
  output
• Result both small large firms can exist in
  same industry
• LAC of small hospitals is 29 more than for large
  ones ? declining LAC

• Industry LACsm/LAClg
• hospitals 129
• electric power 112 banking 102
• airlines 100
• trucking 95
• Result small banks big banks exist

19
Market Structure Long-run AC curves
LAC ()
D industry
LAC tech 1
LAC tech 2
30K
.05Qtotal
Output
.5Qtotal
Qtotal

• Minimum Efficient Scale is the production scale
  at which AC is a minimum.
• This will vary by industry because production
  technology differs and technology is in part
  responsible for declining LAC.
• Key question Where does LAC reach minimum
  compared to total demand?
• If very low (.05Qtotal), then lots of firms in
  that industry.
• If relatively high (.5Qtotal), then very few
  firms in that industry.
• LAC tech 1 coffee shops, breweries LAC tech 2
  cars, law firms, cola, planes

20
LR vs. SR Cost-Minimization

• Min cost of producing 4 widgets is 80 (assuming
  w r 10), using K L 4 (point a)
• In the Long-run, the min cost way to produce Q
  12 is 120, using K L 6 (point c)
• a and c can also be interpreted as the points of
  max output for cost outlays of 80 or 120
• IRS, DRS, or CRS?
• However, in the Short-run K is fixed at 4
  machines, so the min cost to make Q12 is higher,
  140. Point b is (L,K) (10,4)
• LR costs lt SR costs of production

K
14
12
8
c
6
b
4
Q12
a
Q4
L
8
12
4
14
6
10

• MRTS at b lt w/r ? the rate at which K can be
  substituted for L in production lt rate at which K
  can be substituted for L in the market