Transcript Slide 1

Microeconomics for
International Trade Theory
ECON0301
January 2011
Basic Trade Model
 General equilibrium
 multiple agents (firms and consumers) all optimizing
 The aggregation problem
 One shot
 long run
 income=expenditure => balance of trade
 Market structure
 Constant returns to scale, perfect competition
 Variable returns to scale, market power (new trade
theory in 80s; new new trade theory in this past
decade)
Basic Prerequisites
 Consumption
 Production
 Perfectly Competitive Market
 General Equilibrium
I: Consumption Theory
Consumption Problem
 Consumption choice problem
max U  x, y  subject to I  PxY  PyY
x, y
 Results: Demand functions
x  Dx  Px , Py , I  and y  Dy  Px , Py , I 
 Equalization of marginal utility per dollar
MU x MU y
MU x Px



Px
Py
MU y Py
 Marginal rate of substitution = relative price
Marginal rate of substitution = relative
price
At point A, the budget line and
some IC are tangential to each
other
B
Omitting the negative signs,
A
The slope of the IC = MUx/MUy
C
The slope of the budget line =
Px/Py
Marginal rate of substitution = relative
price
Y
U0
At point C, MUx/MUy=Px/Py. But
obviously not an optimal bundle
U1
A
Some properties on the shapes of ICs
are required and will be assumed to
hold.
C
X
Marginal rate of substitution
 Change in utility
dU 
U ( x, y )
U ( x, y )
dx 
dy
x
y
 Along an IC
dU 0 
U ( x, y )
U ( x, y )
dx 
dy  0
x
y
 The slope of an IC
dy
dx
U U

U / x
MU x

0
U / y
MU y
Cobbs-Douglas Utility Function
U  x, y   x y

 CD utility function:
 Marginal utilities:
MU x 
U  x, y 
MUy  
x
U  x, y 

U  x, y 
x y 
y 

x
x
 1 
x
y
 MRS = relative price
MU x  y Px
Px 


 x 
MU y  x Py
Py y 
 Expenditure shares constant
Py y
Px x



and

I
 
I
 
Cobbs-Douglas Utility Function
 Very nice demand functions

I
 I
Dx  px , p y , I  
and Dy  px , p y , I  
   Px
   Py
 Without loss of generality, assume
   1
 If not, we can always represent the old CD utility
function by a new CD function
U  x, y   x y where a 
a
b

 
and b 

 
Measurability of Utility
V=200
V=2001
V=100
U=30
U=20
U=10
An order-preserving re-labeling of ICs does not alter the preference
ordering.
Positive monotonic (order-preserving)
transformation
 They are called positive monotonic transformation
U  xy
U '  U 2  x2 y 2
U ''  U  6  xy  6
U '''  2U  2 xy
U '' ''  U 2  1  x 2 y 2  1
 They all refer to the same preferences, leading to the
same choice
Two properties of CD utility functions
 Unit income elasticity

1% increase in income => 1% increase in
consumption
 The aggregation problem

Given total income, income distribution does
not affect market demand
The aggregation problem
 Suppose there are N agents, each with the same CD utility
function
 Suppose their incomes are I1,I2, … , IN, summing up to I.
 The total market demand for x equals

I1

   Px


IN
   Px
 I1  I N 
 I




  
Px
    Px

 Given the total income, the income distribution among the
agents does not affect the market demand. As long as the latter
is concerned, we do not need to know income distribution.
 A property need not generally hold for other utility functions
An example where income distribution
matters
 Two goods: necessity (x) and luxury (y)
 Two agents, where I1+I2 = I = 10
 each will consume luxury only after x0 <5 units of necessity is




consumed.
Suppose prices px=1, py=1.
Equal income, the market quantity demanded for x is 2x0; the
market quantity demanded for y is 10 - 2x0
Unequal income; suppose I1 < I2 and I1 < x0. Then market
quantity demanded for x is I1 + x0 < 2x0; market quantity
demanded for y is 10-(x0 + I1).
Unequal income diverts resource to luxuries while basic
necessities are not fully provided => income distribution matters
The aggregation problem
 The Cobb-Douglas utility function is among
the family of utility functions for which
income distribution per se does not affect
market demand
 We can further define the concept of “the
relative demand for x” by an individual
Dx   I    I   p y

 
 / 
Dy     Px      Py   px
 This relative demand is independent of the
individual’s income
The aggregation problem
 The relative demand for x in the whole economy (with
individual incomes I1, I2, …, IN) is just the same as the
relative demand for x by any individual
Dx   I1  I 2 

Dy    
Px
I N    I1  I 2 
 / 
Py
   
I N   py
 
  px
 Take out: With CD utility function, we can talk about


demand for x by the economy without knowing how
income is distributed among individuals
Relative demand for x by the economy without
knowing the total income
II. Production, Perfectly
Competitive Market,
Equilibrium
Production function
Q  total product  f ( K , L)
Q
 average product of labor  APL
L
Q
 marginal product of labor  MPL
L
Q
 average product of capital  APK
K
Q
 marginal product of capital  MPK
K
Production Function
 Output elasticity and returns to scale
dL dK d 
Suppose


, some proportional change in inputs.
L
K

dQ / Q
% change in output
R

d  /  % change in all input
1
i.r.t.s.
1
c.r.t.s.
1
d.r.t.s.
Production function
 When returns to scale do not
change with scale, for any t>1, the
technology exhibits
if
f  tK , tL   tf  K , L 
CRTS if
f  tK , tL   tf  K , L 
DRTS if
f  tK , tL   tf  K , L 
IRTS
Production function
 For CRTS technology, there are two nice
properties


MPK and MPL depend on K/L only, but not on the
absolute scale
 E.g., MPK the same when you hire K=3 and L=5,
compared with when you hire K=6 and L=10.
MPK*K+MPL*L = f(K,L)
 When factors are hired up to r = p*MPK and w =
p*MPL, the profit is just zero!
Production function
 We show the second property:
f  tK , tL   tf  K , L 
(
CRTS)
Differentiating it w.r.t t , we obtain
d
f  tK , tL   f  K , L 
dt
f  tK , tL  tK f  tK , tL  tL

 f  K , L  (chain rule)
tK
t
tL
t
f  tK , tL 
f  tK , tL 
K
L  f  K, L
tK
tL
Now imposing the condition that t  1, it becomes
f  K , L 
K
K
f  K , L 
L
L  f  K , L
Profit maximization problem
 Each firm chooses
max pf  K , L  - rK - wL
K ,L
 If an optimum exists, will hire K and L such that
pMPK  p
pMPL  p
f  K , L 
K
f  K , L 
L
r
w
Cost minimization problem
 Sometimes it is easier to re-phase the problem as a cost
minimization problem, following by the output choice problem
 Given input prices, choose the input combination that
minimizes cost
min rK  wL subject to f  K , L   Q
K ,L
 Cost function
C  r , w, Q  , satisfying
C  tr , tw, Q   tC  r , w, Q  for all t  0
Cost function
 When RTS does not change with scale, for
any t>1, the technology exhibits
IRTS
if
CRTS if
DRTS if
C  r , w, tQ 
tQ
C  r , w, tQ 
tQ
C  r , w, tQ 
tQ



C  r , w, Q 
Q
C  r , w, Q 
Q
C  r , w, Q 
Q
isoquant
 Isoquant – the locus of
K and L such that the
output level is constant
 Bending toward the
origin
K
Optimal input
mix to produce
Q=10
Q=20
Iso-cost line
rK+wL=constant
Q=10
L
Equilibrium condition
 Main assumptions: CRTS
technology, perfect
competition, all inputs
variable (long run
equilibrium).
 The min. costs to produce $1
worth of a good are exactly
$1.
 Given output and input
prices, we can determine the
optimal mix of inputs
 Or, given output price and
K/L ratio, we can determine
the relative input prices
K
Q=1/p’
Iso-cost line
rK+wL=constant
Q=1/p
L
Equilibrium condition
 For CRTS technology and in LR equilibrium,
we cannot tell the output level of a particular
firm, because every output level will lead to
the same profit (which is zero) given fixed
input and output prices
Cobb-Douglas Production Function
Q  f ( K , L)  K  L
Q
MPK 
  K  1 L
K
Q
MPL 
  K  L 1
L
L
(1  ) 1 
When  + =1, MPK   K
L   
K


K
and MPL   K  L1 1    
L
Marginal products depend on the K / L ratio, not on the absolute scale
Cobb-Douglas Production Function
 What does α+β=1 mean?



α+β>1; IRTS
α+β=1; CRTS
α+β<1; DRTS
The technology exhibits CRTS iff  + =1.
f  2K , 2L    2K 

 
 
 
2
L

2
K
L

2
f ( K , L)
 

The aggregation problem
 Consider an industry with all firms having the same
CRTS technology: Q=f(K,L); output & input markets
perfectly competitive; and firms maximizing profits.
Collectively, the industry employs K* and L*.
 What is the total output of the industry?

f(K*,L*). The output produced would be the same as if there
were one single firm employing K* and L*. The “fine
structure” of the industry (i.e., the number of firms, the sizes,
etc.) is irrelevant.
The aggregation problem
 Let K1, K2, …, KN be the amount employed in the N firms.
 Let L1, L2, …, LN be the amount employed in the N firms.
 Cost minimization requires that K1/L1=K2/L2=… =
KN/LN=K*/L*=a.
 Total output
f  K1 , L1   f  K 2 , L2  
 f  aL1 , L1   f  aL2 , L2  
 L1 f  a,1  L2 f  a,1 

 L* f  a,1  f aL* , L*

 f K * , L*


f  K N , LN 
f  aLN , LN 
LN f  a,1
(
CRTS)
(
CRTS)
The aggregation problem
 Question: What is the marginal product of
capital (labor) in each firm?
 Despite possibly different scales, each firm’s
marginal product of capital is simply equal to
∂F(K*,L*)/∂K, i.e., marginal product of capital
of a fictitious firm which employs just K* and
L*.
 Similarly, all firms have the same marginal
product of labor = ∂F(K*,L*)/∂L.
The aggregation problem
 Question: What is the rental rate of capital
paid by each firm? What is the wage rate of
labor paid by each firm?
 Let P be the price of the good produced in the
industry. The rental rate of capital is just
P*∂F(K*,L*)/∂K, while the wage rate of labor is
just P*∂F(K*,L*)/∂L.
 The real rental rate (≡ w/p) and real wage
rate (≡ r/p) should be ∂F(K*,L*)/∂K &
∂F(K*,L*)/∂L.
The aggregation problem
 To know the output of an industry, as well the
wage rate of labor and the rental rate of
capital being paid by firms in the industry,
there is no need to know the internal structure
of the industry.
 We can simply work out the problem by
assuming all inputs (K* are L*) are hired by a
single firm (which is nonetheless price taking
in both input and output markets).
 In this case, we can understand f(K,L) as an
industry production function.
The aggregration problem
 Takeout: given same CRTS technology, & LR
competitive equil, the total K* and L*
employed in the sector fully describe the
output level as well as the real rental rate and
real wage rate.