General Equilibrium

1
General Equilibrium

• Review
• Efficiency condition (Consumption)
• MRSAxy MRSBxy PX/PY
• 2. Efficiency condition (Production)
• MRTSX MRTSY MPLX/MPKX MPLY/MPKY
• 3. Efficiency condition (Product mix)
• MRTXY - DY/DX MPLY/MPLX MPKY/MPKX
  PX/PY

2
(1) Consumer Optimization

• Perfectly competitive markets
• U U(X,Y) subject to I PXX PYY
• Optimization
• where marginal utility equals price ratio
• Single consumer
• Ux/PX UY/PY
• Ux/UY PX/PY MRS

3
Consumer Optimization

• Indifference Curve All possible combinations of
  good X and good Y which yield the same level of
  satisfaction
• Budget line I PXX PYY Y I/PY
  PX/PYX

Y
I/PY
IC3
IC2
IC1
X
4
Consumer Optimization

• Edgeworth Box
• At A MRSPaul gt MRSSam
• At B MRSPaul MRSSam PX/PY

0Sam
QY
A
B
PX/PY
0paul
QX
5
(2) Producer Optimization

• Perfectly competitive markets single producer
• Maximize Qx F(K,L) subject to C0 wL rK
• Thus
• optimization when wage-rental ratio is equal to
  the slope of an isoquant -DK/DL MPL/MPK w/r
• (Production Efficiency Condition)

6
Producer Optimization

• Isoquant All possible combinations of capital
  and labour that could be used to produce a
  particular quantity of X
• Qx MPLXL MPKXK
• Isocost C0 wL rK K C0/r w/rL

KX
Implies at equilibrium MPLX/MPKX w/r
C0/r
X 200
X 150
X 100
LX
7
Producer Optimization

• At H MRTSX gt MRTSY MPLX/MPKX gt
  MPLY/MPKY
• X has too much capital, too little labour, Y has
  too much labour , too little capital gtgtgt
  reallocation of capital labour can increase
  output
• At E MRTSX MRTSY w/r MPLX/MPKX
  MPLY/MPKY

0Y
K
200
240
180
H
A
100
C
80
D
150
200
240
0X
L
8
(3) Product Mix

• Contract curve (Producer Optimization) and PPF
  (Production Possibility Frontier)

Y
Slope - DY/DX or opportunity cost of producing 1
more unit of X in terms of Y forgone
A
240
B
200
150
C
- DY/DX
80
D
X
100
240
180
200
9
(3) Product Mix

• Producing an efficient output mix involves
  balancing the subjective wants, or preferences
  (MRS of consumers) with the objective conditions
  of production

10
(3) Product Mix

• Perfectly competitive markets Two-good,
  two-factor model
• Qx Fx(Kx,Lx) QY FY(KY,LY)
• Factor market equilibrium (supply demand)
• K Kx KY and L Lx LY
• Profit maximisation condition
• w MPLXPX r MPKXPX (1)
• w MPLYPY r MPKYPY (2)
• Rearrange (1) into (2) PX/PY MPLY/MPLX
  MPKY/MPKX
• But MPLY DY/DLY where DLY -DLX
• MPLX DX/DLX
• Thus MPLY/MPLX -DY/DX MRT

PX/PY
11
(3) Product Mix

• What is the incentive to shift from A to C under
  Px/Py 5 ?

Y
Px/Py5
A
240
B
200
150
C
- DY/DX
80
D
X
100
240
180
200
12
(3) Product Mix
(1) Producer optimisation MRTSXMRTSY (i.e. on
contract curve)
Y
A
240
B
200
150
C
80
D
X
100
240
180
200
13
Additional slides
14
Production function

• Qx F(K,L) Certain technology (F), which
  requires a combination of capital (K) and labour
  (L), can produce some quantity of good X (Qx).
• Assume fixed capital (K0) gtgt Law of Diminishing
  Returns

F(K0,L)
Qx
L
MPLX
L
15
Producer Optimization

• Marginal Rate of Technical Substitution
• MRTSA MPL1/MPK1 gt MPL2/MPK2 MRTSB

K
A MPL1gt MPK1
K1
B MPL2lt MPK2
K2
L
L2
L1
16
Producer Optimization

• Optimization where isoquant is tangent to
  isocost
• MRTS MPL/MPK w/r
• Isoquant
• DQX (MPL)DL (MPK)DK With DQX 0
• (MPL)DL - (MPK)DK MPL/MPK -DK/DL (1)

K
Isocost C0 wL rK DC0 wDL rDK With DC0
0 wDL - rDK w/r -DK/DL (2)
w/r MPL/MPK
Kop
L
Lop