Transcript Spatial Equilibrium

Lecture 3
Towards a spatially and socially explicit
Chinese agricultural policy model:
A welfare approach
M.A. Keyzer
Presentation available:
www.sow.vu.nl/downloadables.htm
www.ccap.org.cn
Overview of the lecture
1.
Introduction
2.
Welfare economics, AGE-modeling, CHINAGRO model
3.
New algorithm to solve very large partial equilibrium
welfare program with transportation
4.
Conclusion
1. Introduction
Part I : Welfare economics, AGE-modeling, CHINAGRO model
2.1 Welfare optimization and competitive equilibrium
(justifies welfare approach)
2.2 CHINAGRO general equilibrium welfare model
Part II : Algorithm to solve spatially explicit partial equilibrium
Prototype for next generation model
(Check on transport flows and price margins in CHINAGRO)
2.1 Welfare & competitive equilibrium
Consumers are indexed
i  1,...,m
Commodities are indexed
k  1,...,r
Consumers have concave increasing utility functions ui ( xi )
where xi is consumption vector with elements xik
.
Exchange economy:
Consumers obtain an income hi  pi from given endowments
2.1 Welfare & equilibrium: Definitions
Competitive exchange equilibrium: Consumption and prices
maxxi 0 { ui ( xi )| pxi  hi }, for hi  pi , all i
i xik  i ik ,
for all k
Welfare optimization: Consumption solving, given weights  i
maxxi 0,all i i iui ( xi )
subject to
i xi i i
(p)
2.1 Welfare & equilibrium: Theorems
First Welfare Theorem:
“A competitive equilibrium is Pareto efficient”
(no consumer can be made better off
without making some other consumer worse off)
Second Welfare Theorem:
“Every Pareto efficient allocation, including the welfare
optimum, is a competitive equilibrium with transfers”
(lumpsum transfers are efficient for income redistribution)
2.1 Welfare & equilibrium: Theorems (2)
Negishi Theorem:
“There exist welfare weights such that a welfare optimum is a
competitive equilibrium without transfers”
The Negishi-weights reflect marginal utilities of income.
A competitive equilibrium without transfers is a welfare
optimum where consumers with a high marginal utility of
income have a low welfare weight.
2.1 Welfare & equilibrium: Theorems (end)
The three basic theorems of welfare economics equally apply
when production takes place and consumers obtain income
from given endowments and from shares in the profit of
producers indexed j :
pxi  pi  ij py j
The welfare program then reads :
maxxi 0,all i,y j ,all j i i ui ( xi )
subject to
i xi i i + j y j
y j Y j
(p)
2.1 Welfare & equilibrium: Institutions
Institutional requirements :
1)
all goods in the economy are priced (no free use)
2)
no one can manipulate prices (no monopoly)
3)
all consumers pay the price of what they use, and receive
the price for what they sell (no crime).
4)
producers maximize profits independently of preferences
(shareholder value principle).
2.2 CHINAGRO model: point of departure
Point of departure:
Static equilibrium welfare model from the previous lecture:
maxv
j 0;q
;cs ,g s ,ys 0;z s ,z s 0
 s us ( cs )   C ( v 1 ,...,v
 
 
g
,q
)

(

z


z
)

p



sS
s s gs
L
s s
s s
subject to
 sS cs   j v
j
 q
(p )
q   j v j   sS ( ys  zs  zs )
cs  zs  ys  zs
ys  f s ( g s ,es )
(ps )
2.2 CHINAGRO full model
Modifications:
1)
Consider all goods simultaneously; linear trade technology.
(variables become vectors; product becomes inner product)
2)
Open economy, trading with the outside world at given
prices.
3)
Incorporate balance of payments constraint.
4)
Conversion from utility to money metric utility through
welfare weights.
5)
Detailed component for agricultural production.
2.2 CHINAGRO full model (2)
Implication for modeling:
1)
Inputs agriculture subsumed under net supplies of site s ;
For transport requirements  j , s , s
g   j jv
:
 
 


z



sS s s
j
s zs
2,3) Balance of payments with exports, imports w ,w and
world market prices p   p  :
 
 
 (p w  p w )  B
where B is the total of non-trade transactions.
4)
Write  s  sus ( cs ) for  s us ( cs ) .
5)
Write Fs ( ys ,es )  0 for ys  Fs ( gs ,es ).
2.2 CHINAGRO full model (end)
Full CHINAGRO general equilibrium welfare model :
maxv
 
 
j 0;g 0;cs ,ys 0;z s ,z s 0;w ,w 0
 s  sus ( cs )
subject to
 sS cs  g   j v
g   j jv
j

+



w

v

(
y

z

z
)

w


j
s

S
j
j
s
s
s
(p )
  sS  s zs   s zs
 
 
 (p w  p w )  B
cs  zs  ys  zs
Fs ( ys ,es )  0
(ps )
3. Partial equilibrium with transportation
CHINAGRO model is suited to represent a complex economic
system in a transparent way.
Nonetheless, it assumes that all transportation cost within
counties are truly incurred. As explained in the previous
lecture, this assumption would need to be relaxed.
Therefore, as a background check on transport flows and price
margins in CHINAGRO, and as a prototype for next
generation models, consider again the single-commodity
partial equilibrium approach.
3. Spatially explicit equilibrium model
Recall, from lecture 2, the model that maximized
the sum of money-metric utilities minus transport costs
subject to commodity balance at every site.
Demand + Outflow = Production + Inflow
Outflow from site s to r = Inflow into site r from s
maxvsr 0;qs ,cs 0
subject to
 s us ( cs )   s Cs ( vs1 ,...,vsS ,qs )
cs   r vsr  qs
qs   r vrs  es
(ps )
3. Spatial model: transport cost
Work focused on transport cost along main highways,
railways, and waterways, and along secondary roads.
Spatially explicit data were collected for rice and wheat.
The resulting map of transport costs per ton-kilometer is
shown on the next sheet.
3. Spatial model: transport cost
3. Spatial model: solution
Objective :
Find equilibrium
supply, demand, flows and price on a map
Tool :
A new algorithm to solve a large scale,
spatially explicit welfare program
Advantage :
Integration between disciplines
3. Spatial model: what are the costs?
Costs over formal infrastructure(waterways, railways
and highways) relatively low:
But these are only a small fraction of the consumer price.
We must also allow for storage cost, cost of changing
from the informal mode of transportation to the formal
and cost at both ends of the chain: collection and retail
distribution
3. Spatial equilibrium models
Spatial equilibrium models

Connect districts, or nodes in a network

Not spatially explicit
3. Spatially explicit model
Allow for all possible flows on the Union Jack grid
3. Partial equilibrium: new algorithm



Key algorithmic principle:
Gravity :
Transport :
gravity driven flow
water does not flow uphill
goods never flow to lower price
Low price
High price
3. Partial equilibrium: new algorithm (2)

Two step algorithm:
Step 1
Solve gravity constrained welfare program
Impose gravity rule: exclude flows from high to low prices

Per site from low to high price:
(a) update availability = production + inflow
(b) maximize utility of site + value of sales
subject to
consumption + outflow = given availability

Per site from high to low price:
update sales price on basis of customer’s value
3. Partial equilibrium: new algorithm (3)
Step 2
Improvement achieved?
Yes:
Update gravity ordering on basis of prices of gravityconstrained program and go to Step 1
No:
Otherwise, end (optimum is found)
3. Partial equilibrium: new algorithm (4)

Application to spatially explicit welfare model for China

Exogenous variables

production map cereals

population map

tariffs and world market prices cereals

freight costs per ton

Study world market price penetration

Grid of cells of 10-by-10 km = 93125 cells (markets)
3. New algorithm: zoom in on results
Preliminary results for rice
Price
Preliminary results for rice
Flow
Preliminary results for rice
Production
Preliminary results for rice
Consumption
Preliminary results for rice
Joint
Price
Production
Flow
Consumption
Preliminary results for wheat
Price
Preliminary results for wheat
Flow
Preliminary results for wheat
Production
Preliminary results for wheat
Consumption
Preliminary results for wheat
Joint
Price
Production
Flow
Consumption
4. Conclusion
CHINAGRO: Multicommodity general equilibrium welfare
model with spatially explicit partial equilibrium models
in the background.
General equilibrium model: work in progress to be
discussed further tomorrow.
Partial equilibrium model: preliminary results show that it
is possible to generate meaningful spatially explicit
equilibrium, with “very large” number of geographical
units to represent transport flows and price margins in
China.
A next, challenging partial equilibrium application will be
the pork industry considering the meat and feed
markets simultaneously