Lecture 3.1 An Introduction to General Equilibrium Policy Modelingdwrh/FAO_ECTAD_FMD_Cambodia/... · 2013-01-16 · equilibrium (columns equal rows), but it is a static equilibrium

Lecture 3.1

An Introduction to General Equilibrium Policy Modeling

David Roland-Holst, Sam Heft-Neal, and Anaspree Chaiwan UC Berkeley and Chiang Mai University

Training Workshop Economywide Assessment of High Impact Animal Disease

14-18 January 2013 InterContinental Hotel, Phnom Penh, Cambodia

Roland-Holst 2 16 January 2013

SAMs

ACT COM VA HH GOV INV ROW TOTALS

ACT Gross Output

Receipts

COM Int. Use Household Consumption

Government Expenditure

Gross Investment

Exports Demand

VA GDP at Factor Cost

Factor Income

HH GDP at Factor Cost

ROW Trans. to HH

Household Income

GOV Net Indirect Taxes

Household Taxes

Government Borrowing

Government Revenue

INV Household Saving

Government Saving

Current account balance

Savings

ROW Imports ROW

TOTALS Payments Supply Factor Allocation

Household Expenditure

Government Expenditure

Investment ROW


SAM to CGE

•  The SAM provides a snapshot of the economy at equilibrium (columns equal rows), but it is a static equilibrium with fixed prices, no substitution, and typically average behavior.

•  On the contrary, in many cases what we are interested in examining is how economic actors respond to changes in relative prices.

•  CGE allows for flexible prices, substitution, and marginal behavior, at the same time meeting the accounting constraints enforced by SAM structure.


SAM to CGE

•  To put this another way, CGE models overcome the shortcomings of a SAM by specifying a functional form for every cell in the SAM.

•  Each cell in the SAM can be represented by a price and quantity, so the model must be able to determine both prices and quantities.

•  Let’s start with a VERY simple CGE model, then work our way to something a bit more complicated.


Very Basic CGE

•  To see how we go from a SAM to a CGE model, let’s begin with a 2-sector, 2-factor really really simple SAM (RRSS):

Producers Factors Institutions ROWSUM

AG OTH L K HH

AG 150 150

OTH 500 500

L 100 200 150

K 50 300 150

HH 300 350 650

COLSUM 150 500 300 350 650


Our Simple Economy

•  Note that the government is not an economic actor, the economy is closed, factor costs are the only input to production, and households spend all their income.

•  In this case, we have three economic actors –  Producers (2; AG and OTH) –  Factors (2; L and K) – Households (1)

•  Let’s further assume that labor and capital are fully mobile across sectors (1 wage and rental rate).


Side Note

•  (Let’s maintain our convention of having i be rows and j be columns; this means that i will reflect the income side of the economy and j will reflect the expenditure side of the economy).


Supply

•  On the supply side, at a minimum we need to specify how producers behave (e.g., minimize costs), how they choose inputs (factor demands), and how their decisions determine aggregate supply. Using a Cobb-Douglas form, we can describe production within our economy as:

Total Supply

Labor Demand

Capital Demand


Demand

•  On the demand side, we need to specify the level of household income, and how households decide to spend that income. Household income is the sum of factor incomes:

(Remember that we are decomposing SAM transactions into prices and, in this case, volumes.)


Demand

•  Household consumption is modeled with a constant elasticity of substitution (CES) utility function:

Maximizing U s.t. a budget constraint gives us the two reduced form consumption functions:


Equilibrium

•  Lastly, we need to define some sort of equilibrium conditions for the economy, which in our case we can represent by supply = demand in product and factor markets.

Commodity Market

Labor Market

Capital Market


Endogenous Variables

•  In 13 equations we have built a simple general equilibrium model.

•  Our 13 endogenous variables include: –  Pi – prices for AG and OTH goods –  r – rate of return on capital – w – wage rate –  LDj – labor demand for AG and OTH producers –  KDj – capital demand for AG and OTH producers –  XSj – aggregate supply – Ci – household consumption of AG and OTH goods –  Y – household income


Exogenous Variables

•  We have left 2 variables exogenous: –  LS – Aggregate labor supply –  KS – Aggregate capital supply


Initializing Prices

•  Prices are going to be endogenous in our simple CGE model, but we are going to represent prices in a price index rather than as absolute values. Prices can be initialized to any level, but 1 is generally the most obvious choice. –  PAG = 1 POTH = 1

•  We select PAG as the numeraire, which fixes our economy-wide relative price as –  P = POTH / PAG


Initializing Prices

•  We represent factor prices in the same way (as an index). In contrast to goods, however, we might want to initialize wages and rental rates at different levels to represent a factor price ratio that differs from unity – w = 0.8 –  r = 1 (i.e., capital is more expensive in relative terms than labor)


Initializing Endogenous Variables

•  We can assign values to endogenous variables based our SAM: –  LDAG0 = 100 LDOTH0 = 200 –  KDAG0 = 50 KDOTH0 = 300 –  XSAG0 = 150 XSOTH0 = 500 – CAG0 = 150 COTH0 = 150 –  Y0 = 650


SAM Check

Producers Factors Institutions ROWSUM

AG OTH L K HH

AG 150 150

OTH 500 500

L 100 200 150

K 50 300 150

HH 300 350 650

COLSUM 150 500 300 350 650


Initializing Endogenous Variables

•  LD, KD, XS, and C are volumes, so we need to convert them to volumes by dividing by the appropriate initialized price –  LDAG0/w0 = 100/0.8 =125 LDOTH0/w0 = 200/0.8 = 250 –  KDAG0/r0 = 50/1 = 50 KDOTH0/r0 = 300/1 =

300 –  XSAG0/pAG0 = 150/1 = 150 XSOTH0/pOTH0 = 500/1

= 500 – CAG0/pAG0 = 150/1 = 150 COTH0/pOTH0 = 500/1

= 500

•  We can also initialize LS and KS volumes –  LS0 = LDAG0+ LDOTH0

–  KS0 = KDAG0+ KDOTH0


Model Calibration

•  We can use SAM data to determine the baseline values of some of our parameters; in this case: – Cobb-Douglas scaling factors (Aj) – Cobb-Douglas share parameters (αj) – CES utility function share parameters (δ)


Model Calibration

•  From our aggregate output equation

•  We can calculate the Cobb-Douglas scaling factors as


Model Calibration

•  Similarly, from labor demand

we can calculate the Cobb-Douglas share parameters as


Model Calibration

•  The CES share parameters are derived from

with a less than tidy result of


Model Calibration

•  Alternatively, the CES utility function’s substitution elasticity (σ) cannot be determined with SAM data.

•  We can either specify σ heuristically (e.g., a 0 if we determine that the goods are perfect complements, or a high value if they are perfect substitutes) or through econometrics.

•  In this case, let’s arbitrarily assign σ with a value of 0.3.


Model Simulation

•  Let’s walk through what happens when we perturb one of the exogenous variables in the model. Say we have an exogenous increase in labor supply (LS). From

we know this exogenous increase in LS will be accompanied by an increase in aggregate LD.


Model Simulation

•  But it isn’t clear how this change will affect our other variables:


To a New Equilibrium

•  We need a way to move from our initial equilibrium, in which all of our model equations held (i.e., our markets cleared), to a new equilibrium, in which all of our equations hold again. This shift from an old equilibrium to a new equilibrium is what is usually meant by “adjustment.”

•  To find our new equilibrium solution, our endogenous variables will have to adjust so that both our equations hold, and our exogenous shock is accounted for.


Model Solutions and Consistency

•  CGE models require numerical solutions, which means that you will need to use some sort of solver package to generate a solution.

•  To ensure that the model is consistent and you have not made errors in coding, in general your first step after building a CGE model is to make sure that you can reproduce the base solution (i.e., with no exogenous shock).


A Quick Thought on Model Building

•  Before we get into more complex models, a bit of advice. It is always useful to start any research project with a quick theoretical model that maps relationships among the variables that you wish to examine.

•  By doing this kind of exercise, you can get a good sense of where you can make simplifications, where you should be more detailed, and how much you can leave out of your model.


Toward more Complex Models

•  Our next model will be significantly more complex, but still simple as far as CGE models go.

•  MINI_CGE will address several of the oversimplifications of our previous model: –  Producers typically have non-factor intermediate

inputs and non-uniform substitution elasticities – Households are more complex than CES utility

describes – Most economies have an active government and

capital markets – Most economies have ROW interactions


1. Producer Behavior

•  Producers choose inputs to minimize costs; with two inputs we can represent this mathematically as: Min(wL+rK) s.t. V=F(K,L) where w and r are the wage and rental rates, L and K are labor and capital, and V is the level of output. Producers choose K and L; w, r, and V are typically determined by market equilibrium conditions.


Producer Behavior

•  The Lagrangean for the producer’s optimization problem is L = wL + rK + P[V – F(K,L)]

•  Setting the partial derivatives with respect to K, L and P equal to zero, we have the following three first order conditions:

∂∂

∂∂

LK

r P FK

= ⇒ =0

LFPw

L ∂∂

∂∂

=⇒= 0L ( )LKFVP

,0 =⇒=∂∂L


CES Production

•  Let’s take this one step further by assigning a functional form to F: a CES (constant elasticity of substitution) function, the most ubiquitous functional form used in GE models.

•  The primal form of the CES function is

where the coefficients al and ak are called the labour and capital share parameters, respectively, and ρ is the CES exponent (which will be related to the CES substitution elasticity).

( ) [ ]V F K L a L a Kl k= = +,/ρ ρ ρ1


CES First Order Conditions

•  Differentiating the primal form CES yields

•  Substituting back into our original problem, this implies

[ ]∂

∂ ρρρ ρ ρ ρ

ρFL

a L a K a L a LVl k l l= + =⎛⎝⎜

⎞⎠⎟

− −−1 1 1 11

/

∂∂

ρFK

a KVk= ⎛⎝⎜

⎞⎠⎟

−1

1−

⎟⎠

⎞⎜⎝

⎛==ρ

∂∂

VLaP

LFPw l

1−

⎟⎠

⎞⎜⎝

⎛==ρ

∂∂

VKaP

KFPr k


CES Factor Demand

•  Three simplifications:

then give us the following derived factor demands

σ

ρρ

σσ

=−

⇔ =−1

11

( )α ρ σl l la a= =−1 1/

( )α ρ σk k ka a= =−1 1/

L Pw

V

K Pr

V

l

k

= ⎛⎝⎜

⎞⎠⎟

= ⎛⎝⎜

⎞⎠⎟

α

α

σ

σ

(i.e., the relationship between the CES exponent and CES substitution elasticity.)


CES Unit Cost and Pricing

•  Using the total cost function

And substituting reduced for expressions for L and K Gives us the unit price-cost equivalence from duality

PV wL rK= +

[ ]PV w P

wV r P

rV VP w rl k l k=

⎛⎝⎜

⎞⎠⎟ +

⎛⎝⎜

⎞⎠⎟ = +− −α α α α

σ σσ σ σ1 1

[ ] ( )P w rl k= +− − −

α ασ σ σ1 1 1 1/


Generalized CES

•  The CES optimization problem can be generalized to i = 2,…,n inputs as

where Xi are the inputs to production and Pi are their prices. A is a uniform shift parameter that can be applied to all inputs, and λ is an input-specific shift parameter. So, for instance, neutral productivity growth could be applied by shifting the A parameter. Hicks neutral productivity growth could be applied by shifting the λ parameter.

min P Xi ii∑

( )V A a Xi i i

i

=⎡

⎣⎢

⎤

⎦⎥∑ λ

ρρ1/

s.t.


Generalized CES

•  The generalized CES has first order conditions

where the shift and share parameters have been merged so that

111

1

−−−

−

=⎥⎦

⎤⎢⎣

⎡= ∑ ρρρ

ρρ

ρiiii

iiii XcVPXcXcPP

ρρ

1

⎥⎦

⎤⎢⎣

⎡= ∑

iii XcV

( )ρλiii Aac =


Generalized CES

•  We can rewrite as

substituting back into the second FOC gives

and with a bit of manipulation, we get unit costs

VPPcXi

ii

ρ−

⎥⎦

⎤⎢⎣

⎡=

11

11 −− ρρii XcVP

ρ

ρρρ /1

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡= ∑

−

i i

ii VPPccV

( ) ( ) ( )σσ

σ

σσ

σσ

σσ

λλ

−−−−−

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡= ∑∑∑

1/111/111/11 1

i i

ii

i i

ii

iii

PaAA

PaPcP


Generalized CES

•  So our unit cost of production is determined by

•  Again using the relationship between the CES exponent and CES substitution elasticities, we get reduced form input demands

( )X A PP

Vi i ii

=⎛

⎝⎜

⎞

⎠⎟

−α λ

σσ

1

( )

PA

Pi

i

ii

=⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

− −

∑11 1 1

αλ

σ σ/


Generalized CES

•  In most applications, A is typically set to 1, and the exponent on the share parameter is merged into the primal share parameter to yield

and

VPPXi

iii

σ

σλα ⎟⎟⎠

⎞⎜⎜⎝

⎛= −1

( )σσ

λα

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

1/11

i i

iiPP


Generalized CES

•  The first of these two equations

defines the CES dual price, which is an average of the input prices. The CES dual price function is the aggregator, with the share and productivity parameters providing the appropriate weights.

( )σσ

λα

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

1/11

i i

iiPP


Generalized CES

•  The second equation

represents optimal demand for each input. Individual demand equals a constant share of the level of output, V, adjusted by a term in the relative price of the input (compared to the aggregate cost of inputs). Hence, if an input’s price increases (relative to overall costs), then demand for that factor will decrease. The percentage decrease will depend on the elasticity of substitution.

VPPXi

iii

σ

σλα ⎟⎟⎠

⎞⎜⎜⎝

⎛= −1


CES Substitution Elasticities

•  By dividing input demands, we can calculate the ratio of demand for any two inputs (e.g., i and j) as:

•  Taking the partial derivative of the above with respect to the ratio Pj/Pi and multiplying the resulting expression by (Pi/Pj)/(Xi/Xj) yields the elasticity of substitution, or…

σ

σ

σ

λαλα

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

−

i

j

jj

ii

j

i

PP

XX

1

1


CES Substitution Elasticities

•  Which is the percent change in the ratio of two inputs with respect to a percentage change in their relative prices.

σ−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

j

i

j

i

j

i

j

i

XXPP

PP

XX


Special Case: Leontief

•  In the case where σ = 0, there is no input substitution and we have the Leontief fixed coefficient, linear production technology that we used in I/O analysis

•  Output price is simply a weighted average of input prices

VA

Xi

ii λ

α=

V A

Xi

i

i=

⎛

⎝⎜

⎞

⎠⎟min λ

α

ii i

i PA

P ∑=λα1


Stratified Input Substitution

•  Up until now we have been assuming a single elasticity of substitution among all input types, which is obviously not realistic.

•  To get around this problem, we can group inputs into sub-groups that each have different substitution elasticity properties.


Nested CES

•  A common solution is to use nested CES production, which uses stratified input substitution

σ = 0

σm σk

σv

σp XP

ND VA

L KF

K F

XAp

XM XDd


CES Nest

•  The top nest determines demand for intermediate demand and value added

with unit cost of production

σ is the ND-VA substitution, which is set to 0 in THAIMINI.

ii

indii XPPNDPXND

piσ

α ⎟⎟⎠

⎞⎜⎜⎝

⎛= i

i

ivaii XPPVAPXVA

piσ

α ⎟⎟⎠

⎞⎜⎜⎝

⎛=

[ ] ( )pipi

pi

ivaii

ndii PVAPNDPX

σσσ αα−

−− +=1/111


CES Nest

•  The value added sub-nest includes three factors

( ) iil

ili

di VA

WPVAL

viv

i

σσ

λα ⎟⎠

⎞⎜⎝

⎛=−1

ii

ikfii VAPKFPXKF

viσ

α ⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )vivi

vi

ikfil

i

lii PKFWPVA

σ

σ

σ

αλ

α

−

−

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

1/1

11


2. Household Behavior

•  THAIMINI uses a linear expenditure system (LES), also known as a Stone-Geary demand system, which is widely used in CGE models.

•  LES is also a fairly general functional form that is based on a number of tenuous assumptions (e.g., all goods are gross substitutes, no good is inferior).

•  Nevertheless, LES has 3 advantages: – Relatively small number of parameters to calibrate

(2n) –  Flexible income elasticities –  Ease of use


LES Utility

•  The LES utility function is given by

or equivalently

with a condition that

( )∑ −=i

iii xxU θµ ln)(

( )∏=

−=n

iii

ixU1

µθ Yxpn

iii =∑

=1s.t.

Yxpn

iii =∑

=1s.t.

11

=∑=

n

iiµ


LES Utility

•  The Lagrangean for the LES utility function is

•  The LES Lagrangean has FOCs

rearranging dL/dx gives the LES demand function

µθ

λi

i iixp

−− = 0 Y p xi i

i

− =∑ 0

xpi ii

i= +θ

µλ

( )L = − + −

⎛

⎝⎜

⎞

⎠⎟∑∑µ θ λi i i i i

ii

x Y p xln


LES Utility

•  To express the LES as a reduced form in prices, note that

(remember our condition on µ)

and the Lagrange multiplier can be expressed as

Y p

pp p

ppi i

i

iii i

ii

i

iii i

i

= +⎛

⎝⎜

⎞

⎠⎟ = +

⎛

⎝⎜

⎞

⎠⎟ = +∑ ∑ ∑ ∑θ

µλ

θµλ λ

θ1

∑−=

jjjpY θ

λ1


LES Elasticities

Income elasticities can be obtained directly from

and, by extension

In words, LES income elasticities are the ratios of marginal (µi) to average (si) expenditure shares.

Price elasticities follow from

∂∂

µxY pi i

i=

η

∂∂

µ µi

i

i

i

i i

i

i

xYYx p

Yx s

= = =

( )∂

∂µ µ

θµ

θxp p

Yp p

Yp

i

i

i

i

i

ii

i

i ii= − + − = − +

⎛

⎝⎜

⎞

⎠⎟2

**

( ) ( ) ( )

ε∂∂

µθ

µθ

µ θθ

θ µi

i

i

i

i

i

i

i

ii

i

i

ii i

i i

i ii i

i i

i

xppx

px p p

px

x xx

x= = − + −

⎛

⎝⎜

⎞

⎠⎟ = − − − =

−−

1 1 11


3. Other Final Demand – Government

•  We assume that the volume of government expenditure is fixed, i.e.,

•  Government is assumed to have a CES expenditure function

•  Where government expenditure price is given by

0XGXG =

( ) XGPA

PGXAgg

iitgi

gii

σ

τα ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+=

1

( )[ ]( )g

g

ii

itgi

gi PAPG

σσ

τα−

−

⎥⎦

⎤⎢⎣

⎡+= ∑

1/111


3. Other Final Demand – Investment

•  Investment is savings determined. The investment-savings closure rule is given by

where XI is the aggregate volume of investment, PI is an investment price deflator, Sh and Sg represent domestic savings, ER.Sf is foreign savings adjusted by the exchange rate (which we’ll discuss later), and DeprY is a depreciation allowance term.

DeprYSERSSXIPI fgh +++= ..


Investment

•  As with government expenditure, a CES expenditure function is assumed to allocate aggregate investment into sectoral demand XAi

( ) XIPA

PIXAii

iitii

iii

σ

τα ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+=

1

( )[ ]( )i

i

ii

itii

ii PAPI

σσ

τα−

−

⎥⎦

⎤⎢⎣

⎡+= ∑

1/111


4. Trade

•  Trade is the final key component of demand. •  If there are no differences between imports and

domestic products, imports are a residual between domestic production and domestic demand.

•  In reality, there are few commodities (e.g., oil) that are truly homogeneous, and most models assume some degree of differentiation between imports and domestically produced goods.


Trade Stratification

•  Demand is thought to combine domestic and imported goods in each product category with a nested CES aggregation

•  Output is modeled symmetrically with a dual nested CET structure

Imports/Exports Domestic Goods

Aggregate Demand/Supply

CES/CET


Trade Analytically – Imports

Denoting domestic demand by XD and imports by XM, total demand is modeled with the CES preference function

Min(PD•XD+PM•XM) subject to

where PD and PM denote prices for domestic and imported goods and XA is aggregate demand. Passing over derivations from the production analytics, we have the following reduced forms

where and denotes the price index of XA.

[ ]XA a XD a XMd m= +ρ ρ ρ1/

XAPDPAXD d

σ

α ⎟⎠

⎞⎜⎝

⎛= XM PA

PMXAm=

⎛⎝⎜

⎞⎠⎟ασ

[ ] ( )PA XD XMd m= +− − −

α ασ σ σ1 1 1 1/

α σd da=

α σm ma=

ρ

σσ

=−1


Trade Analytically – Exports

Denoting domestic supply by XD and export supply by XE, total supply is modeled with the CET production frontier

Max(PD•XD+PE•XE) subject to

where PD and PM denote prices for domestic and imported goods and XA is aggregate demand. Passing over derivations from the production analytics, we have the following reduced forms

where and denotes the price index of XP.

XD PD

PPXPd= ⎛

⎝⎜

⎞⎠⎟γν

[ ] ωωω /1XEgXDgXP ed +=

XE PE

PPXPe= ⎛

⎝⎜

⎞⎠⎟γν

[ ] ( )( )PP XD XE PD XD PE XE XPd e= + = ++ + +

γ γν ν ν1 1 1 1/. . /

γ νd dg=

−

γ νe eg=

−

ω

νν

=+1


Trade Schematically

Domestic Goods/Services

Exports

PPF

slope=-PD/PE

CET

Domestic Goods/Services

Impor t s

Indifference Curve

slope=-PD/PM

CES

Lecture 3.1 An Introduction to General Equilibrium Policy Modelingdwrh/FAO_ECTAD_FMD_Cambodia/... · 2013-01-16 · equilibrium (columns equal rows), but it is a static equilibrium

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