An Example of Calibration

Shoven and Whalley's calibration can be illustrated using a simple general equilibrium model with consumption and production. A single consumer is endowed with two factors of production. These factors combine to produce two goods using CES technology, and the consumer has Cobb-Douglas preferences over the two goods. The

6 Calibration can only be undertaken if the equations in G satisfy the conditions of implicit functions, that is, if the equations of F are continuously differentiable with respect to Y, w, and a and if at Ÿ, ft, and à ,, the determinant of the Jacobian matrix given by the derivatives of F

consumer generates demands for the goods by maximizing utility subject to a budget constraint.

The vector of endogenous variables in this model, Y, is comprised of: X, the

consumer's demand for good Q„ the quantity produced of good /'; P„ the price of output /'; Fj, the demand for factor j in the production of good /'; and w1 the price of factor 1. The vector of exogenous variables, w, includes the E1, the consumer's endowment of factor j , and the price o f factor 2, w2 which is set to 1. It is chosen arbitrarily as a numeraire since only relative prices matter in the model.

f inally, the vector of parameters to be calibrated, a 2, is comprised of the eight parameters, (J„ aj, X, where the p, are the shares of goods in the consumer's utility function, the a{ are the CES share parameters of factor / in the production of good i, and the X, are the scale parameters in the production function for good /'. The vector a , consists of o„ the two elasticities of substitution in the CES production functions.

The model, F, can be described by the eight conditions:

i) factor markets clear: LF/ - E> = 0 0 = 1 ,2 ), (

2

4

ii) Goods markets clear: X r Q , - 0 0 = 1,2), (

2

5

iii) Production sectors make zero profits: i\q, - i y f f = o 0 = 1 ,2 ), (

2

6

iv) Household exhibits budget balance: YjWEt- £, PtX, = 0, (

2

7

V) Fixing o f a numeraire: w2= 1. (

2

8

The Shoven-Whalley calibration o f this model uses equilibrium data to find the values of the parameters which comprise the vector o 2. To be used in calibration,

however, the data must represent a solution to the model, that is, they must satisfy the model's equilibrium conditions given by equations (2.4) - (2.8). Table 2.1 provides an

Table 2.1

An Example of a Microconsistent Data Set

Transactions Values in Units o f Currency

Expenditures

Factor 1 Factor 2 Production Production

of Good I of Good 2

Receipts

Use of Factor 1 in production (inputs) 12 10

Use of Factor 2 in production (inputs) 8 16

Production of Good 1 (sales) Production of Good 2 (sales)

Consumer's endowments of factors 22 24

Purchases by Consumer

20

example of such data. The row entries in Table 2.1 denote receipts and the column entries give expenditures, so that together the data are microconsistent in value terms: the value of inputs equals the value of outputs in each sector, the value o f consumption equals that o f production of each good, and the consumer is on her budget constraint.

If the units convention due to Harberger (1962) is adopted under which the quantities of both goods and services are defined as that amount which sells for one unit o f currency, all base case prices in the economy can be set to 1. This convention implies that the value of transactions in Table 2.1 also denotes quantities transacted and that the market clearing conditions also hold.

Cobb-Douglas demands are given by

For specified values of E ' and known solution values X„ /’„ and vv\ the calibration of

the demand parameters is undertaken by calculating

(2.9)

P , = ^ Q > W .

On the production side, the CES factor demand functions are

"L

The first step in the calibration of the production parameters is to set values for the elasticity parameters, a,. Suppose that either econometric estimation or a literature search yielded the elasticity values o, = 1.2, and o2= 0.8. First order conditions from cost minimization allow calibration of the share parameters of factors in production as

w JF { C-L) ( J - ) £ w' F / ° ' j (2.12)

Substituting the o/ into the production function allows the calibration of the scale parameters X(, Qt (0, -1)

[E

«/F/

I'“'

0

2

13

The calibrated parameter values using the data from Table 2.1 and the specified elasticities, are given in Table 2.2.

A modeller would typically substitute the calibrated parameter values set out in Table 2.2 into the model given by equations (2.4) - (2.8), using the functional specifications (2.9) and (2.11), to ensure that the equilibrium solution values are the same as those given by the data in Table 2 .1. This replication test provides assurance that no errors are present, either in the calibration calculations or in the model coding.

The possibility exists that the model has multiple equilibria and that the replication test might fail because the model solves for an equilibrium other than that

Table 2.2

Calibrated Parameter Values for the Example Model Using the Data in Table 2.1

utility function share parameters Pi * 0.43

P

2

production function share parameters i 0.58

V * 0.42

ai - 0.36

* 0.64

production function scale parameters - 1.97

of the base case data. Numerical examples of multiple equilibria have been constructed by Kehoe (1985) for simple Cobb-Douglas economies with a small number of production activities. However, where smooth production functions of the Cobb-

Douglas or CES variety are used, uniqueness is the more likely outcome.7 A d hoc tests,

undertaken with applied models, seem to confirm this view.* *

Although this example is simple, the same calibration approach can be used for large scale models. Piggott and Whalley (1985) use a model of the UK with 100 households, 33 productive sectors, and 29 traded goods. Including the intermediate production structure, the model uses around 20,000 parameter values. Models o f these dimensions are not exceptional. An even larger model, the ORANI model of the Australian economy described in Dixon, Parmenter, Sutton and Vincent (1982), identifies 115 commodities and 113 industries in its base period input-output data.