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URL http://www.gams.com/solvers/mpsge/cesfun.htm CES Demand Functions: Hints and Formulae

Constant Elasticity of Substitution

Functions: Some Hints and Useful Formulae

Notes prepared for GAMS General Equilibrium Workshop
held December, 1995 in Boulder Colorado

Thomas F. Rutherford
University of Colorado

Contents

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The Basics

In many economic textbooks the constant elasticity of substitution (CES) utility function is defined as:

eq001

It is a fairly routine but tedious calculus excercise to demonstrate that the associated demand functions are:

eq002

and

eq003 .

The corresponding indirect utility function has is:

eq004

Note that U(x,y) is linearly homogeneous:

eq005

This is a convenient cardinalization of utility, because percentage changes in U are equivalent to percentage Hicksian equivalent variations in income.

Because U is linearly homogeneous, V is homogeneous of degree one in M and degree -1 in p.

In the representation of technology, we have an analogous set of relationships, based on the cost and compensated demand functions. If we have a CES production function of the form:

eq006 ,

the unit cost function then has the form:

eq007 ,

and associated demand functions are:

eq090 ,

and

eq091 .

In most large-scale applied general equilibrium models, we have many function parameters to specify with relative ly few observations. The conventional approach is to calibrate functional parameters to a single benchmark equilibrium. For example, if we have benchmark estimates for output, labor, capital inputs and factor prices , we calibrate function coefficients by inverting the factor demand functions:

eq010

and

eq011 .

To the begining

The Calibrated Share Form

Calibration formulae for CES functions are messy and difficult to remember. Consequently, the specification of function coefficients is complicated and error-prone. For applied work using calibrated functions, it is much easier to use the "calibrated share form" of the CES function. In the calibrated form, the cost and demand functions explicitly incorporate

  • benchmark factor demands
  • benchmark factor prices
  • the elasticity of substitution
  • benchmark cost
  • benchmark output
  • benchmark value shares
In this form, the production function is written:
eq012

The only calibrated parameter, eq013 , represents the value share of capital at the benchmark point. The corresponding cost functions in the calibrated form is written:

eq092

where eq093 and the compensated demand functions are:

eq015

and

eq016 .

Normalizing the benchmark utility index to unity, the utility function in calibrated share form is written:

eq017

Defining the unit expenditure function as:

eq018

the indirect utility function is:

eq019

and the demand functions are:

eq020

and

eq021

The calibrated form extends directly to the n-factor case. An n-factor production function is written:

eq022

and has unit cost function:

eq023

and compensated factor demands:

eq024

To the begining

Excercises:

(i) Show that given a generic CES utility function:
eq025

can be represented in share form using:

eq026

for any value of t > 0.

(ii) Consider the utility function defined:

eq027

A benchmark demand point with both prices equal and demand for y equal to twice the demand for x. Find values for which are consistent with optimal choice at the benchmark. Select these parameters so that the income elasticity of demand for x at the benchmark point equals 1.1.

(iii) Consider the utility function:

eq028

which is maximized subject to the budget constraint:

eq029

in which M is interpreted as non-wage income, w is the market wage rate. Assume a benchmark equilibrium in which prices for x and L are equal, demands for x and L are equal, and non-wage income equals one-half of expenditure on x. Find values of eq030 and eq031 consistent with these choices and for which the price elasticity of labor supply equals 0.2.

(iv) Consider a consumer with CES preferences over two goods. A price change makes the benchmark consumption bundle unaffordable, yet the consumer is indifferent. Graph the choice. Find an equation which determines the elasticity of substitution as a function of the benchmark value shares. (You can write down the equation, but it cannot be solved in closed form.)

(v) Consider a model with three commodities, x, y and z. Preferences are CES. Benchmark demands and prices are equal for all goods. Find demands for x, y and z for a doubling in the price of x as a function of the elasticity of substitution.

(iv) Consider the same model in the immediately preceeding question, except assume that preferences are instead given by:

eq032

Determine eq033 from the benchmark, and find demands for x, y and z if the price of x doubles.


To the begining

Flexibility and Non-Separable CES functions

We let eq034 denote the user price of the ith input, and let eq035 be the cost-minizing demand for the ith input. The reference price and quantities are eq036 and eq037 . One can think of set i as {K,L,E,M} but the methods we employ may be applied to any number of inputs. Define the reference cost, and reference value share for ith input by eq038 and eq039 , where
eq040

and

eq041 .

The single-level constant elasticity of substitution cost function in "calibrated share form" is written:

eq042

Compensated demands may be obtained from Shephard's lemma:

eq043

Cross-price Allen-Uzawa elasticities of substitution (AUES) are defined as:

eq044

where

eq045

For single-level CES functions:

eq046 .

The CES cost function exibits homogeneity of degree one, hence Euler's condition applies to the second derivatives of the cost function (the Slutsky matrix):

eq047

or, equivalently:

eq048

The Euler condition provides a simple formula for the diagonal AUES values:

eq049

As an aside, note that convexity of the cost function implies that all minors of order 1 are negative, i.e. eq050 . Hence, there must be at least one positive off-diagonal element in each row of the AUES or Slutsky matrices. When there are only two factors, then the off-diagonals must be negative. When there are three factors, then only one pair of negative goods may be complements.

Let:
kbe the reference the index of second-level nest
eq051 denote the fraction of good i inputs assigned to the kth nest
eq052 denote the benchmark value share of total cost which enters through the kth nest
eq053 denote the top-level elasticity of substitution
eq054 denote the elasticity of substitution in the kth aggregate
eq055 denote the price index associated with aggregate k, normalized to equal unity in the benchmark, i.e.:

eq056

The two-level nested, nonseparable constant-elasticity-of-substitution (NNCES) cost function is then defined as:

eq057 .

Demand indices for second-level aggregates are needed to express demand functions in a compact form. Let eq058 denote the demand index for aggregate k, normalized to unity in the benchmark; i.e.

eq059

Compensated demand functions are obtained by differentiating eq060 . In this derivative, one term arise for each nest in which the commodity enters, so:

eq061

Simple differentiation shows that benchmark cross-elasticities of substitution have the form:

eq062

Given the benchmark value shares eq039 and the benchmark cross-price elasticities of substitution, eq063 , we can solve for values of eq051 , eq052 , eq054 and eq053 . We compute these parameters using a constrained nonlinear programming algorithm, CONOPT, which is available through GAMS, the same programming environment in which the equilibrium model is specified. Perroni and Rutherford (EER, 1994) prove that calibration of the NNCES form is possible for arbitrary dimensions whenever the given Slutsky matrix is negative semi-definite. The two-level (NxN) function is flexible for three inputs; and although we have not proven that it is flexible for 4 inputs, the only difficulties we have encountered have resulted from indefinite calibration data points.

Two GAMS programs are listed below. The first illustrates two analytic calibrations of the three-factor cost function. The second illustrates the use of numerical methods to calibrate a four-factor cost function.


To the begining

$TITLE Two nonseparable CES calibrations for a 3-input cost function.


* ========================================================================
*       Model-specific data defined here:


SET     I  Production input aggregates / A,B,C /;  ALIAS (I,J);

PARAMETER

    THETA(I)    Benchmark value shares /A 0.2, B 0.5, C 0.3/

    AUES(I,J)   Benchmark cross-elasticities (off-diagonals) /
                        A.B     2
                        A.C   -0.05
                        B.C     0.5 /;
* ========================================================================


*       Use an analytic calibration of the three-factor CES cost
*       function:

ABORT$(CARD(I) NE 3) "Error: not a three-factor model!";

*       Fill in off-diagonals:

AUES(I,J)$AUES(J,I)  =  AUES(J,I);

*       Verify that the cross elasticities are symmetric:

ABORT$SUM((I,J), ABS(AUES(I,J)-AUES(J,I))) " AUES values non-symmetric?";

*       Check that all value shares are positive:

ABORT$(SMIN(I, THETA(I)) LE 0) " Zero value shares are not valid:",THETA;

*       Fill in the elasticity matrices:

AUES(I,I) = 0; AUES(I,I) = -SUM(J, AUES(I,J)*THETA(J))/THETA(I); DISPLAY AUES;

SET     N       Potential nesting /N1*N3/
        K(N)    Nesting aggregates used in the model
        I1(I)   Good fully assigned to first nest
        I2(I)   Good fully assigned to second nest
        I3(I)   Good split between nests;

SCALAR  ASSIGNED /0/;

PARAMETER
        ESUB(*,*)       Alternative calibrated elasticities
        SHR(*,I,N)      Alternative calibrated shares
        SIGMA(N)        Second level elasticities
        S(I,N)          Nesting assignments (in model)
        GAMMA           Top level elasticity (in model);

*       First the Leontief structure:

ESUB("LTF","GAMMA") = SMAX((I,J), AUES(I,J));
ESUB("LTF",N) = 0;
LOOP((I,J)$((AUES(I,J) EQ ESUB("LTF","GAMMA"))*(NOT ASSIGNED)),
        I1(I) = YES;
        I2(J) = YES;
        ASSIGNED = 1;
);

I3(I) = YES$((NOT I1(I))*(NOT I2(I)));
DISPLAY I1,I2,I3;
LOOP((I1,I2,I3),
        SHR("LTF",I1,"N1") = 1;
        SHR("LTF",I2,"N2") = 1;
        SHR("LTF",I3,"N1") = THETA(I1)*(1-AUES(I1,I3)/AUES(I1,I2)) /
                     ( 1 - THETA(I3) * (1-AUES(I1,I3)/AUES(I1,I2)) );
        SHR("LTF",I3,"N2") = THETA(I2)*(1-AUES(I2,I3)/AUES(I1,I2)) /
                     ( 1 - THETA(I3) * (1-AUES(I2,I3)/AUES(I1,I2)) );
        SHR("LTF",I3,"N3") = 1 - SHR("LTF",I3,"N1") - SHR("LTF",I3,"N2");
);
ABORT$(SMIN((I,N), SHR("LTF",I,N)) LT 0) "Benchmark AUES is indefinite.";

*       Now, the CES function:

ESUB("CES","GAMMA") = SMAX((I,J), AUES(I,J));
ESUB("CES","N1") = 0;
LOOP((I1,I2,I3),
        SHR("CES",I1,"N1") = 1;
        SHR("CES",I2,"N2") = 1;
        ESUB("CES","N2") = (AUES(I1,I2)*AUES(I1,I3)-AUES(I2,I3)*AUES(I1,I1)) /
                           (AUES(I1,I3)-AUES(I1,I1));
        SHR("CES",I3,"N1") =
                (AUES(I1,I2)-AUES(I1,I3)) / (AUES(I1,I2)-AUES(I1,I1));
        SHR("CES",I3,"N2") = 1 - SHR("CES",I3,"N1");
);
ABORT$(SMIN(N, ESUB("CES",N)) LT 0)   "Benchmark AUES is indefinite?";
ABORT$(SMIN((I,N), SHR("CES",I,N)) LT 0) "Benchmark AUES is indefinite?";


PARAMETER       PRICE(I)        PRICE INDICES USING TO VERIFY CALIBRATION
                AUESCHK(*,I,J)  CHECK OF BENCHMARK AUES VALUES;

PRICE(I) = 1;

$ontext

$MODEL:CHKCALIB

$SECTORS:
        Y       ! PRODUCTION FUNCTION
        D(I)

$COMMODITIES:
        PY      ! PRODUCTION FUNCTION OUTPUT
        P(I)    ! FACTORS OF PRODUCTION
        PFX     ! AGGREGATE PRICE LEVEL

$CONSUMERS:
        RA

$PROD:Y  s:GAMMA  K.TL:SIGMA(K)
        O:PY            Q:1
        I:P(I)#(K)      Q:(THETA(I)*S(I,K))     K.TL:

$PROD:D(I)
        O:P(I)  Q:THETA(I)
        I:PFX   Q:(THETA(I)*PRICE(I))

$DEMAND:RA
        D:PFX
        E:PFX   Q:2
        E:PY    Q:-1
$OFFTEXT
$SYSINCLUDE mpsgeset CHKCALIB

SCALAR DELTA /1.E-5/;

SET     FUNCTION /LTF, CES/;

ALIAS (I,II);

LOOP(FUNCTION,

        K(N) = YES$SUM(I, SHR(FUNCTION,I,N));
        GAMMA = ESUB(FUNCTION,"GAMMA");
        SIGMA(K) = ESUB(FUNCTION,K);
        S(I,K) = SHR(FUNCTION,I,K);

        LOOP(II,

          PRICE(J) = 1; PRICE(II) = 1 + DELTA;

$INCLUDE CHKCALIB.GEN
        SOLVE CHKCALIB USING MCP;

        AUESCHK(FUNCTION,J,II) = (D.L(J)-1) / (DELTA*THETA(II));

));

AUESCHK(FUNCTION,I,J) = AUESCHK(FUNCTION,I,J) - AUES(I,J);
DISPLAY AUESCHK;

*       Evaluate the demand functions:

$LIBINCLUDE qadplot
SET PR Alternative price levels /PR0*PR10/;

PARAMETER
        DEMAND(FUNCTION,I,PR)   Demand functions
        DPLOT(PR,FUNCTION)      Plotting output array;

LOOP(II,
        LOOP(FUNCTION,
          K(N) = YES$SUM(I, SHR(FUNCTION,I,N));
          GAMMA = ESUB(FUNCTION,"GAMMA");
          SIGMA(K) = ESUB(FUNCTION,K);
          S(I,K) = SHR(FUNCTION,I,K);
          LOOP(PR,
            PRICE(J) = 1;
            PRICE(II) = 0.2 * ORD(PR);
$INCLUDE CHKCALIB.GEN
            SOLVE CHKCALIB USING MCP;
            DEMAND(FUNCTION,II,PR) = D.L(II);
            DPLOT(PR,FUNCTION) = D.L(II);
          );
        );

$LIBINCLUDE qadplot DPLOT PR FUNCTION

);

DISPLAY DEMAND;

To the begining

A Comparison of Locally-Identical Functions

Figure 1: Demand Function Comparison -- Good A

Figure 2: Demand Function Comparison -- Good B

Figure 3: Demand Function Comparison -- Good C


To the begining

$TITLE Numerical calibration of NNCES given KLEM elasticities

SET  I  Production input aggregates / K, L, E, M/;  ALIAS (I,J);

* ========================================================================
*       Model-specific data defined here:

PARAMETER
    THETA(I)    Benchmark value shares /K 0.2, L 0.4, E 0.05, M 0.35/

    AUES(I,J)   Benchmark cross-elasticities (off-diagonals) /
                        K.L     1
                        K.E     -0.1
                        K.M     0
                        L.E     0.3
                        L.M     0
                        E.M     0.1 /;
* ========================================================================

SCALAR EPSILON          Minimum value share tolerance /0.001/;

*       Fill in off-diagonals:

AUES(I,J)$AUES(J,I)  =  AUES(J,I);

*       Verify that the cross elasticities are symmetric:

ABORT$SUM((I,J), ABS(AUES(I,J)-AUES(J,I))) " AUES values non-symmetric?";

*       Check that all value shares are positive:

ABORT$(SMIN(I, THETA(I)) LE 0) " Zero value shares are not valid:",THETA;

*       Fill in the elasticity matrices:

AUES(I,I) = 0; AUES(I,I) = -SUM(J, AUES(I,J)*THETA(J))/THETA(I); DISPLAY AUES;

* ========================================================================

*       Define variables and equations for NNCES calibration:

SET     N       Nests within the two-level NNCES function /N1*N4/,
        K(N)    Nests which are in use;

VARIABLES
        S(I,N)        Fraction of good I which enters through nest N,
        SHARE(N)        Value share of nest N,
        SIGMA(N)        Elasticity of substitution within nest N,
        GAMMA           Elasticity of substitution at the top level,
        OBJ             Objective function;

POSITIVE VARIABLES S, SHARE, SIGMA, GAMMA;

EQUATIONS
        SDEF(I)         Nest shares must sum to one,
        TDEF(N)         Nest share in total cost,
        ELAST(I,J)      Consistency with given AUES values,
        OBJDEF          Maximize concentration;

ELAST(I,J)$(ORD(I) GT ORD(J))..

                AUES(I,J) =E=   GAMMA +

                        SUM(K, (SIGMA(K)-GAMMA)*S(I,K)*S(J,K)/SHARE(K));

TDEF(K)..       SHARE(K) =E= SUM(I, THETA(I) * S(I,K));

SDEF(I)..       SUM(N, S(I,N)) =E= 1;

*       Maximize concentration at the same time keeping the elasticities
*       to be reasonable:

OBJDEF..        OBJ =E= SUM((I,K),S(I,K)*S(I,K))

                        - SQR(GAMMA) - SUM(K, SQR(SIGMA(K)));

MODEL   CESCALIB /ELAST, TDEF, SDEF, OBJDEF/;


*       Apply some bounds to avoid divide by zero:

SHARE.LO(N) = EPSILON;

SCALAR  SOLVED  Flag for having solved the calibration problem /0/
        MINSHR  Minimum share in candidate calibration;

SET     TRIES   Counter on the number of attempted calibrations /T1*T10/;

*       We use the random number generator to select starting points,
*       so it is helpful to initialize the seed so that the results
*       will be reproducible:

OPTION SEED=0;

LOOP(TRIES$(NOT SOLVED),

*       Initialize the set of active nests and the bounds:

        K(N) = YES;
        S.LO(I,N) = 0;          S.UP(I,N) = 1;
        SHARE.LO(N) = EPSILON;  SHARE.UP(N) = 1;
        SIGMA.LO(N) = 0;        SIGMA.UP(N) = +INF;

*       Install a starting point:

        SHARE.L(K)  = MAX(UNIFORM(0,1), EPSILON);
        S.L(I,K)    = UNIFORM(0,1);
        GAMMA.L     = UNIFORM(0,1);
        SIGMA.L(K)  = UNIFORM(0,1);

*       Drop any basis information so that we start from scratch:

        SDEF.M(I)     = 0; TDEF.M(K)     = 0; ELAST.M(I,J)  = 0;

        SOLVE CESCALIB USING NLP MAXIMIZING OBJ;

        SOLVED = 1$(CESCALIB.MODELSTAT LE 2);

*       We have a solution -- now see if it is not on a bound:

        IF (SOLVED,

          MINSHR = SMIN(K, SHARE.L(K)) - EPSILON;

          IF (MINSHR EQ 0,

*       Drop nests which have shares equal to EPSILON in the current
*       solution:

            K(N)$(SHARE.L(N) EQ EPSILON) = NO;

            S.FX(I,N)$(NOT K(N)) = 0;
            SHARE.FX(N)$(NOT K(N)) = 0;
            SIGMA.FX(N)$(NOT K(N)) = 0;

            DISPLAY "Recalibrating with the following nests:",K;

            SOLVE CESCALIB USING NLP MAXIMIZING OBJ;

            IF (CESCALIB.MODELSTAT GT 2, SOLVED = 0;);
            MINSHR = SMIN(K, SHARE.L(K)) - EPSILON;
            IF (MINSHR EQ 0, SOLVED = 0;);
          );
        );
);

IF (SOLVED,
        DISPLAY "Function calibrated:",GAMMA.L,SIGMA.L,SHARE.L,S.L;
ELSE
        DISPLAY "Function calibration fails!";
);

$ONTEXT

*==========================================================================

Solution from MINOS5 obtained on the second try, following an
ITERATION INTERRUPT on the first:

----    151 Function calibrated:


----    151 VARIABLE  GAMMA.L              =        0.300 Elasticity of
                                                          substitution at the
                                                          top level


----    151 VARIABLE  SIGMA.L       Elasticity of substitution within nest N

N3 7.804


----    151 VARIABLE  SHARE.L       Value share of nest N

N1 0.604,    N2 0.266,    N3 0.030,    N4 0.100


----    151 VARIABLE  S.L           Fraction of good I which enters through
                                    nest N

           N1          N2          N3          N4

K                   0.797       0.069       0.133
L       0.960                   0.040
E                                           1.000
M       0.630       0.304                   0.067


*==========================================================================

The following solution is obtained by CONOPT on the second try,
following a LOCALLY INFEASIBLE termination on the first problem.
Notice that it is identical to the MINOS5 solution except that the
nesting assigments have been permuted:

----    149 Function calibrated:


----    149 VARIABLE  GAMMA.L              =        0.300 Elasticity of
                                                          substitution at the
                                                          top level


----    149 VARIABLE  SIGMA.L       Elasticity of substitution within nest N

N4 7.804


----    149 VARIABLE  SHARE.L       Value share of nest N

N1 0.100,    N2 0.604,    N3 0.266,    N4 0.030


----    149 VARIABLE  S.L           Fraction of good I which enters through
                                    nest N

           N1          N2          N3          N4

K       0.133                   0.797       0.069
L                   0.960                   0.040
E       1.000
M       0.067       0.630       0.304

*==========================================================================

$OFFTEXT

$TITLE	Numerical Verification of the the AUES Calibration Results

PARAMETER	PRICE(I)	PRICE INDICES USING TO VERIFY CALIBRATION
		AUESCHK(I,J)	CHECK OF BENCHMARK AUES VALUES;

PRICE(I) = 1;

$ontext

$MODEL:CHKCALIB

$SECTORS:
	Y	! PRODUCTION FUNCTION
	D(I)

$COMMODITIES:
	PY	! PRODUCTION FUNCTION OUTPUT
	P(I)	! FACTORS OF PRODUCTION
	PFX	! AGGREGATE PRICE LEVEL

$CONSUMERS:
	RA

$PROD:Y  s:GAMMA.L  K.TL:SIGMA.L(K)
	O:PY		Q:1
	I:P(I)#(K)	Q:(THETA(I)*S.L(I,K))	K.TL:

$PROD:D(I)
	O:P(I)	Q:THETA(I)
	I:PFX	Q:(THETA(I)*PRICE(I))

$DEMAND:RA
	D:PFX
	E:PFX	Q:2
	E:PY	Q:-1
$OFFTEXT
$SYSINCLUDE mpsgeset CHKCALIB

CHKCALIB.ITERLIM = 0;
$INCLUDE CHKCALIB.GEN
SOLVE CHKCALIB USING MCP;
CHKCALIB.ITERLIM = 2000;

SCALAR DELTA /1.E-5/;

ALIAS (I,II);
LOOP(II,

	PRICE(J) = 1;	PRICE(II) = 1 + DELTA;

$INCLUDE CHKCALIB.GEN
	SOLVE CHKCALIB USING MCP;

	AUESCHK(J,II) = (D.L(J)-1) / (DELTA*THETA(II));

);
DISPLAY AUES, AUESCHK;

To the begining

Analytic Calibration of the GEMTAP Final Demand System

This material was published in the MUG newsletter, 8/95.

Following Ballard, Fullerton, Shoven and Whalley (BFSW), we consider a representative agent whose utility is based upon current consumption, future consumption and current leisure. Changes in "future consumption" in this static framework are associated with changes in the level of savings. There are three prices which jointly determine the price index for future consumption. These are:
PI the composite price index for investment goods
PK the composite rental price for capital services
PC the composite price of current consumption.

All of these prices equal unity in the benchmark equilibrium.

Capital income in each future year finances future consumption, which is expected to cost the same as in the current period, PC (static expectations). The consumer demand for savings therefore depends not only on PI, but also on PK and PC, namely:

eq064

The price index for savings is unity in the benchmark period. In a counter-factual equilibrium, however, we would expect generally that eq065 . When these price indices are not equal, there is a "virtual tax payment" associated with savings demand.

Following BFSW, we adopt a nested constant-elasticity-of-substitution function to represent preferences. In this function, at the top level demand for savings (future consumption) trades off with a second CES aggregate of leisure and current consumption. These preferences can be summarized with the following expenditure function:

eq066
Preferences are homothetic, so we have defined PU as a linearly homogeneous cost index for a unit of utility. We conveniently scale this price index to equal unity in the benchmark. In this definition, eq030 is the benchmark value share for current consumption (goods and leisure). PH is a compositive price for current consumption defined as:
eq067
in which eq033 is the benchmark value share for leisure within current consumption.

Demand functions can be written as follows:

eq068 ,
eq069 ,
and
eq070

Demands are written here in terms of their benchmark values (S0, C0 and eq071 ) and current and benchmark income (I and I0).

There are four components in income. The first is the value of labor endowment (E), defined inclusive of leisure. The second is the value of capital endowment (K). The third is all other income (M). The fourth is the value of "virtual tax revenue" associated with differences between the shadow price of savings and the cost of investment.

eq072

The following parameter values are specified exogenously:

  1. eq073 is the ratio of labor endowment:
    eq074
    where L0 is the benchmark labor supply. Given eq075 and L0 we have:
    eq076
  2. eq077 is the uncompensated elasticity of labor supply with respect to the net of tax wage, i.e.
    eq078
  3. eq079 is the elasticity of savings with respect to the return to capital:
    eq080

    Shephard's lemma applied at benchmark prices provides the following identities which are helpful in deriving expressions for eq081 and eq082 :

    eq083
    It is then a relatively routine application of the chain rule to show that:
    eq084
    and
    eq085

    The expression for eq081 does not involve eq086 , so we may first solve for eq087 and use this value in determining eq086 :

    eq088
    and
    eq089

    To the begining