Lecture Notes on Constant Elasticity Functions

1 Lecture Notes on Constant Elasticity FunctionsThomas F. RutherfordUniversity of ColoradoNovember, 20021 CES UtilityIn many economic textbooks the Constant - Elasticity -of-substitution (CES) utility function isdefined as:U(x, y) = ( x + (1 )y )1/ It is a tedious but straight-forward application of Lagrangian calculus to demonstrate that theassociated demand Functions are:x(px, py, M) =( px) M p1 x+ (1 ) p1 yandy(px, py, M) =(1 py) M p1 x+ (1 ) p1 corresponding indirect utility function has is:V(px, py, M) =M( p1 x+ (1 ) p1 y)1 1 Note thatU(x, y) is linearly homogeneous:U( x, y) = U(x, y)This is a convenient cardinalization of utility, because percentage changes inUareequivalent to percentage Hicksian equivalent variations in income. BecauseUis linearlyhomogeneous,Vis homogeneous of degree one inM:V(px, py, M) = V(px, py, M)andVis homogeneous of degree -1 inp:V( px, py, M) =V(px, py, M) .Furthermore, linear homogeneity permits us to form an exact price index corresponding tothe cost of a unit of utility:e(px, py) =( p1 x+ (1 ) p1 y)11 The indirect utility function can then be written:V(px, py, M) =Me(px, py)1 Conceptually, this equation states that the utility which can be realized with incomeMandpricespxandpyis equal to the income level divided by the unit cost of utility.

2 The key idea isthat when the underlying is linearly homogeneous, utility can be represented like any other goodin the economy. Put another way, without loss of generality, we can thing of each consumerdemanding only one CES TechnologyIn the representation of technology, we have a set of relationships which are directly analogous tothe CES utility function. These relationships are based on the cost and compensated demandfunctions. If we have a CES production function of the form:y(K, L) = ( K + (1 )L )1/ the unit cost function then has the form:c(pK, pL) =1 ( p1 K+ (1 ) p1 L)11 and associated demand Functions are:K(pK, pL, y) =(y )( c(pK, pL)pK) andL(pK, pL, y) =(y )((1 ) c(pK, pL)pL) .In most large-scale applied general equilibrium models, we have many function parametersto specify with relatively few observations. The conventional approach is tocalibratefunctionalparameters to a single benchmark equilibrium. For example, if we have benchmark estimates foroutput, labor, capital inputs and factor prices , we calibrate function coefficients by inverting thefactor demand Functions :1 = pK K pK K+ pL L, = 1 , = pK K1/ pK K1/ + pL L1/ and = y[ K + (1 ) L ] 1/ 1I wish to thank Professor Olivier de La Grandville for correcting an error in the expression for in an earlierversion of these Mikki once lived in Boulder and spent 30% of her income for rent, 10% for food and 60%for skiing.

3 She then moved to Georgetown where rent and food prices are identical toBoulder. In Georgetown, however, Mikki discovered that the quality-adjusted cost of skiingwas ten-times the cost of skiing in Boulder. She adopted a lifestyle in which she spend only30% of her income on skiing. Suppose that her preferences are characterized by a CESutility function. What values of and describe Mikki s utility function?2. What fraction of Mikki s income does she spend on rent in Georgetown?3. How much larger would Mikki s income need to be to compensate for the higher cost ofskiing such that she would be indifferent between living in Boulder or The Calibrated Share FormCalibration formulae for CES Functions are messy and difficult to remember. Consequently, thespecification of function coefficients is complicated and error-prone. For applied work usingcalibrated Functions , it is much easier to use the calibrated share form of the CES function. Inthe calibrated form, the cost and demand Functions explicitly incorporate benchmark factordemands benchmark factorprices the Elasticity of substitution benchmarkcost benchmarkoutput benchmarkvalue sharesIn this form, the production function is written:y= y[ (K K) + (1 )(L L) ]1/ The onlycalibratedparameter, , represents the value share of capital at the benchmarkpoint, = pK K pK K+ pL LThe corresponding cost Functions in the calibrated form is written:c(pK, pL) = c[ (pK pK)1 + (1 )(pL pL)1 ]11 where c= pL L+ pK Kand the compensated demand Functions are:K(pK, pL, y) = Ky y( pKcpK c) andL(pK, pL, y) = Ly y(c pL c pL) Normalizing the benchmark utility index to unity, the utility function in calibrated shareform is written:U(x, y) =[ (x x) + (1 )(y y) ]1/ The unit expenditure function can be written:e(px, py) =[ (px px)1 + (1 )(py py)]11 ,5the indirect utility function is:V(px, py, M) =M M e(px, py),and the demand Functions are.

4 X(px, py, M) = x V(px, py, M)(e(px, py) pxpx) andy(px, py, M) = y V(px, py, M)(e(px, py) pypy) .The calibrated form extends directly to then-factor case. Ann-factor production function iswritten:y=f(x) = y[ i i(xi xi) ]1/ and has unit cost function:C(p) = C[ i i(pi pi)1 ]11 and compensated factor demands:xi= xiy y(C pi C pi) 6 Exercises1. Show that given a generic CES utility function:U(x, y) = ( + (1 )y )1/ can be represented in share form using: x= 1, y= 1, px=t , py=t(1 ), M= any value oft > Consider the utility function defined:U(x, y) = (x a) (y b)1 A benchmark demand point with both prices equal and demand foryequal to twice thedemand forx. Find values for which are consistent with optimal choice at the these parameters so that the income Elasticity of demand forxat the benchmarkpoint equals Consider the utility function:U(x, L) = ( L + (1 )x )1/ which is maximized subject to the budget constraint:pxx=M+w( L L)in whichMis interpreted as non-wage income,wis the market wage rate.

5 Assume abenchmark equilibrium in which prices forxandLare equal, demands forxandLareequal, and non-wage income equals one-half of expenditure onx. Find values of and consistent with these choices and for which the price Elasticity of labor supply equals Consider a consumer with CES preferences over two goods. A price change makes thebenchmark consumption bundle unaffordable, yet the consumer is indifferent. Graph thechoice. Find an equation which determines the Elasticity of substitution as a function of thebenchmark value shares. (You can write down the equation, but it cannot be solved inclosed form.)5. Consider a model with three commodities,x,yandz. Preferences are CES. Benchmarkdemands and prices are equal for all goods. Find demands forx,yandzfor a doubling inthe price ofxas a function of the Elasticity of Consider the same model in the immediately preceeding question, except assume thatpreferences are instead given by:U(x, y, z) = ( min(x, y) + (1 )z )1/ Determine from the benchmark, and find demands forx,yandzif the price Consider a two-period model in which consumers maximizes the discounted present value ofutility:U(c1, c2) =c1 11 + c1 21 7subject to the budget constraint:c1+c21 +r= 1 +11 +rin which is the discount factor, is the intertempoal Elasticity parameter andris thegiven interest the calibrated share formulation to show (on inspection) that the equivalent variationof a change in the interest rate from rtoris equal to:EV=M/ M 1 =(2 +r2 + r)(1 + r1 +r)(1 + 1/ (1 +r)1/ 11 + 1/ (1 + r)1/ 1) /(1 ) 184 Flexibility and Non-Separable CESWe let idenote the user price of theith input, and letxi( ) be the cost-minizing demand fortheith input.

6 The reference price and quantities are iand xi. One can think of setias{K, L, E, M}but the methods we employ may be applied to any number of inputs. Define thereference cost, and reference value share forith input by Cand i, where C i i xiand i i xi CThe single-level Constant Elasticity of substitution cost function in calibrated form is written:C( ) = C( i i( i i)1 )11 Compensated demands may be obtained from Shephard s lemma:xi( ) = C i Ci= xi(C( ) C i i) Cross-price Allen-Uzawa elasticities of substitution (AUES) are defined as: ij CijCCiCjwhereCij 2C( ) i j= xi j= xj iFor single-level CES Functions : ij= i6=jThe CES cost function exibits homogeneity of degree one, hence Euler s condition applies tothe second derivatives of the cost function (the Slutsky matrix): jCij( ) j= 0or, equivalently: j ij j= 0 The Euler condition provides a simple formula for the diagonal AUES values: ii= j6=i ij j iAs an aside, note that convexity of the cost function implies that all minors of order 1 arenegative, ii<0 i.

7 Hence, there must beat least onepositive off-diagonal element in eachrow of the AUES or Slutsky matrices. When there are only two factors, then the off-diagonalsmust be negative. When there are three factors, then only one pair of negative goods may :9kindex a second-level nestsikdenote the fraction of goodiinputs assigned to thekth nest kdenote the benchmark value share of total cost which enters through thekth nest denote the top-level Elasticity of substitution kdenote the Elasticity of substitution in thekth aggregatepk( ) denote the price index associated with aggregatek, normalized to equal unity in thebenchmark, :pk( ) =[ isik i k i i)1 k]11 kThe two-level nested, nonseparable Constant - Elasticity -of-substitution (NNCES) costfunction is then defined as:C( ) = C( k kpk( )1 )11 Demand indices for second-level aggregates are needed to express demand Functions in acompact form. Letzk( ) denote the demand index for aggregatek, normalized to unity in thebenchmark; ( ) =(C( ) C1pk( )) Compensated demand Functions are obtained by differentiatingC( ).

8 In this derivative, oneterm arise for each nest in which the commodity enters, so:xi( ) = xi kzk( )(pk( ) i i) k= xi k(C( ) C1pk( )) (pk( ) i i) kSimple differentiation shows that benchmark cross-elasticities of substitution have the form: ij= + k( k )siksjk kGiven the benchmark value shares iand the benchmark cross-price elasticities ofsubstitution, ij, we can solve for values ofsik, k, kand . A closed-form solution of thecalibration problem is not always practical, so it is convenient to compute these parameters usinga constrained nonlinear programming algorithm, CONOPT, which is available through GAMS,the same programming environment in which the equilibrium model is specified. Perroni andRutherford [1995] prove that calibration of the NNCES form is possible for arbitrary dimensionswhenever the given Slutsky matrix is negative semi-definite. The two-level (N N) function isflexible for three inputs; and although we have not proven that it is flexible for 4 inputs, the onlydifficulties we have encountered have resulted from indefinite calibration data GAMS programs are listed below.

9 The first illustrates two analytic calibrations of thethree-factor cost function. The second illustrates the use of nonlinear programming to calibrate afour-factor cost function.(See Rutherford [1999] for an introduction to MPSGE.)10$TITLE Two nonseparable CES calibrations for a 3-input cost function.* Model-specific data defined here:SET i Production input aggregates / A,B,C /; ALIAS (i,j);PARAMETER theta(i) Benchmark value shares /A , B , C (i,j) Benchmark cross-elasticities (off-diagonals) /;* Use an analytic calibration of the three-factor CES cost* function:ABORT$(CARD(i) <> 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$(SMIN(i, theta(i)) <= 0) " Zero value shares are not valid:",THETA;* Fill in the Elasticity matrices:aues(i,i) = 0; aues(i,i) = -SUM(j, aues(i,j)*theta(j))/theta(i); DISPLAY aues;SET n Potential nesting /N1*N3/k(n) Nesting aggregates used in the modeli1(i) Good fully assigned to first nesti2(i) Good fully assigned to second nesti3(i) Good split between nests;SCALAR assigned /0/;PARAMETER esub(*,*) Alternative calibrated elasticitiesshr(*,i,n) Alternative calibrated sharessigma(n) Second level elasticitiess(i,n) Nesting assignments (in model)gamma Top level Elasticity (in model);* First the Leontief structure:esub("LTF","GAMMA") = SMAX((i,j), aues(i,j));esub("LTF",n) = 0;LOOP((i,j)$((aues(i,j) = esub("LTF","GAMMA"))*(NOT assigned)),11i1(i) = YES;i2(j) = YES;assigned = 1;);i3(i) = YES$((NOT i1(i))*(NOT i2(i)));DISPLAY i1,i2,i3.

10 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$(SMIN((i,n), shr("LTF",i,n)) < 0) "Benchmark AUES is indefinite.";* Now specify the two-level CES function:esub("CES","GAMMA") = SMAX((i,j), aues(i,j));ESUB("CES","N1") = 0;LOOP((i1,i2,i3),shr("CES",i1,"N1") = 1;shr("CES",i2,"N2") = 1;esub("CES","N2") = (aues(i1,i2)*aues(i1,i3)-aues(i2,i3)*aue s(i1,i1)) /(aues(i1,i3)-aues(i1,i1));shr("CES",i3, "N1") =(aues(i1,i2)-aues(i1,i3)) / (aues(i1,i2)-aues(i1,i1));shr("CES",i3," N2") = 1 - shr("CES",i3,"N1"););ABORT$(SMIN(n, esub("CES",n)) < 0) "Benchmark AUES is indefinite?";ABORT$(SMIN((i,n), shr("CES",i,n)) < 0) "Benchmark AUES is indefinite?";PARAMETER price(i) Price indices used to verify calibration,aueschk(*,i,j) Check of benchmark AUES values;price(i) = 1;$ontext$MODEL:CHKCALIB$SECTORS:Y !