aa r X i v : . [ ec on . T H ] O c t An Application of Hölder’s Inequality to Economics
James OttersonCongressional Budget Office (CBO) ∗ Solow introduced CES functions in Solow (1956), see also Arrow et al. (1961), to develop a growth modelwithout the Harrod-Domar model assumption, which leads to model instability, that capital and labor are notsubstitutes. Since its appearance in the seminal papers, CES functions have become an essential economicmodeling tool. We recall some of their basic properties here to fix notation. By definition, a CES functionwith parameter r ≤ , r = 0 assigns to a vector x = ( x , . . . , x n ) with positive entries the number: n X i =1 x ri ! r . (1)When r > , the CES function describes inputs that are substitutes, or perfect substitutes when r = 1 .When r < , the inputs of the CES function are complements, or perfect complements in the limiting case r = −∞ , which is a Leontief function. If θ is a weight vector, we can replace x with θx = ( θ x , . . . , θ n x n ) inFormula (1) and take the limit r → to get a Cobb-Douglas function. We can also change Formula (1) bymultiplying it by a factor or raising it to a power to change homogeneity. The CES functional form, whenincluding all extra parameters and limiting cases, is the only function with the CES between any pair ofinputs (see Arrow et al., 1961).Given their versatility, CES functions are the building blocks of several models. For example, manyComputable General Equilibrium models consist of families of nested CES functions. A drawback of CESfunctions, however, is solving optimization problems involving them. As stated in Rutherford (2002), theusual Lagrangian calculus approach to solving such optimization problems can lead to long derivations. Weclaim that, by recasting CES functions as a type of norm of a vector spaces, with the use of the reverseHölder’s inequality it is easy to solve these optimization problems. Similarly, the ( n -stage) Armingtonfunctions of Armington (1969) can be viewed as the a type of norm on a direct sum of “normed” vectorspaces. With a version of Hölder’s inequality applicable to that setting, it is still possible to get simplesolutions methods of optimization problems involving these functions. L p -spaces and Hölder’s inequality A norm on a vector space V is a real valued function k · k satisfying: (1) k · k is finite and convex, (2) k αv k = | α | k v k for any vector v of V and scalar α , and (3) k · k is zero only at the zero vector. The convexityassumption is usually replaced by the triangular inequality. If we allow k · k to be zero away from zero, thenit is called a pseudo-norm. If k · k only satisfies the triangular inequality up to a constant multiple , then itis called a quasinorm. L p -spaces, where p is a number larger than , are a particular rich family of normedvector spaces that have been the subject of intense study (see, for example, Rudin, 1991). We will denotethese spaces by L p ≥ . Their norms are defined as: k f k p = (cid:18)Z | f | p (cid:19) p . ∗ This paper has not been subject to CBO’s regular review and editing process. The views expressed here should not beinterpreted as CBO’s. i.e., k x + y k ≤ K ( k x k + k y k ) for some K > . f is, roughly, a real valued function on a measurable space. L p ≥ -spaces have a wide range ofapplications. For example, the state spaces in quantum mechanics are L -spaces. If the measure is aprobability measure, then f is a random variable and k f k q is its q -moment. To simplify our arguments, wewill restrict to L p ( R n ) , the set of vectors x = ( x , . . . , x n ) of R n with norm: k x k p = n X i =1 | x i | p ! p . (2)It is clear that the definition of CES functions, Equation (1), and the definition of the L p -norms, Equation(2), have the same functional form. The only difference is that CES functions depend on a parameter r ≤ and for L p -spaces one usually assumes that p ≥ . For now on we use p for price and use r instead of p for L p -spaces, even when p > . From the viewpoint of functional analysis, the case when r < has anumber of undesirable properties. For example, for the infinite dimensional case, the dual space of L
We begin by showing that if a and b are positive numbers, and for r and s satisfying the lemma’sassumptions, then ab ≥ a r r + b s s . (3)(This is an extension of Young’s inequality, see Rudin (1991)). To check this, consider the function y = x r − and its inverse x = y s − . Both functions are decreasing since r, s < . By evaluating the right hand side ofthe equation below, it is easy to show that: a r r + b s s = 12 (cid:18)Z ab s − x r − dx (cid:19) + 12 Z ba r − y s − dy ! + 12 b s − ( b − a r − ) + 12 a r − ( a − b s − ) + a r − b s − . (4)Either a ≥ b s − or a ≤ b s − in the domain of the first integral of the above equation. If we assumethe former case, then the right-hand side of Equation (4) calculates the area of shaded region of Figure 1.Since this region is contained in a rectangle with area equal to ab , the statement follows. We can give analgebraic proof of the statement by starting with a ≥ x ≥ b s − , and raising these numbers by r − to get b ≥ x r − ≥ a r − . Setting y = x r − , we can raise the last inequality by s − to get a ≥ y s − ≥ b s − . Hence, Z ab s − x r − dx ≤ Z ab s − b dx = b ( a − b s − ) , (5) Namely, under these assumptions, if φ is a continuous linear function on L r then φ = 0 . For a proof, see Conrad (2002).In particular, if x : [0 , → R is a continuum of goods with prices p : [0 , → R then the linear functional p ( x ) = R p x is notcontinuous with respect to a utility function defined by k · k r . For example, if we set prices to be equal to for every goodand consider the sequence of consumption bundles x n = n χ [ , n ] , where χ is an indicator function, then k x n k r = n r − → as n → ∞ , ( x n approaches the zero bundle as n increases) but p ( x n ) = 1 for all n whereas p (0) = 0 . When r < , it is natural to assume that k x k r = 0 whenever x has a zero entry. We can write any vector v of L r< ( R n ) as asum of two vectors v , v each of which with a zero entry. If k · k r was a pseudo-quasinorm, then k v k r ≤ K ( k v k r + k v k r ) = 0 for any vector v , a contradiction. Z ba r − y s − dy ≤ Z ba r − a dy = a ( b − a r − ) . (6)By plugging in Equations (5), (6) into Equation (4) we get that a r r + b s s ≤ ab . Notice that the same inequalitieshold true if a ≤ b s − (and hence b ≤ a r − ) since now the integrals of Equation (4) become negative numbers.Visually, as depicted in Figure 1, are now subtracting from the are of a rectangle an amount larger than thearea of a region whose complement has area ab . Hence, we still have that a r r + b s s ≤ ab .With the extended Young’s inequality (Equation (3)) hand, we can now finish the proof of the lemma.Suppose x = ( x , . . . , x n ) and y = ( y , . . . , y n ) are two vectors with positive entries. Then: x k x k r · y k y k s := n X i =1 x i k x k r · y i k y k s ≥ r X x ri k x k rr + 1 s X y si k y k ss = 1 r + 1 s = 1 . Where the inequality in the above equation follows from Equation (3). Hence x · y ≥ k x k r k y k s . (cid:3) a r − ab b s − y = x r − a a p − b q − b y = x r − Figure 1: When a > b s − , Equation (4) computes the shaded area in the left rectangle with area ab . When a < b s − , Equation (4) subtracts an from the area of the right side rectangle a number larger than theshaded are shaded.For the case when r = 0 , Hardy et al. (1952) shows that y · x > n n Y i y i n Y j x j n , where n is the dimension of the non-negative vectors x and y . Corollary 1 gives a slight generalization ofthis to include the case of weight vectors that are not evenly distributed. Corollary 1 (Hölder’s inequality for L -spaces) . Suppose θ = ( θ , . . . , θ n ) is a weight vector, that is, avector with positive entries where P ni =1 θ i = 1 . For any two vectors x = ( x , . . . , x n ) and y = ( y , . . . , y n ) with positive entries, we have that: x · y ≥ n Y i =1 x θ i i n Y i =1 ( θ − i y i ) θ i = k x k , θ k θ − y k , θ . Where k x k ,θ = Q ni =1 x θ i i and θ − y = ( θ − i y , . . . , θ − n y n ) . Remark 1 (Versions of L -spaces) . To clarify, we are considering L θ ( R n ) as R n together with the Cobb-Douglas function Q | x i | θ i . There are other natural candidates for L -spaces such as the F-norm of Banach(2009), the counting norm of Donoho and Elad (2002), and the example in Kalton et al. (1984) of functionswith the convergence in measure topology. 3 roof: For any r ≤ , r = 0 , and given θ , x and y satisfying the Corollary’s assumptions, we have, byLemma 1: x · y = ( θ /r x ) · ( θ − /r y ) ≥ k θ /r x k r . k θ /s ( θ − y ) k s . In the limit: x · y ≥ lim r → k θ /r x k r k θ /s ( θ − y ) k s . = Y x θ i i Y (cid:0) θ − i y i (cid:1) θ i . (cid:3) To cover the Armington case, we need to consider a version of Hölder’s inequality for the direct sum of L r ≤ -spaces. When r ≥ , there is no obvious best candidate for a norm on a direct sum of L r -spaces.Banach considered the following norms for such spaces in Banach (2009). Let L r ( R n ) , . . . , L r m ( R n m ) bespaces with r i ≥ , i = 1 , . . . , m . We can use the given norms to define a function on their direct sum: N : R n × · · · × R n m −→ R m , ( x , . . . , x m ) ( k x k r , . . . , k x m k r m ) . We can endow R m , the target space of N , with an L r norm, for any r ≥ . Composing the function N withany such norm defines a new function on the direct sum. The new function is still a norm and the space isstill a Banach space.If in the previous construction we assume instead that r i ≤ for all cases i , then the constructed “norm”is an Armington function (usually the norm picked over R m is the L one). A version of Hölder’s inequalityalso exists on this setting. Corollary 2 (Hölder’s inequality for direct sums of L r ≤ -spaces) . Fix a finite set L r ( R n ) , . . . , L r m ( R n m ) where r i ≤ , i = 1 , . . . , m . For r ≤ and s satisfying r + 1 s = 1 , and for x = ( x , . . . , x m ) and y = ( y , . . . , y m ) vectors in R n × . . . × R n m where the vectors x i and y i havenon-negative entries, we have x · y ≥ k X k r k Y k s , if r = 0 ,x · y ≥ k X k , θ k θ − Y k , θ , if r = 0 and any weight vector θ. Where X = ( k x k r , . . . , k x m k r m ) , Y = ( k y k s , . . . , k y m k s m ) , and s i = r i r i − . (Notice the slight abuse ofnotation: whenever r i = 0 , we assume a weight vector θ i has being fixed and we replace k x i k r i and k y i k s i by k x i k , θ i and k θ − i y i k , θ i , respectively.) Proof: by Lemma 1 or Corollary 1, we have that: X · Y ≥ k X k r k Y k s , if r = 0 X · Y ≥ k X k , θ k θ − Y k , θ if r = 0 and any weight vector θ. By definition, X · Y = P mi =1 X i Y i . We can apply again Lemma 1 or Corollary 1, to X i Y i to get: x i · y i ≥ X i Y i , for i = 1 , . . . , m . Since x · y = P mi =1 x i · y i , the proof is complete. (cid:3) In this section we go over some concrete calculations.4 xample 1 (CES Utility Function) . Let x = ( x , . . . , x n ) be a bundle of goods (a vector in R n with positiveentries), and U an utility function over such bundles. Assuming that U is a CES function then, from theprevious section, we think of the bundle x as a point in a L r ( R n ) -space for some r ≤ , r = 0 . Similarly,from the perspective of the previous section, it is natural to consider a price vector p (a vector of positiveentries in R n ) of the bundle x as a point in L s ( R n ) where r + 1 s = 1 . (In other words, a “ r -norm” k x k r on the space of goods induces a “ s -norm” k p k s on the space of prices.) Letus start with the problem of minimizing expenditure given utility and price levels. Suppose that U ( x ) = u ,from Lemma 1 (reverse Hölder’s inequality), we always have that: x · p = n X i =1 p i x i ≥ k x k r k p k s = u k p k s . Since the above inequality is true for any bundle with utility level u , we get a lower bound to the expenditurefunction: e ( u, p ) ≥ u k p k s . To show that this lower bound is sharp, we show that there is a feasible consump-tion bundle x (a vector with positive entries) satisfying the lower bound. If we had that e ( u, p ) = u k p k s ,then, since the utility function is locally nonsatiated and strictly convex, the Hicksian demand functionequals to: x ( u, p ) = ∇ p e ( u, p ) = u k p k σs p − σ . Where σ = 1 − s = − r and p − σ = ( p − σ , . . . , p − σn ) . Since all terms on the left-hand side of the above equationare positive, x ( u, p ) is a feasible bundle and the lower bound under consideration is sharp. Furthermore,since the lower bound is sharp, we can view k p k s as the cost of a unit of utility for a given price p , and hencethe price index is: P K ( p , p , u ) = e ( u, p ) e ( u, p ) = k p k s k p k s . We now consider the dual problem of maximizing utility given price and wealth constraints. This can besolved by noting that if the expenditure function is e ( u, p ) = u k p k s then the indirect utility function is equalto: ν ( m, p ) = m k p k − s . By Roy’s identity, the Marshallian demand function equals to: x ( m, p ) = − ∇ p ν ( p, m ) ∇ m ν ( p, m ) = m k p k σs p − σ k p k s = m p s − k p k ss . Hence x i p i = m p si k p k ss and p si / k p k s is the share of the budget spent on sector i . Remark 2.
The above example avoided solving expenditure minimization problems via Lagrange multipliermethods by getting a lower bound to the expenditure function using ideas from functional analysis andshowing that the bound is sharp. Related utility maximization problems were then solved by using resultsthat are applicable to general settings.
Remark 3 (CES with weights and Cobb-Douglas) . We can adapt Example 1 to the CES with weights orthe Cobb-Douglas case by starting from the inequalities (see Corollary 1): x · p = ( θ /r x ) · ( θ − /r p ) ≥ k θ /r x k r . k θ /s ( θ − p ) k s ,x · p ≥ Y x θ i i Y (cid:0) θ − i p i (cid:1) θ i = k x k ,θ k θ − p k ,θ . The key calculations for these two cases are: ∇ p k θ /s ( θ − p ) k s = k θ /s ( θ − p ) k σs ( θ − p ) − σ , ∇ p k θ − p k ,θ = k θ − p k ,θ ( θ − p ) − . This is true cost-of-living index (or Konüs expenditure-based cost-of-living index). It measures the necessary compensationto fix a consumer’s utility level after a movement in prices. xample 2 (Armington functions) . Suppose x i is a bundle of n i goods for i = 1 , . . . , m and let x =( x , . . . , x m ) . For each x i fix an CES utility function k · k r i and let k · k s i be the corresponding CES functionon prices p i where s i = r i − r i . Let X = ( k x k r , . . . , k x m k r m ) ,P = ( k p k s , . . . , k p m k s m ) . For a given r < , r = 0 and s = rr − let: k x k = k X k r , and k p k • = k P k s . These are Armington functions.By Corollary 2, we have x · p ≥ k X k r k P k s . Hence, e ( u, p ) ≥ u k P k s . By the chain rule, ∇ p u k P k s is a productof positive numbers, hence the lower bound is sharp. In particular, the Hicksian demand functions are: x i ( u, p ) = ∇ p i e ( u, p ) = u k P k σs P − σi k p i k σ i s i p − σ i i . Where σ = 1 − s , and σ i = 1 − s i are the elasticities. As in Example 1, k · k • is the cost of a unit of utilityand the price index is: P K ( p ′ , p, u ) = k p ′ k • k p k • = (cid:0)P mi =1 k p ′ i k ss i (cid:1) /s (cid:0)P mi =1 k p i k ss i (cid:1) /s . Since the indirect utility function is ν ( m, p ) = m k P k − s , by Roy’s lemma the Marshallian demand functionsare: x i ( m, p ) = − ∇ p i ν ( p, m ) ∇ m ν ( p, m ) = m k P k σs P − σi k p i k σ i s i p − σ i i k P k s = m k P k σs P − σi ( P i P − i ) k p i k σ i s i p − σ i i k P k s = m P si k P k ss p s i − i k p i k s i s i . As in Example 1, we can interpret the fractions of the last formula in terms of shares of allocated budget.
Remark 4.
As in Remark 3, we can introduce weights θ to the previous Armington function example andtake the limit r → to an Armington utility function where the aggregate function is a Cobb-Douglasfunction. In this case, we have: e ( u, p ) = u Q (cid:0) θ i − k p i k s i (cid:1) θ i = u k θ − P k ,θ . The price index is then: P K ( p ′ , p, u ) = Y (cid:18) k p ′ i k s i k p i k s i (cid:19) θ i . The indirect utility function is ν ( m, p ) = m k P k − ,θ . By Roy’s identity, x i ( m, p ) = m θ i P i ∇ p i P i = mθ i = mθ i p s i − i k p i k s i s i . As before, we can interpret the last formula in terms of budget shares.
Remark 5 (Armington with several stages) . We can introduce more stages to the Armington function byaggregating finite sets of Armington functions using Cobb-Douglas or CES functions. These process can berepeated n times. Hölder’s inequality, as stated in Corollary 2, can be extended to these cases by induction.Example 2 can be adapted to these cases. In particular, the Marshallian demand function is again a productof factors that can be understood as budget shares. In this paper, we consider CES, Cobb-Douglas and Armington functions as L r ≤ ( R n ) -spaces, or direct sums ofthese spaces. We use a central result of L r ≥ -spaces, reverse Hölder’s inequality, to obtain simple derivationsof main economic formulas such as Hicksian and Marshallian demand functions. Going in the other direction,economic theory suggests a new family of L -spaces. To aid intuition, see Table 1 for a dictionary of guidingconcepts and Figure 2 for a depiction of sample solution sets k x k r = 1 , for −∞ ≤ r ≤ ∞ .6able 1: Dictionary between L r ≥ -spaces and L r ≤ -spaces. L r -spaces, r ≥ L r -spaces r ≤ upper bounds lower bounds L , L ∞ pairing L , L −∞ pairing L , Hilbert Spaces L , Cobb-Douglas spaces L r -spaces, ≤ r < complementary goods L r -spaces, < r substitute goods k·k r and k·k s where r − + s − = 1 Utility functions on space of goods ( k · k r ), cost of a an unit ofutility on the space of prices ( k · k s ). For a vector of prices p , theshare p si / k p k ss is the share of the budget allocated to item i .Banach’s norms on directsums of L r -spaces Armington functionsFigure 2: Regions of solutions sets of k x k r = 1 . r < r = 00 < r ≤ Note: For r < the solution set lies in the outer corner squares; the special case when r = −∞ is traced by the boundaries of these squares that lie inside the picture. For r = 0 the solution set lies inside the outer rectangles. For < r ≤ the solution set liesinside the inner diamond; the solution set is the edge of the diamond when r = 1 . For r > the solution set lies in the non-shaded region; the solution set is the boundaryof square containing the non-shaded region when r = ∞ .7 eferences Armington, Paul (1969) “A theory of demand for products distinguished by place of production” IMF StaffPapers , 159–178.Arrow, K. J.; Chenery, H. B.; Minhas, B. S.; Solow, R. M. (1961) “Capital-labor substitution and economicefficiency” Review of Economics and Statistics , 225–250.Banach, Stefan (2009) “Theory of Linear Operations” Dover: New York.Barthe, Franck (1998) “Optimal Young’s inequality and its converse: a simple proof” Geometric & FunctionalAnalysis , 234-242.Burqan, A., Khandaqji, M. (2015) “Reverses of Young type inequalities” Journal of Mathematical Inequalities , 113-120.Conrad, Keith (2012) “ L p -Spaces For < p < ” Lecture notes, Donoho, D., Elad, M. (2003) “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ minimization” Proceedings of the National Academy of Sciences , 2197-2202.Hardy, G.H.; Littlewood J.E.; Pólya, G. (1952) “Inequalities” Second Edition, Cambridge University Press:Cambridge.Kalton, N.; Peck, N.; Roberts, J. (1984) “An F-space Sampler” London Mathematical Society Lecture NoteSeries 89.Rudin, Walter (1991) “Functional Analysis” Second Edition, McGraw-Hill Inc.: New York.Rutherford, Thomas (2002) “Lecture notes on constant elasticity functions” Solow, Robert (1956) “A contribution to the theory of economic growth” The Quarterly Journal of Economics70