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A note on the integrability of partial-equilibrium measures of the welfare costs of inflation
Journal of Banking & Finance 26 (2002) 2357–2363 www.elsevier.com/locate/econbase
A note on the integrability of partial-equilibrium measures of the welfare costs of inflation q Rubens Penha Cysne
1
Fundacßa~o Getulio Vargas, Rio de Janeiro, Brazil Received 19 January 2001; accepted 6 June 2001
Abstract Multidimensional measures of the welfare costs of inflation have been employed in the literature without an explicit concern of how the demand for the respective monetary assets are generated and without an investigation of the respective integrability conditions. This note establishes conditions under which such welfare measures are well defined. Ó 2002 Elsevier Science B.V. All rights reserved. JEL classification: E40; E60 Keywords: Inflation; Welfare; Integrability
1. Introduction This work investigates the integrability of multidimensional partial-equilibrium measures of the welfare costs of inflation. Such measures, which in a certain way
q
A previous version of this paper circulated with the title ‘‘Integrability and the demand for monetary assets: an alternative approach to an old problem’’. E-mail address: [email protected] (R.P. Cysne). 1 Rubens Penha Cysne is a professor of economics at the Fundacß~ao Getulio Vargas (FGV) and a researcher at the IBRE/FGV. 0378-4266/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 0 1 ) 0 0 2 1 3 - 8
2358
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
extend Bailey’s (1956) seminal contribution, are particularly aimed at economies where more than one asset perform monetary functions. They have been used in the literature (Marty and Chaloupka, 1988; Marty, 1994, 1999; Baltensperger and Jordan, 1997), though, without an explicit concern about the issue of path independence, as well as the issue of how the demands for the respective monetary assets are derived. By showing how this can be done and under what conditions these measures are well defined, their applicability is strengthened. This is even more important due to the fact that, elsewhere, Lucas (2000), for the one-dimensional case, as well as Simonsen and Cysne (2001) and Cysne (2001), for the n-dimensional case, have shown that such partial-equilibrium measures can be regarded as very good approximations, especially for moderate rates of inflation, of their general-equilibrium counterparts. Our procedure can be understood as paralleling the search, in classical static consumer theory, of particular specifications of the utility function which assured the integrability of Marshallian demands (the two classic textbook cases being those of homothetic preferences and of parallel preferences with respect to the numeraire). Likewise, we look for conditions that a certain transacting technology must fulfill in order to make a specific welfare measure well defined. Our basic conclusion is that, in a shopping-time framework, the partial-equilibrium measure of the welfare costs of inflation is well defined (path independent) if and only if the transacting technology is blockwise-weakly separable with respect to the monetary variables and shopping time. Expressed differently, the marginal rates of substitution between the two monetary assets must be independent of the total shopping time. Our derivation of the demands for the monetary assets is based on Simonsen and Cysne (2001) model, which, in turn, draws on Lucas (1993, 2000) previous analysis of welfare costs of inflation in shopping-time economies. To simplify, we present a model using currency and one kind of alternative monetary asset, which we call deposits. Our basic results, though, can be easily extended to the case when more than two assets perform monetary functions. This note presents two more sections. Section 2 develops the basic model. Section 3 presents the partial-equilibrium measure of the welfare costs of inflation, establishes conditions for its path independence and displays two contrasting examples.
2. The model This section draws heavily on the exposition of the model in Simonsen and Cysne (2001). Households are assumed to maximize discounted utility subject to their budget constraint and to the constraint that the total time spent producing the consumption good and shopping must sum to one. A transacting technology specifies how currency and deposits permit agents to economize on the amount of time spent on transactions in the goods market.
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
2359
Household preferences are determined by Z
1
egt U ðcÞ dt
ð1Þ
0
where U : X0 ! R, X0 Rþ , is a strictly concave function of the consumption at instant t and g > 0. The household is endowed with one unit of time that can be used to transact or to produce the consumption good with constant returns to scale: y þ s ¼ 1:
ð2Þ
Here, y stands for the production of the consumption good and s for the fraction of the initial endowment spent as transacting time. Households can accumulate three assets: currency ðMÞ, bonds ðBÞ and deposits ðX Þ. In their maximization, households take as given the nominal interest rate of bonds, i, and the opportunity cost of holding deposits, j ¼ i ix , where ix is the interest rate paid by X and 0 < j < i. Letting P ¼ P ðtÞ be the price of the consumption good, the household faces the budget constraint _ þ B_ þ X_ ¼ iB þ ix X þ P ð y cÞ þ H : M H indicates the flow of currency transferred to the household by the government and the dot over the variable its time derivative. Making a ¼ ðM þ B þ X Þ=P , p ¼ P_ =P (inflation rate), m ¼ M=P , x ¼ X =P and h ¼ H =P , and taking Eq. (2) into account, the budget constraint reads a_ ¼ ði pÞa þ 1 s im jx c þ h:
ð3Þ
Compared to currency, or to deposits, bonds are obviously preferable as a reserve of value. However, currency, as well as deposits, are useful because they save transaction time, as the transacting function describes: c ¼ F ðm; x; sÞ:
ð4Þ
F is supposed to be differentiable and strictly increasing in each of its variables, with decreasing marginal returns. The household maximizes Eq. (1) subject to the budget constraint (3) and subject to the time-transacting technology (4). The steady state is characterized by m_ ¼ b_ ¼ x_ ¼ 0. Also, in steady state, the rate of inflation is determined so that the seigniorage matches the real value of the net _ =M). transfers made by the government (h ¼ pm, where p ¼ M In steady state, Euler’s equations lead to the equilibrium relations: i ¼ p þ g; Fm ¼ iFs ; Fx ¼ jFs :
ð5Þ ð6Þ
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R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
In equilibrium, since the consumption good is non-storable and since all households are equal, y ¼ c. Using Eqs. (2) and (4), we get the fourth equation that completes the model: 1 s ¼ F ðm; x; sÞ:
ð7Þ
We use Eqs. (5)–(7) to determine, locally, i, j and s, as functions of m and x. In this model, the shopping-time variable ðsÞ can be considered as a general-equilibrium measure of the welfare costs of inflation (Simonsen and Cysne, 2001; Lucas, 2000). Indeed, since fiat money can be produced with a zero social cost, the totality of the transacting services demanded in the economy could be produced at no cost. Time dedicated to shopping is therefore a waste. In what follows, though, we will be solely interested in the partial-equilibrium measure of the welfare costs of inflation PE, to be defined in the next section. Simonsen and Cysne (2001) show, in the particular case when j is constant, that PE ðBðiB Þ in their paperÞ is an upper bound to s. Cysne (2001) extends this result, under certain conditions on the path followed by the parameters of the model, by relaxing the assumption of a constant value of j. 3. Integrability We will be interested in the evaluation of iðm; xÞ and jðm; xÞ along paths CðtÞ ¼ ðmðtÞ, xðtÞÞ, CðtÞ V , V an open set of R2þþ and a 6 t 6 b. With currency and interest-bearing deposits, the partial-equilibrium measure of the welfare costs of inflation is given by the line integral Z PE ¼ K dr ð8Þ C
where K ¼ ðiðm; xÞ; jðm; xÞÞ and dr ¼ ðdm; dxÞ. PE can be interpreted as a generalization of the area under a demand curve, although it is a different object from the mathematical point of view. For PE to be well defined as a welfare measure, it must take a unique value for different paths of m and x, when the initial and final points are the same (path independence). It is a well-known result from Calculus, sometimes referred to as the ‘‘potential function theorem’’ (see e.g. Apostol, 1957; Lang, 1987), that such path independence happens iff K is the gradient of some function b in all points where K is defined. Locally, this is equivalent to having: oi oj ðm; xÞ ¼ ðm; xÞ ox om
ð9Þ
in all points of V. The formal definition and an encompassing analysis of separability can be found in Leontief (1947). For our purposes, the function F ðm; x; sÞ is said to be blockwiseweakly separable when there are functions G and H such that F ðm; x; sÞ ¼
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
2361
H ðGðm; xÞ; sÞ. It also follows from the analysis made by Leontief that this condition is equivalent to having the marginal rate of substitution between m and x independent of s: o ðFm =Fx Þ ¼ 0: os The following proposition uses this definition to establish our main result: Proposition 1. The welfare measure PE is well defined (path independent) if and only if the transacting technology F ðm; x; sÞ is blockwise-weakly separable with respect to the monetary aggregator and the shopping-time variable, s. Proof. Considering Eq. (4) and the equilibrium conditions, taking the partial derivatives of Eqs. (5)–(7) and making 2
Fs D ¼ det 4 0 0
0 Fs 0
3 iFss Fms jFss Fxs 5 ¼ Fs2 ð1 þ Fs Þ > 0: 1 þ Fs
We have D D
oi ¼ Fs ½ðiFsx Fmx Þð1 þ Fs Þ ðiFss Fms ÞFx ; ox
oj ¼ Fs ½ðjFsm Fxm Þð1 þ Fs Þ ðjFss Fxs ÞFm : om
Therefore, Fs ð1 þ Fs Þ
oi oj ox om
¼ ðiFsx þ jFsm Þð1 þ Fs Þ þ Fss ðiFx jFm Þ þ ðFm Fxs Fx Fms Þ:
Substituting Fm =Fs and Fx =Fs , respectively, for i and j, in the above expression, one concludes that oi=ox ¼ oj=om for any values of m and x if and only if Fms Fx ¼ Fm Fxs () oðFm =Fx Þ=os ¼ 0. Example 1. Consider an economy with transacting technology: F ðm; x; sÞ ¼ Aðm1=2 þ sÞx1=2 ;
A > 0:
This technology is not weakly separable, since Fm =Fx 6¼ Fms =Fxs . Given the first order conditions, Eqs. (5)–(7), we have the functions
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R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
iðm; xÞ ¼
1 ; 2m1=2
jðx; mÞ ¼
1 þ m1=2 ; 2xð1 þ Ax1=2 Þ
and the partial-equilibrium welfare measure Z 1 1 þ m1=2 PE ¼ dx: dm þ 1=2 2xð1 þ Ax1=2 Þ C 2m
ð10Þ
Since in this example we are interested only in comparing the values of PE along different paths, we make A ¼ 1 to simplify the calculations. Let us then suppose that this economy presents an initial value of ðm; xÞ given by ð0:04; 0:04Þ, and a final value of ð0:01; 0:01Þ. We consider two different paths. In the first, C1 , the economy moves from ð0:04; 0:04Þ to ð0:01; 0:01Þ along the straight line x ¼ m ¼ t. The second path is a union of two intermediate paths. In the first, C21 , in which m is always kept constant, the economy moves from ð0:04; 0:04Þ to ð0:04; 0:01Þ. In the second, C22 , in which x is always kept constant in its new level, 0:01, the economy moves from ð0:04; 0:01Þ to ð0:01; 0:01Þ. By first calculating the value of PE for C1 , making x ¼ m ¼ t and dx ¼ dm ¼ dt in Eq. (10), we get PE1 ¼ 0:1 þ log 2. On the other hand, along C22 , PE22 ¼ 0:1, whereas along C21 we have Z 0:04 1 dx: PE21 ¼ 0:6 xð1 þ x1=2 Þ 0:01 Since PE2 ¼ PE21 þ PE22 , PE1 ¼ PE2 () PE21 ¼ ln 2. However, this does not occur, as the inequality below shows: PE21 ¼ 0:6
Z
0:04 0:01
1 dx > 0:6 xð1 þ x1=2 Þ
Z
0:04
0:01
1 dx ¼ 0:5 1:2x
Z
0:04
0:01
1 dx x
¼ log 2: Any evaluation of PE for this economy would therefore have to consider the path followed by the variables. Example 2. Here we consider the technology F ðm; x; sÞ ¼ Gðm; xÞs ¼ Bma x1a s, B > 0. This technology is separable, since Fm =Fx ¼ Fms =Fxs ¼ ða=ð1 aÞÞðx=mÞ. Along a path C, this technology leads to the following measure of PE: PE ¼
Z C
a 1 1a 1 dm þ dx: m Bma x1a þ 1 x Bma x1a þ 1
ð11Þ
We prove path independence by showing that this problem admits a potential function. By making Gðm; xÞ ¼ Bma x1a and solving the partial differential equations
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
2363
oCðm; xÞ a 1 ¼ ; om m 1þG oCðm; xÞ 1 a 1 ¼ : ox x 1þG One easily finds the potential function to be Cðm; xÞ ¼ logðG=ð1 þ GÞÞ. Hence, for initial and final values of m and x given by, respectively, ðm1 ; x1 Þ and ðm2 ; x2 Þ: PE ¼ ðCðm2 ; x2 Þ Cðm1 ; x1 ÞÞ whatever the path taken by ðm; xÞ between these two points.
Acknowledgements This work benefitted from conversations with Robert Lucas Jr. Humberto Moreira, Paulo Klinger Monteiro, Samuel Pessoa and two anonymous referees made valuable comments and suggestions. Remaining errors are my responsibility. The work was carried out while I was visiting the Department of Economics of the University of Chicago. I wish to thank the Department for its hospitality and the financial support from Capes.
References Apostol, T., 1957. Mathematical Analysis. Addison Wesley, Reading, MA. Bailey, M.J., 1956. Welfare cost of inflationary finance. Journal of Political Economy 64, 93–110. Baltensperger, E., Jordan, T.J., 1997. Seigniorage, banking, and the optimal quantity of money. Journal of Banking and Finance 21, 781–796. Cysne, R., 2001. Divisia indexes, inflation and welfare. Working paper 397, FGV, and the University of Chicago. Annals of the 2001, Summer Meeting of the North American Econometric Society. Lang, S., 1987. Calculus of Several Variables, third ed. Springer, Berlin. Leontief, W., 1947. Introduction to a theory of the internal structure of functional relationships. Econometrica 15 (4), 361–373. Lucas Jr., R.E., 1993. The welfare costs of inflation. Working paper. University of Chicago, Chicago. Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68, 247–274. Marty, A.L., 1994. The inflation tax and the marginal welfare cost in a world of currency and deposits. The Federal Reserve Bank of St. Louis Review 76, 67–71. Marty, A.L., 1999. The welfare cost of inflation: A critique of Bailey and Lucas. Federal Bank of St. Louis Review, No. 1. Marty, A.L., Chaloupka, F., 1988. Optimal inflation rates: A generalization. Journal of Money, Credit, and Banking 20, 141–144. Simonsen, M.H., Cysne, R.P., 2001. Welfare costs of inflation and interest-bearing deposits. Journal of Money, Credit, and Banking 33-1, 90–101.
A note on the integrability of partial-equilibrium measures of the welfare costs of inflation q Rubens Penha Cysne
1
Fundacßa~o Getulio Vargas, Rio de Janeiro, Brazil Received 19 January 2001; accepted 6 June 2001
Abstract Multidimensional measures of the welfare costs of inflation have been employed in the literature without an explicit concern of how the demand for the respective monetary assets are generated and without an investigation of the respective integrability conditions. This note establishes conditions under which such welfare measures are well defined. Ó 2002 Elsevier Science B.V. All rights reserved. JEL classification: E40; E60 Keywords: Inflation; Welfare; Integrability
1. Introduction This work investigates the integrability of multidimensional partial-equilibrium measures of the welfare costs of inflation. Such measures, which in a certain way
q
A previous version of this paper circulated with the title ‘‘Integrability and the demand for monetary assets: an alternative approach to an old problem’’. E-mail address: [email protected] (R.P. Cysne). 1 Rubens Penha Cysne is a professor of economics at the Fundacß~ao Getulio Vargas (FGV) and a researcher at the IBRE/FGV. 0378-4266/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 0 1 ) 0 0 2 1 3 - 8
2358
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
extend Bailey’s (1956) seminal contribution, are particularly aimed at economies where more than one asset perform monetary functions. They have been used in the literature (Marty and Chaloupka, 1988; Marty, 1994, 1999; Baltensperger and Jordan, 1997), though, without an explicit concern about the issue of path independence, as well as the issue of how the demands for the respective monetary assets are derived. By showing how this can be done and under what conditions these measures are well defined, their applicability is strengthened. This is even more important due to the fact that, elsewhere, Lucas (2000), for the one-dimensional case, as well as Simonsen and Cysne (2001) and Cysne (2001), for the n-dimensional case, have shown that such partial-equilibrium measures can be regarded as very good approximations, especially for moderate rates of inflation, of their general-equilibrium counterparts. Our procedure can be understood as paralleling the search, in classical static consumer theory, of particular specifications of the utility function which assured the integrability of Marshallian demands (the two classic textbook cases being those of homothetic preferences and of parallel preferences with respect to the numeraire). Likewise, we look for conditions that a certain transacting technology must fulfill in order to make a specific welfare measure well defined. Our basic conclusion is that, in a shopping-time framework, the partial-equilibrium measure of the welfare costs of inflation is well defined (path independent) if and only if the transacting technology is blockwise-weakly separable with respect to the monetary variables and shopping time. Expressed differently, the marginal rates of substitution between the two monetary assets must be independent of the total shopping time. Our derivation of the demands for the monetary assets is based on Simonsen and Cysne (2001) model, which, in turn, draws on Lucas (1993, 2000) previous analysis of welfare costs of inflation in shopping-time economies. To simplify, we present a model using currency and one kind of alternative monetary asset, which we call deposits. Our basic results, though, can be easily extended to the case when more than two assets perform monetary functions. This note presents two more sections. Section 2 develops the basic model. Section 3 presents the partial-equilibrium measure of the welfare costs of inflation, establishes conditions for its path independence and displays two contrasting examples.
2. The model This section draws heavily on the exposition of the model in Simonsen and Cysne (2001). Households are assumed to maximize discounted utility subject to their budget constraint and to the constraint that the total time spent producing the consumption good and shopping must sum to one. A transacting technology specifies how currency and deposits permit agents to economize on the amount of time spent on transactions in the goods market.
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
2359
Household preferences are determined by Z
1
egt U ðcÞ dt
ð1Þ
0
where U : X0 ! R, X0 Rþ , is a strictly concave function of the consumption at instant t and g > 0. The household is endowed with one unit of time that can be used to transact or to produce the consumption good with constant returns to scale: y þ s ¼ 1:
ð2Þ
Here, y stands for the production of the consumption good and s for the fraction of the initial endowment spent as transacting time. Households can accumulate three assets: currency ðMÞ, bonds ðBÞ and deposits ðX Þ. In their maximization, households take as given the nominal interest rate of bonds, i, and the opportunity cost of holding deposits, j ¼ i ix , where ix is the interest rate paid by X and 0 < j < i. Letting P ¼ P ðtÞ be the price of the consumption good, the household faces the budget constraint _ þ B_ þ X_ ¼ iB þ ix X þ P ð y cÞ þ H : M H indicates the flow of currency transferred to the household by the government and the dot over the variable its time derivative. Making a ¼ ðM þ B þ X Þ=P , p ¼ P_ =P (inflation rate), m ¼ M=P , x ¼ X =P and h ¼ H =P , and taking Eq. (2) into account, the budget constraint reads a_ ¼ ði pÞa þ 1 s im jx c þ h:
ð3Þ
Compared to currency, or to deposits, bonds are obviously preferable as a reserve of value. However, currency, as well as deposits, are useful because they save transaction time, as the transacting function describes: c ¼ F ðm; x; sÞ:
ð4Þ
F is supposed to be differentiable and strictly increasing in each of its variables, with decreasing marginal returns. The household maximizes Eq. (1) subject to the budget constraint (3) and subject to the time-transacting technology (4). The steady state is characterized by m_ ¼ b_ ¼ x_ ¼ 0. Also, in steady state, the rate of inflation is determined so that the seigniorage matches the real value of the net _ =M). transfers made by the government (h ¼ pm, where p ¼ M In steady state, Euler’s equations lead to the equilibrium relations: i ¼ p þ g; Fm ¼ iFs ; Fx ¼ jFs :
ð5Þ ð6Þ
2360
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
In equilibrium, since the consumption good is non-storable and since all households are equal, y ¼ c. Using Eqs. (2) and (4), we get the fourth equation that completes the model: 1 s ¼ F ðm; x; sÞ:
ð7Þ
We use Eqs. (5)–(7) to determine, locally, i, j and s, as functions of m and x. In this model, the shopping-time variable ðsÞ can be considered as a general-equilibrium measure of the welfare costs of inflation (Simonsen and Cysne, 2001; Lucas, 2000). Indeed, since fiat money can be produced with a zero social cost, the totality of the transacting services demanded in the economy could be produced at no cost. Time dedicated to shopping is therefore a waste. In what follows, though, we will be solely interested in the partial-equilibrium measure of the welfare costs of inflation PE, to be defined in the next section. Simonsen and Cysne (2001) show, in the particular case when j is constant, that PE ðBðiB Þ in their paperÞ is an upper bound to s. Cysne (2001) extends this result, under certain conditions on the path followed by the parameters of the model, by relaxing the assumption of a constant value of j. 3. Integrability We will be interested in the evaluation of iðm; xÞ and jðm; xÞ along paths CðtÞ ¼ ðmðtÞ, xðtÞÞ, CðtÞ V , V an open set of R2þþ and a 6 t 6 b. With currency and interest-bearing deposits, the partial-equilibrium measure of the welfare costs of inflation is given by the line integral Z PE ¼ K dr ð8Þ C
where K ¼ ðiðm; xÞ; jðm; xÞÞ and dr ¼ ðdm; dxÞ. PE can be interpreted as a generalization of the area under a demand curve, although it is a different object from the mathematical point of view. For PE to be well defined as a welfare measure, it must take a unique value for different paths of m and x, when the initial and final points are the same (path independence). It is a well-known result from Calculus, sometimes referred to as the ‘‘potential function theorem’’ (see e.g. Apostol, 1957; Lang, 1987), that such path independence happens iff K is the gradient of some function b in all points where K is defined. Locally, this is equivalent to having: oi oj ðm; xÞ ¼ ðm; xÞ ox om
ð9Þ
in all points of V. The formal definition and an encompassing analysis of separability can be found in Leontief (1947). For our purposes, the function F ðm; x; sÞ is said to be blockwiseweakly separable when there are functions G and H such that F ðm; x; sÞ ¼
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
2361
H ðGðm; xÞ; sÞ. It also follows from the analysis made by Leontief that this condition is equivalent to having the marginal rate of substitution between m and x independent of s: o ðFm =Fx Þ ¼ 0: os The following proposition uses this definition to establish our main result: Proposition 1. The welfare measure PE is well defined (path independent) if and only if the transacting technology F ðm; x; sÞ is blockwise-weakly separable with respect to the monetary aggregator and the shopping-time variable, s. Proof. Considering Eq. (4) and the equilibrium conditions, taking the partial derivatives of Eqs. (5)–(7) and making 2
Fs D ¼ det 4 0 0
0 Fs 0
3 iFss Fms jFss Fxs 5 ¼ Fs2 ð1 þ Fs Þ > 0: 1 þ Fs
We have D D
oi ¼ Fs ½ðiFsx Fmx Þð1 þ Fs Þ ðiFss Fms ÞFx ; ox
oj ¼ Fs ½ðjFsm Fxm Þð1 þ Fs Þ ðjFss Fxs ÞFm : om
Therefore, Fs ð1 þ Fs Þ
oi oj ox om
¼ ðiFsx þ jFsm Þð1 þ Fs Þ þ Fss ðiFx jFm Þ þ ðFm Fxs Fx Fms Þ:
Substituting Fm =Fs and Fx =Fs , respectively, for i and j, in the above expression, one concludes that oi=ox ¼ oj=om for any values of m and x if and only if Fms Fx ¼ Fm Fxs () oðFm =Fx Þ=os ¼ 0. Example 1. Consider an economy with transacting technology: F ðm; x; sÞ ¼ Aðm1=2 þ sÞx1=2 ;
A > 0:
This technology is not weakly separable, since Fm =Fx 6¼ Fms =Fxs . Given the first order conditions, Eqs. (5)–(7), we have the functions
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R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
iðm; xÞ ¼
1 ; 2m1=2
jðx; mÞ ¼
1 þ m1=2 ; 2xð1 þ Ax1=2 Þ
and the partial-equilibrium welfare measure Z 1 1 þ m1=2 PE ¼ dx: dm þ 1=2 2xð1 þ Ax1=2 Þ C 2m
ð10Þ
Since in this example we are interested only in comparing the values of PE along different paths, we make A ¼ 1 to simplify the calculations. Let us then suppose that this economy presents an initial value of ðm; xÞ given by ð0:04; 0:04Þ, and a final value of ð0:01; 0:01Þ. We consider two different paths. In the first, C1 , the economy moves from ð0:04; 0:04Þ to ð0:01; 0:01Þ along the straight line x ¼ m ¼ t. The second path is a union of two intermediate paths. In the first, C21 , in which m is always kept constant, the economy moves from ð0:04; 0:04Þ to ð0:04; 0:01Þ. In the second, C22 , in which x is always kept constant in its new level, 0:01, the economy moves from ð0:04; 0:01Þ to ð0:01; 0:01Þ. By first calculating the value of PE for C1 , making x ¼ m ¼ t and dx ¼ dm ¼ dt in Eq. (10), we get PE1 ¼ 0:1 þ log 2. On the other hand, along C22 , PE22 ¼ 0:1, whereas along C21 we have Z 0:04 1 dx: PE21 ¼ 0:6 xð1 þ x1=2 Þ 0:01 Since PE2 ¼ PE21 þ PE22 , PE1 ¼ PE2 () PE21 ¼ ln 2. However, this does not occur, as the inequality below shows: PE21 ¼ 0:6
Z
0:04 0:01
1 dx > 0:6 xð1 þ x1=2 Þ
Z
0:04
0:01
1 dx ¼ 0:5 1:2x
Z
0:04
0:01
1 dx x
¼ log 2: Any evaluation of PE for this economy would therefore have to consider the path followed by the variables. Example 2. Here we consider the technology F ðm; x; sÞ ¼ Gðm; xÞs ¼ Bma x1a s, B > 0. This technology is separable, since Fm =Fx ¼ Fms =Fxs ¼ ða=ð1 aÞÞðx=mÞ. Along a path C, this technology leads to the following measure of PE: PE ¼
Z C
a 1 1a 1 dm þ dx: m Bma x1a þ 1 x Bma x1a þ 1
ð11Þ
We prove path independence by showing that this problem admits a potential function. By making Gðm; xÞ ¼ Bma x1a and solving the partial differential equations
R.P. Cysne / Journal of Banking & Finance 26 (2002) 2357–2363
2363
oCðm; xÞ a 1 ¼ ; om m 1þG oCðm; xÞ 1 a 1 ¼ : ox x 1þG One easily finds the potential function to be Cðm; xÞ ¼ logðG=ð1 þ GÞÞ. Hence, for initial and final values of m and x given by, respectively, ðm1 ; x1 Þ and ðm2 ; x2 Þ: PE ¼ ðCðm2 ; x2 Þ Cðm1 ; x1 ÞÞ whatever the path taken by ðm; xÞ between these two points.
Acknowledgements This work benefitted from conversations with Robert Lucas Jr. Humberto Moreira, Paulo Klinger Monteiro, Samuel Pessoa and two anonymous referees made valuable comments and suggestions. Remaining errors are my responsibility. The work was carried out while I was visiting the Department of Economics of the University of Chicago. I wish to thank the Department for its hospitality and the financial support from Capes.
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