- Email: [email protected]
On the missing equation: Comment
DOUGLAS R. SHALLER Rutgers Uniuersity
On the Missing Equation:
Comment
Canto and Miles (CM) present and analyze graphically “a third, alternative method for closing the IS-LM system of equations”-the ‘wedge model’. According to CM, and “the two traditional alternatives “the result is a more general set of results” are only special cases.” The CM model, however, is logically inconsistent. Because CM implicitly assume simultaneous capital market equilibrium and disequilibrium, CM’s assumptions are mutually inconsistent. Thus, none of the reported results logically follow, Furthermore, the CM model is not very general. This paper proves analytically that GNP is determined without the IS and LM equations both in the CM model and in the obvious logical patchups of the CM model. Therefore, for example, in the CM model exogenous changes in business-fixed investment or consumption have no predicted effect on national income.
In a recent paper in this journal, Canto and Miles (CM) develop a macromodel that, according to the authors, is “sufficiently general to incorporate as special cases” both the neoclassical and neo-Keynesian versions of the IS-LM model. They describe their model as a new solution to “the missing equation problem.” Unfortunately, there are some serious logical problems in the construction of the model and with its presentation. The macromodel presented by Canto and Miles is not logically consistent because the assumptions of the model are contradictory. Since the capital market is assumed to be simultaneously in equilibrium and out of equilibrium, Canto and Miles have not proved any of the propositions they discuss. Comparative-static conclusions derived from the model, if accepted, must be accepted on faith. The authors state on page 251 that “c and Lt represent the market sector derived demand for capital and labor services” and in footnote 3 that “at any point in time the supply of capital is a given quantity 8.” Capital market equilibrium is defined by c(Y’, w/r) = Z? [their Equation (4)]. The capital market equilibrium condition states that desired capital stock equals actual capital stock. What is called the IS curve [Equation (l’)] is specified on page 250 as Cd(Y,,w) + Z(r) = Y5, where Z(r) represents the flow demand for investment. But, in a stock-flow macromodel, the flow investment function comes into play only when the stock of capital does ]ournal Copyright
of
Macroeconomics, 6
1985
by Wayne
Winter State
1964, Vol. 6, No. University Press.
1, pp.
97-102
97
Douglas R. Shaller not equal the aggregate desired stock of capital.’ In the CM model however, the flow investment function is specified along with th specification of equality of the desired stock of capital goods wit the actual stock of capital goods. Thus, the market for capital good is specified to be both out of equilibrium and in a state of equilib rium at the same point in time, a logical contradiction. Furthermore, without regard to the logical consistency issue the CM macromodel can not possibly contain a neo-Keynesian IS LM model as a special case. A major characteristic of any distinc tively neo-Keynesian IS-LM model is that an exogenous shift ii any component of aggregate demand shifts GNP [see Branson (1979) for example, for a standard textbook IS-LM model]. However, ir the CM model, GNP is determined independently of the IS ant LM equations. (For a proof of this proposition, see PROPOSITION 1 of this paper.) Neo-Keynesian macromodels of all makes are famous for being demand driven. For example, the well-known IS-LM illustration oi the onset of the Great Depression commences with an exogenous decrease in business-fixed investment causing the IS curve to fal backward. The standard prediction of the IS-LM model is a decrease in equilibrium GNP and a concomitant decline in the interest rate. In the CM model, however, an exogenous decrease in business-fixed investment has no effect on equilibrium GNP. Of course, there are some special cases of the CM model where the problem of contradictory assumptions can be avoided. For example, if the flow investment demand function equals zero, then the model is a special case of Tobin’s (1955) dynamic aggregative model-in this case, a classical model with a market for used capital (see COROLLARY 1 of this paper). Alternatively, if the capital market equilibrium condition is discarded (see COROLLARY 2) then the model is a variant of the textbook short-period classical model.
1. The Model Following CM, equilibrium is specified solution to the following equations: Cd(Yf, w) + Z(r) - Y” = 0 Li(Y’, w/r) ‘See assumes
98
Gould costs
as the simultaneous
(IS curve) ;
- T - Ld(Yf, W) = 0
(labor market equilibrium)
(1968) for a standard derivation of adjustment for a firm seeking
of a flow to expand
;
(2)
investment function that its productive capacity.
On the Missing J&Y”, w/r) where,
- K = 0
Equation:
(capital market equilibrium)
Comment
;
(3)
by definition,
Y, = Y” + wLd(Y,, Y” = F(&,
L,,,)
The notation
= = = = =
Li(Y”, w/r) = Ld(Yf, W) = w/d
?’ = = 2? =
(%.rll income”)
;
(aggregate production
is defined
w r Cd(Yfi w) I(r) r
KP”,
w)
(4)
function)
.
@>
as follows:
real wage rate; rate of interest; consumption function; investment function; output of market goods market-derived national income; labor demand by firms; demand for nonmarket time (leisure) by workers; total supply of workers’ time; capital demand by firms; and stock of capital.
2. A Formal Analysis
of the CM Model
Canto and Miles points. Since I subject it more convenient to tion. Substitute (4) into compactly as’
use only graphical illustrations to make their their model to a more formal analysis, I find write the model using more compact nota(1) and (2) so that the system can be written
Eg
(Y”,
W/T)
=
EL(Ys, w, w/r)
0 (IS curve) ;
(6)
= 0 (labor market equilibrium)
Ek(Y”, w/r)
= 0 (capital market equilibrium)
Y” = F(&,
I+,,) (aggregate production
K,,, = e(Y’,
w/r)
(capital demanded
*The truly exogenous variables forms of (61, (7), and (6).
such
;
(7)
;
function)
03) ;
(9)
by firms) ;
as K have
been
subsumed
(10) in the
functional
Douglas R. Shaller L,,, = L$Y’, w/r) (labor demanded
by firms) ;
(11)
PROPOSITION 1. In the CM model equilibrium GNP, Y*, is determined independently of the IS curve Equation (6). (Y* is in&pendent of Aggregate demand.) Proof. Under the usual regularity conditions, (7) and (8) can be collapsed into quasi-reduced
form equations:
w/r = $(YS) .
(12)
Therefore substituting (12) into (10) and (ll), and then substituting (10) and (11) into (9) yields equilibrium Y*. Q.E.D. Thus, by PROPOSITION 1 the CM model as specified determines GNP without the IS and LM equations. The following
corollaries
indicate that the basic conclusion of when the CM model is modified to remove either of the two logically inconsistent capital market assumptions:
PROPOSITION1 remains unchanged
COROLLARY 1. When Z(r) = 0, Y* is independent of aggregate demand. Proof. The proof follows from the fact that Z(r) affects only (6). Q.E.D. This model is clearly a version of Tobin’s (1955) dynamic aggregative model [see, e.g., Sargent (1979), 74-831.
COROLLARY 2. When there does not exist a market for used capital, then Y* is independent of aggregate demand. Proof. When (8) is no longer binding then from a choice-theoretic point of view, we have
Km= K;
(10”
L, = Ld(Y”, w) ;
(11’)
to replace (10) and (11). Thus, this model is a variant of the textbook short-period classical model. To see this, substitute (13) into (ll’), then (10’) and (11) into (9) to yield Y* which is independent of aggregate demand. (Clearly, all other endogenous variables solve recursively.) Q. E. D. 100
On the Missing
Equation:
Comment
3. Conclusion Canto and Miles (p. 247) d escribe with the following statement:
the purpose
of their
paper
In this paper, a third alternative method for closing the ISLM system of equations is developed by systematically incorporating the economy’s work-leisure decision. By including this additional decision constraint, the interactions between product and factor markets can be explicitly considered. The result is a more general set of results regarding the effects of economic policy on the determination of output, employment, and the price level of which the two traditional alternatives are only special cases. The CM model, however, can not possibly contain any model resembling the neo-Keynesian IS-LM macromodel as a special case. PROPOS~ON 1 of this paper proves that the CM model determines GNP independently of the IS and LM equations. Therefore, in contrast to the neo-Keynesian version of IS-LM model, the CM model implies that exogenous changes in propensities to save or invest have no effect on aggregate output determination. Furthermore, the macromodel presented by Canto and Miles explicitly postulates the existence of a market for used capital goods. For the purpose of solving the model, they assume that the capital market is in a state of equilibrium defined by the equality of desired and actual stocks of capital [their Equation (4)]. Simultaneously, however, the model builders assume that the capital market is out of equilibrium in that they specify a flow demand for investment [their Equation (l’)]. The assumptions are contradictory except in the special case not considered by CM when desired investment I(r) equals zero. Therefore, not one of the conclusions derived from the model by Canto and Miles follows from the assumptions of the model. A version of the model along the lines of COROLLARY 1 or COROLLARY 2 is necessary for the comparative-statics experiments to have meaning. Receiued: March 1982 Final version receiued:
April
1983
References Branson, W. H. Macroeconomic Harper and Row, 1979.
Theory and Policy. 2d ed. New York:
101
Douglas R. Shaller Canto, V.A., and M.A. Miles. “The Missing Equation: The Wedge Model Alternative.” Journal of Macroeconomics 3 (Spring 1981): 247-69. Gould, J.P. “Adjustment Costs in the Theory of Investment of the Firm.” Review of Economic Studies 35 (January 1968): 47-55. Sargent, T.J. Macroeconomic Theory. New York: Academic Press, 1979. Tobin, J. “A Dynamic Aggregative Model.” The journal of Political Economy 63 (February 1955): 103-15.
102
On the Missing Equation:
Comment
Canto and Miles (CM) present and analyze graphically “a third, alternative method for closing the IS-LM system of equations”-the ‘wedge model’. According to CM, and “the two traditional alternatives “the result is a more general set of results” are only special cases.” The CM model, however, is logically inconsistent. Because CM implicitly assume simultaneous capital market equilibrium and disequilibrium, CM’s assumptions are mutually inconsistent. Thus, none of the reported results logically follow, Furthermore, the CM model is not very general. This paper proves analytically that GNP is determined without the IS and LM equations both in the CM model and in the obvious logical patchups of the CM model. Therefore, for example, in the CM model exogenous changes in business-fixed investment or consumption have no predicted effect on national income.
In a recent paper in this journal, Canto and Miles (CM) develop a macromodel that, according to the authors, is “sufficiently general to incorporate as special cases” both the neoclassical and neo-Keynesian versions of the IS-LM model. They describe their model as a new solution to “the missing equation problem.” Unfortunately, there are some serious logical problems in the construction of the model and with its presentation. The macromodel presented by Canto and Miles is not logically consistent because the assumptions of the model are contradictory. Since the capital market is assumed to be simultaneously in equilibrium and out of equilibrium, Canto and Miles have not proved any of the propositions they discuss. Comparative-static conclusions derived from the model, if accepted, must be accepted on faith. The authors state on page 251 that “c and Lt represent the market sector derived demand for capital and labor services” and in footnote 3 that “at any point in time the supply of capital is a given quantity 8.” Capital market equilibrium is defined by c(Y’, w/r) = Z? [their Equation (4)]. The capital market equilibrium condition states that desired capital stock equals actual capital stock. What is called the IS curve [Equation (l’)] is specified on page 250 as Cd(Y,,w) + Z(r) = Y5, where Z(r) represents the flow demand for investment. But, in a stock-flow macromodel, the flow investment function comes into play only when the stock of capital does ]ournal Copyright
of
Macroeconomics, 6
1985
by Wayne
Winter State
1964, Vol. 6, No. University Press.
1, pp.
97-102
97
Douglas R. Shaller not equal the aggregate desired stock of capital.’ In the CM model however, the flow investment function is specified along with th specification of equality of the desired stock of capital goods wit the actual stock of capital goods. Thus, the market for capital good is specified to be both out of equilibrium and in a state of equilib rium at the same point in time, a logical contradiction. Furthermore, without regard to the logical consistency issue the CM macromodel can not possibly contain a neo-Keynesian IS LM model as a special case. A major characteristic of any distinc tively neo-Keynesian IS-LM model is that an exogenous shift ii any component of aggregate demand shifts GNP [see Branson (1979) for example, for a standard textbook IS-LM model]. However, ir the CM model, GNP is determined independently of the IS ant LM equations. (For a proof of this proposition, see PROPOSITION 1 of this paper.) Neo-Keynesian macromodels of all makes are famous for being demand driven. For example, the well-known IS-LM illustration oi the onset of the Great Depression commences with an exogenous decrease in business-fixed investment causing the IS curve to fal backward. The standard prediction of the IS-LM model is a decrease in equilibrium GNP and a concomitant decline in the interest rate. In the CM model, however, an exogenous decrease in business-fixed investment has no effect on equilibrium GNP. Of course, there are some special cases of the CM model where the problem of contradictory assumptions can be avoided. For example, if the flow investment demand function equals zero, then the model is a special case of Tobin’s (1955) dynamic aggregative model-in this case, a classical model with a market for used capital (see COROLLARY 1 of this paper). Alternatively, if the capital market equilibrium condition is discarded (see COROLLARY 2) then the model is a variant of the textbook short-period classical model.
1. The Model Following CM, equilibrium is specified solution to the following equations: Cd(Yf, w) + Z(r) - Y” = 0 Li(Y’, w/r) ‘See assumes
98
Gould costs
as the simultaneous
(IS curve) ;
- T - Ld(Yf, W) = 0
(labor market equilibrium)
(1968) for a standard derivation of adjustment for a firm seeking
of a flow to expand
;
(2)
investment function that its productive capacity.
On the Missing J&Y”, w/r) where,
- K = 0
Equation:
(capital market equilibrium)
Comment
;
(3)
by definition,
Y, = Y” + wLd(Y,, Y” = F(&,
L,,,)
The notation
= = = = =
Li(Y”, w/r) = Ld(Yf, W) = w/d
?’ = = 2? =
(%.rll income”)
;
(aggregate production
is defined
w r Cd(Yfi w) I(r) r
KP”,
w)
(4)
function)
.
@>
as follows:
real wage rate; rate of interest; consumption function; investment function; output of market goods market-derived national income; labor demand by firms; demand for nonmarket time (leisure) by workers; total supply of workers’ time; capital demand by firms; and stock of capital.
2. A Formal Analysis
of the CM Model
Canto and Miles points. Since I subject it more convenient to tion. Substitute (4) into compactly as’
use only graphical illustrations to make their their model to a more formal analysis, I find write the model using more compact nota(1) and (2) so that the system can be written
Eg
(Y”,
W/T)
=
EL(Ys, w, w/r)
0 (IS curve) ;
(6)
= 0 (labor market equilibrium)
Ek(Y”, w/r)
= 0 (capital market equilibrium)
Y” = F(&,
I+,,) (aggregate production
K,,, = e(Y’,
w/r)
(capital demanded
*The truly exogenous variables forms of (61, (7), and (6).
such
;
(7)
;
function)
03) ;
(9)
by firms) ;
as K have
been
subsumed
(10) in the
functional
Douglas R. Shaller L,,, = L$Y’, w/r) (labor demanded
by firms) ;
(11)
PROPOSITION 1. In the CM model equilibrium GNP, Y*, is determined independently of the IS curve Equation (6). (Y* is in&pendent of Aggregate demand.) Proof. Under the usual regularity conditions, (7) and (8) can be collapsed into quasi-reduced
form equations:
w/r = $(YS) .
(12)
Therefore substituting (12) into (10) and (ll), and then substituting (10) and (11) into (9) yields equilibrium Y*. Q.E.D. Thus, by PROPOSITION 1 the CM model as specified determines GNP without the IS and LM equations. The following
corollaries
indicate that the basic conclusion of when the CM model is modified to remove either of the two logically inconsistent capital market assumptions:
PROPOSITION1 remains unchanged
COROLLARY 1. When Z(r) = 0, Y* is independent of aggregate demand. Proof. The proof follows from the fact that Z(r) affects only (6). Q.E.D. This model is clearly a version of Tobin’s (1955) dynamic aggregative model [see, e.g., Sargent (1979), 74-831.
COROLLARY 2. When there does not exist a market for used capital, then Y* is independent of aggregate demand. Proof. When (8) is no longer binding then from a choice-theoretic point of view, we have
Km= K;
(10”
L, = Ld(Y”, w) ;
(11’)
to replace (10) and (11). Thus, this model is a variant of the textbook short-period classical model. To see this, substitute (13) into (ll’), then (10’) and (11) into (9) to yield Y* which is independent of aggregate demand. (Clearly, all other endogenous variables solve recursively.) Q. E. D. 100
On the Missing
Equation:
Comment
3. Conclusion Canto and Miles (p. 247) d escribe with the following statement:
the purpose
of their
paper
In this paper, a third alternative method for closing the ISLM system of equations is developed by systematically incorporating the economy’s work-leisure decision. By including this additional decision constraint, the interactions between product and factor markets can be explicitly considered. The result is a more general set of results regarding the effects of economic policy on the determination of output, employment, and the price level of which the two traditional alternatives are only special cases. The CM model, however, can not possibly contain any model resembling the neo-Keynesian IS-LM macromodel as a special case. PROPOS~ON 1 of this paper proves that the CM model determines GNP independently of the IS and LM equations. Therefore, in contrast to the neo-Keynesian version of IS-LM model, the CM model implies that exogenous changes in propensities to save or invest have no effect on aggregate output determination. Furthermore, the macromodel presented by Canto and Miles explicitly postulates the existence of a market for used capital goods. For the purpose of solving the model, they assume that the capital market is in a state of equilibrium defined by the equality of desired and actual stocks of capital [their Equation (4)]. Simultaneously, however, the model builders assume that the capital market is out of equilibrium in that they specify a flow demand for investment [their Equation (l’)]. The assumptions are contradictory except in the special case not considered by CM when desired investment I(r) equals zero. Therefore, not one of the conclusions derived from the model by Canto and Miles follows from the assumptions of the model. A version of the model along the lines of COROLLARY 1 or COROLLARY 2 is necessary for the comparative-statics experiments to have meaning. Receiued: March 1982 Final version receiued:
April
1983
References Branson, W. H. Macroeconomic Harper and Row, 1979.
Theory and Policy. 2d ed. New York:
101
Douglas R. Shaller Canto, V.A., and M.A. Miles. “The Missing Equation: The Wedge Model Alternative.” Journal of Macroeconomics 3 (Spring 1981): 247-69. Gould, J.P. “Adjustment Costs in the Theory of Investment of the Firm.” Review of Economic Studies 35 (January 1968): 47-55. Sargent, T.J. Macroeconomic Theory. New York: Academic Press, 1979. Tobin, J. “A Dynamic Aggregative Model.” The journal of Political Economy 63 (February 1955): 103-15.
102