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Is higher confidence of fiat money necessarily desirable?
Economics Letters 95 (2007) 211 – 216 www.elsevier.com/locate/econbase
Is higher confidence of fiat money necessarily desirable? Jumpei Tanaka ⁎ Department of Economics, The University of Kitakyusyu, 4-2-1 Kitagata, Kokura-minami, Kitakyusyu, 802-8577, Japan Received 15 November 2005; received in revised form 3 October 2006; accepted 11 October 2006 Available online 2 March 2007
Abstract We investigate whether higher confidence of fiat money is desirable from the viewpoint of dynamic resource allocation. Using a simple overlapping generations model, we demonstrate that higher confidence of fiat money depresses economic growth and harms the welfare of future generations born sufficiently later. © 2006 Elsevier B.V. All rights reserved. Keywords: Fiat money; Overlapping generations JEL classification: E52; O41; O42
1. Introduction It is needless to say that high confidence of fiat money is indispensable for efficient exchange in the market economy. Does this mean that higher confidence of it is desirable for all the agents of the economy? The purpose of this paper is to examine this problem and clarify the social cost of fiat money with high confidence. Weil (1987) is the first to investigate the role of the confidence of fiat money under the framework of the neoclassical growth model with overlapping generations.1 He defined the degree of confidence as the exogenous probability with which fiat money is still valued in the next period, and demonstrated that fiat money is positively valued in equilibrium only when the probability is large enough. In his paper, ⁎ Tel.: +81 93 964 4135. E-mail address: [email protected]. 1 The famous previous works of Samuelson (1958), Wallace (1980), and Tirole (1985) and so on did not consider the concept of the degree of confidence, while the concept itself had been introduced by Blanchard (1979). 0165-1765/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.10.005
212
J. Tanaka / Economics Letters 95 (2007) 211–216
however, it is not discussed what effect a change in the degree of confidence has on the economy. In this paper we investigate the effects of such a change using a simpler framework presented by Grossman and Yanagawa (1993) 2 , and demonstrate in particular that higher confidence of fiat money depresses economic growth and harms the welfare of future generations born sufficiently later. This implies that higher confidence of fiat money is not necessarily desirable from the viewpoint of dynamic resource allocation.
2. The model We consider a one-sector production economy with overlapping generations in which agents live for two periods. The population of each generation is assumed to be L and constant, so the population growth rate is zero. The formulation of a representative household is basically the same as that of Weil (1987), so we mention it only briefly. In this economy there are two methods of savings: physical capital and fiat money. We suppose that fiat money is a risky asset in that it is valued in the next period with exogenous probability q and is not valued with probability 1 − q, and also that if fiat money is not valued in some period it will never be valued in the following future periods. So the probability q represents the degree of confidence of fiat money, and higher q means higher confidence. At period t a representative household of generation t solves the following problem: Max Ut ¼ logc yt þ b½qlogcotþ1 ð1Þ þ ð1−qÞlogcotþ1 ð2Þ s:t: c yt þ st þ pt mt ≤wt ;
cotþ1 ð1Þ≤ð1 þ rtþ1 Þst þ ptþ1 mt ;
cotþ1 ð2Þ≤ð1 þ rtþ1 Þst
o o where β, cty, ct+1 (1), ct+1 (2), st, wt, rt+1, pt, mt denote, respectively, subjective discount factor, young period consumption, old period consumption when money is still valued, old period consumption when money is not valued, quantity of investment in physical capital, labor income, return rate of physical capital, price of one unit of money in terms of goods, and quantity of money purchased. After some calculations we can derive the optimal plans for consumptions and savings as follows.
c yt ¼
1 wt 1þb
ð1Þ
cotþ1 ð1Þ ¼
q b ð1 þ rtþ1 Þ wt /t 1þb
cotþ1 ð2Þ ¼
1−q b wt ð1 þ rtþ1 Þ 1−/t 1þb
where /t u
pt ð1 þ rtþ1 Þ ptþ1
ð2Þ
ð3Þ
2 The paper of Grossman and Yanagawa (1993) is an extension of the work of Tirole (1985) to the framework of endogenous growth theory. The reason why we use their framework is that their framework is simple enough to investigate fully what effect a change in the degree of confidence has on the welfare of all the current and future generations in the steady state. A similar investigation under the framework of the neoclassical growth model will be an important question for future research.
J. Tanaka / Economics Letters 95 (2007) 211–216
st ¼
1−q b wt 1−/t 1 þ b
pt mt ¼
213
ð4Þ
q−/t b wt 1−/t 1 þ b
ð5Þ
b wt 1þb
ð6Þ
st þ pt mt ¼
From (4), (5) and (6) we can see that under logarithmic utility a change in q does not affect the level of the aggregate savings st + pt mt, and affects only the holding ratio between two assets. On the formulation of firms, we follow Grossman and Yanagawa (1993). There are many identical firms which behave competitively and the production function of a representative firm j is given by y jt ¼ F½k jt ; AðKt Þl jt
ð7Þ
where y tj , k tj Kt and l tj represent output of firm j, physical capital employed by firm j, aggregate capital stock, and labor force employed by firm j, respectively. We assume that the production function exhibits constant returns to scale, the functional form of labor productivity is A(Kt) = aKt, and the total labor force employed at each period is L. Under these assumptions perfect competition yields rt ¼ r ¼ F1 ½1; aL;
wt ¼ aF2 ½1; aL Kt :
ð8Þ
The market equilibrium conditions of physical capital and fiat money are given respectively by Ktþ1 ¼ st L
ð9Þ
mt L ¼ 1
ð10Þ
where we assume the total nominal supply of fiat money is 1 and fixed over time. Using these conditions we can derive the equilibrium dynamic as btþ1 ¼
ð1 þ rÞbt qxL−bt
ð11Þ
1 þ jt ¼ xL−bt
ð12Þ
qxL−bt xL−bt
ð13Þ
/t u
where we define x, 1 + κt and bt as follows. xu
b aF2 ; 1þb
1 þ jt u
Ktþ1 ; Kt
Bt upt mt L; and bt u
Bt Kt
ð14Þ
214
J. Tanaka / Economics Letters 95 (2007) 211–216
Fig. 1. The phase diagram of bt.
We can easily confirm that our model reduces to the model of Grossman and Yanagawa (1993) when q = 1 holds, so this model includes their model as a special case. Fig. 1 depicts the phase diagram of Eq. (11). From this figure, we can see that the condition that fiat money has positive value in equilibrium is Abtþ1 1þr 1þr ¼ b1 or equivalently; qN ð15Þ qxL xL Abt b ¼0
j t
The condition (15) means that fiat money cannot be valued if the degree of its confidence is too low, which has already been pointed out by Weil (1987). The difference between our result and Weil's one is that in our model fiat money can be valued under the condition that the population growth rate is smaller than the interest rate, but in Weil's model it cannot. This difference stems from the fact that the equilibrium growth rate in an endogenous growth model like ours does not depend on the population growth rate. Finally, the steady state values of 1 + κt, bt, and ϕt are b⁎ ¼ qxL−ð1 þ rÞ /⁎ ¼
1þr ð1−qÞxL þ ð1 þ rÞ
1 þ j⁎ ¼ ð1−qÞxL þ ð1 þ rÞ
ð16Þ ð17Þ ð18Þ
3. The effects of a change in the degree of confidence In this section we investigate the effects of a permanent unexpected change in the degree of confidence (namely, q).3 3
We suppose by such a change the economy instantaneously jumps from the old steady state to the new steady state. This is possible because b is a jump variable.
J. Tanaka / Economics Letters 95 (2007) 211–216
215
First, let us examine the effect of a change in q on the real value of money, the return rate of holding money, and the growth rate of the economy. Differentiating (16), (17), and (18) with respect to q, we have A/⁎ N0; Aq
Ab⁎ N0; Aq
Að1 þ j⁎ Þ b0 Aq
ð19Þ
These results mean that an increase in q has positive effects on the real value of money, and it also has a negative impact on its return rate and economic growth. The reason why these results hold is as follows. As we can easily verify by differentiating (4) and (5) evaluated at the steady state with respect to q, an increase in q raises the demand for fiat money and discourages the demand for physical capital. Clearly the former effect pulls up the current price of fiat money and lowers its return rate, and the latter effect discourages capital accumulation and economic growth. From these results we can conclude that higher confidence of fiat money is harmful to growth. Second, let us investigate the intergenerational welfare effect of a change in q. Suppose that an unexpected and permanent rise in q occurs at the beginning of period t. In such a case the welfare of an individual of generation t − 1 who sells fiat money to generation t is undoubtedly improved, because such a change pulls up its market value. The welfare effect on an individual of generation t measured by the indirect utility function can be calculated as AUt⁎ Aq
qxL ð1 þ rÞ−qxL þ b log 1 þ r ð1−qÞxL þ ð1 þ rÞ
¼
ð20Þ
Since from Eq. (15) the first term in the parenthesis is positive and the second term is negative, the sign of Eq. (20) is generally indeterminate. This is because an increase in q has the following opposite effects. First, such a change reduces the uncertainty of the return of holding money and this has a positive effect on welfare. Second, as we pointed out in Eq. (19) it also reduces the return rate of holding money and this has a negative effect on welfare. Accordingly the sign of Eq. (20) depends on which effect is stronger. Finally, we can also calculate the welfare effect on an individual of generation t + j ( j ≥ 1) by the same step, and the result is eq qxL ð1 þ rÞ−qxL þ þ ð1 þ bÞ j ¼ b log 1 þ r ð1−qÞxL þ ð1 þ rÞ Aq q
⁎ AUtþj
where
eq u
ð21Þ
q dð1 þ j⁎ Þ b0 dq 1 þ j⁎
Since εq in the second term in the RHS of Eq. (21) is negative, the sign of Eq. (21) is negative for sufficiently large j. This means that an increase in q harms the welfare of future generations born sufficiently later. The reason of this result is as follows. In our model the wage level is in proportion to the level of capital stock at that period, as represented in Eq. (8). Since an increase in q depresses the accumulation of physical capital and declines the wage future generations will earn, such a change can harm the welfare of future generations. Thus we can see that an increase in the degree of confidence cannot be Pareto-improving.
216
J. Tanaka / Economics Letters 95 (2007) 211–216
In summary higher confidence of fiat money lowers the investment in physical capital, which has negative impacts on long-run growth and future generations' welfare. Note that such negative impacts will not arise in the standard neoclassical growth model in which the long-run growth rate is exogenous. In other words the above conclusion is characteristic to the case where the engine of economic growth is the productivity growth through an Arrow–Romer learning-by-doing externality. In such a case the underaccumulation of capital arises in a competitive equilibrium because of the discrepancy between private and social return rate of capital and accordingly any change that speeds up capital accumulation can improve the welfare, as Grossman and Yanagawa (1993, p7) have argued. Therefore, an increase in the confidence of fiat money, which does not stimulate but depress capital accumulation, cannot improve the inefficient resource allocation and accordingly cannot enjoy the welfare improvement. Acknowledgement The author would like to thank an anonymous referee for useful comments. Needless to say, all the remaining errors are mine.
References Blanchard, O., 1979. Speculative bubbles, crashes and rational expectations. Economics Letters 3, 387–389. Grossman, G., Yanagawa, N., 1993. Asset bubbles and endogenous growth. Journal of Monetary Economics 31, 3–19. Samuelson, P., 1958. An exact consumption loan model of interest with or without the social contrivance of money. Journal of Political Economy 66, 467–482. Tirole, J., 1985. Asset bubbles and overlapping generations. Econometrica 53, 1499–1528. Wallace, N., 1980. The overlapping generations model of fiat money. In: Kareken, J.H., Wallace, N. (Eds.), Models of Monetary Economies. Federal Reserve Bank of Minneapolis, Minneapolis, pp. 49–82. Weil, P., 1987. Confidence and the real value of money in an overlapping generations economy. Quarterly Journal of Economics 107, 29–42.
Is higher confidence of fiat money necessarily desirable? Jumpei Tanaka ⁎ Department of Economics, The University of Kitakyusyu, 4-2-1 Kitagata, Kokura-minami, Kitakyusyu, 802-8577, Japan Received 15 November 2005; received in revised form 3 October 2006; accepted 11 October 2006 Available online 2 March 2007
Abstract We investigate whether higher confidence of fiat money is desirable from the viewpoint of dynamic resource allocation. Using a simple overlapping generations model, we demonstrate that higher confidence of fiat money depresses economic growth and harms the welfare of future generations born sufficiently later. © 2006 Elsevier B.V. All rights reserved. Keywords: Fiat money; Overlapping generations JEL classification: E52; O41; O42
1. Introduction It is needless to say that high confidence of fiat money is indispensable for efficient exchange in the market economy. Does this mean that higher confidence of it is desirable for all the agents of the economy? The purpose of this paper is to examine this problem and clarify the social cost of fiat money with high confidence. Weil (1987) is the first to investigate the role of the confidence of fiat money under the framework of the neoclassical growth model with overlapping generations.1 He defined the degree of confidence as the exogenous probability with which fiat money is still valued in the next period, and demonstrated that fiat money is positively valued in equilibrium only when the probability is large enough. In his paper, ⁎ Tel.: +81 93 964 4135. E-mail address: [email protected]. 1 The famous previous works of Samuelson (1958), Wallace (1980), and Tirole (1985) and so on did not consider the concept of the degree of confidence, while the concept itself had been introduced by Blanchard (1979). 0165-1765/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.10.005
212
J. Tanaka / Economics Letters 95 (2007) 211–216
however, it is not discussed what effect a change in the degree of confidence has on the economy. In this paper we investigate the effects of such a change using a simpler framework presented by Grossman and Yanagawa (1993) 2 , and demonstrate in particular that higher confidence of fiat money depresses economic growth and harms the welfare of future generations born sufficiently later. This implies that higher confidence of fiat money is not necessarily desirable from the viewpoint of dynamic resource allocation.
2. The model We consider a one-sector production economy with overlapping generations in which agents live for two periods. The population of each generation is assumed to be L and constant, so the population growth rate is zero. The formulation of a representative household is basically the same as that of Weil (1987), so we mention it only briefly. In this economy there are two methods of savings: physical capital and fiat money. We suppose that fiat money is a risky asset in that it is valued in the next period with exogenous probability q and is not valued with probability 1 − q, and also that if fiat money is not valued in some period it will never be valued in the following future periods. So the probability q represents the degree of confidence of fiat money, and higher q means higher confidence. At period t a representative household of generation t solves the following problem: Max Ut ¼ logc yt þ b½qlogcotþ1 ð1Þ þ ð1−qÞlogcotþ1 ð2Þ s:t: c yt þ st þ pt mt ≤wt ;
cotþ1 ð1Þ≤ð1 þ rtþ1 Þst þ ptþ1 mt ;
cotþ1 ð2Þ≤ð1 þ rtþ1 Þst
o o where β, cty, ct+1 (1), ct+1 (2), st, wt, rt+1, pt, mt denote, respectively, subjective discount factor, young period consumption, old period consumption when money is still valued, old period consumption when money is not valued, quantity of investment in physical capital, labor income, return rate of physical capital, price of one unit of money in terms of goods, and quantity of money purchased. After some calculations we can derive the optimal plans for consumptions and savings as follows.
c yt ¼
1 wt 1þb
ð1Þ
cotþ1 ð1Þ ¼
q b ð1 þ rtþ1 Þ wt /t 1þb
cotþ1 ð2Þ ¼
1−q b wt ð1 þ rtþ1 Þ 1−/t 1þb
where /t u
pt ð1 þ rtþ1 Þ ptþ1
ð2Þ
ð3Þ
2 The paper of Grossman and Yanagawa (1993) is an extension of the work of Tirole (1985) to the framework of endogenous growth theory. The reason why we use their framework is that their framework is simple enough to investigate fully what effect a change in the degree of confidence has on the welfare of all the current and future generations in the steady state. A similar investigation under the framework of the neoclassical growth model will be an important question for future research.
J. Tanaka / Economics Letters 95 (2007) 211–216
st ¼
1−q b wt 1−/t 1 þ b
pt mt ¼
213
ð4Þ
q−/t b wt 1−/t 1 þ b
ð5Þ
b wt 1þb
ð6Þ
st þ pt mt ¼
From (4), (5) and (6) we can see that under logarithmic utility a change in q does not affect the level of the aggregate savings st + pt mt, and affects only the holding ratio between two assets. On the formulation of firms, we follow Grossman and Yanagawa (1993). There are many identical firms which behave competitively and the production function of a representative firm j is given by y jt ¼ F½k jt ; AðKt Þl jt
ð7Þ
where y tj , k tj Kt and l tj represent output of firm j, physical capital employed by firm j, aggregate capital stock, and labor force employed by firm j, respectively. We assume that the production function exhibits constant returns to scale, the functional form of labor productivity is A(Kt) = aKt, and the total labor force employed at each period is L. Under these assumptions perfect competition yields rt ¼ r ¼ F1 ½1; aL;
wt ¼ aF2 ½1; aL Kt :
ð8Þ
The market equilibrium conditions of physical capital and fiat money are given respectively by Ktþ1 ¼ st L
ð9Þ
mt L ¼ 1
ð10Þ
where we assume the total nominal supply of fiat money is 1 and fixed over time. Using these conditions we can derive the equilibrium dynamic as btþ1 ¼
ð1 þ rÞbt qxL−bt
ð11Þ
1 þ jt ¼ xL−bt
ð12Þ
qxL−bt xL−bt
ð13Þ
/t u
where we define x, 1 + κt and bt as follows. xu
b aF2 ; 1þb
1 þ jt u
Ktþ1 ; Kt
Bt upt mt L; and bt u
Bt Kt
ð14Þ
214
J. Tanaka / Economics Letters 95 (2007) 211–216
Fig. 1. The phase diagram of bt.
We can easily confirm that our model reduces to the model of Grossman and Yanagawa (1993) when q = 1 holds, so this model includes their model as a special case. Fig. 1 depicts the phase diagram of Eq. (11). From this figure, we can see that the condition that fiat money has positive value in equilibrium is Abtþ1 1þr 1þr ¼ b1 or equivalently; qN ð15Þ qxL xL Abt b ¼0
j t
The condition (15) means that fiat money cannot be valued if the degree of its confidence is too low, which has already been pointed out by Weil (1987). The difference between our result and Weil's one is that in our model fiat money can be valued under the condition that the population growth rate is smaller than the interest rate, but in Weil's model it cannot. This difference stems from the fact that the equilibrium growth rate in an endogenous growth model like ours does not depend on the population growth rate. Finally, the steady state values of 1 + κt, bt, and ϕt are b⁎ ¼ qxL−ð1 þ rÞ /⁎ ¼
1þr ð1−qÞxL þ ð1 þ rÞ
1 þ j⁎ ¼ ð1−qÞxL þ ð1 þ rÞ
ð16Þ ð17Þ ð18Þ
3. The effects of a change in the degree of confidence In this section we investigate the effects of a permanent unexpected change in the degree of confidence (namely, q).3 3
We suppose by such a change the economy instantaneously jumps from the old steady state to the new steady state. This is possible because b is a jump variable.
J. Tanaka / Economics Letters 95 (2007) 211–216
215
First, let us examine the effect of a change in q on the real value of money, the return rate of holding money, and the growth rate of the economy. Differentiating (16), (17), and (18) with respect to q, we have A/⁎ N0; Aq
Ab⁎ N0; Aq
Að1 þ j⁎ Þ b0 Aq
ð19Þ
These results mean that an increase in q has positive effects on the real value of money, and it also has a negative impact on its return rate and economic growth. The reason why these results hold is as follows. As we can easily verify by differentiating (4) and (5) evaluated at the steady state with respect to q, an increase in q raises the demand for fiat money and discourages the demand for physical capital. Clearly the former effect pulls up the current price of fiat money and lowers its return rate, and the latter effect discourages capital accumulation and economic growth. From these results we can conclude that higher confidence of fiat money is harmful to growth. Second, let us investigate the intergenerational welfare effect of a change in q. Suppose that an unexpected and permanent rise in q occurs at the beginning of period t. In such a case the welfare of an individual of generation t − 1 who sells fiat money to generation t is undoubtedly improved, because such a change pulls up its market value. The welfare effect on an individual of generation t measured by the indirect utility function can be calculated as AUt⁎ Aq
qxL ð1 þ rÞ−qxL þ b log 1 þ r ð1−qÞxL þ ð1 þ rÞ
¼
ð20Þ
Since from Eq. (15) the first term in the parenthesis is positive and the second term is negative, the sign of Eq. (20) is generally indeterminate. This is because an increase in q has the following opposite effects. First, such a change reduces the uncertainty of the return of holding money and this has a positive effect on welfare. Second, as we pointed out in Eq. (19) it also reduces the return rate of holding money and this has a negative effect on welfare. Accordingly the sign of Eq. (20) depends on which effect is stronger. Finally, we can also calculate the welfare effect on an individual of generation t + j ( j ≥ 1) by the same step, and the result is eq qxL ð1 þ rÞ−qxL þ þ ð1 þ bÞ j ¼ b log 1 þ r ð1−qÞxL þ ð1 þ rÞ Aq q
⁎ AUtþj
where
eq u
ð21Þ
q dð1 þ j⁎ Þ b0 dq 1 þ j⁎
Since εq in the second term in the RHS of Eq. (21) is negative, the sign of Eq. (21) is negative for sufficiently large j. This means that an increase in q harms the welfare of future generations born sufficiently later. The reason of this result is as follows. In our model the wage level is in proportion to the level of capital stock at that period, as represented in Eq. (8). Since an increase in q depresses the accumulation of physical capital and declines the wage future generations will earn, such a change can harm the welfare of future generations. Thus we can see that an increase in the degree of confidence cannot be Pareto-improving.
216
J. Tanaka / Economics Letters 95 (2007) 211–216
In summary higher confidence of fiat money lowers the investment in physical capital, which has negative impacts on long-run growth and future generations' welfare. Note that such negative impacts will not arise in the standard neoclassical growth model in which the long-run growth rate is exogenous. In other words the above conclusion is characteristic to the case where the engine of economic growth is the productivity growth through an Arrow–Romer learning-by-doing externality. In such a case the underaccumulation of capital arises in a competitive equilibrium because of the discrepancy between private and social return rate of capital and accordingly any change that speeds up capital accumulation can improve the welfare, as Grossman and Yanagawa (1993, p7) have argued. Therefore, an increase in the confidence of fiat money, which does not stimulate but depress capital accumulation, cannot improve the inefficient resource allocation and accordingly cannot enjoy the welfare improvement. Acknowledgement The author would like to thank an anonymous referee for useful comments. Needless to say, all the remaining errors are mine.
References Blanchard, O., 1979. Speculative bubbles, crashes and rational expectations. Economics Letters 3, 387–389. Grossman, G., Yanagawa, N., 1993. Asset bubbles and endogenous growth. Journal of Monetary Economics 31, 3–19. Samuelson, P., 1958. An exact consumption loan model of interest with or without the social contrivance of money. Journal of Political Economy 66, 467–482. Tirole, J., 1985. Asset bubbles and overlapping generations. Econometrica 53, 1499–1528. Wallace, N., 1980. The overlapping generations model of fiat money. In: Kareken, J.H., Wallace, N. (Eds.), Models of Monetary Economies. Federal Reserve Bank of Minneapolis, Minneapolis, pp. 49–82. Weil, P., 1987. Confidence and the real value of money in an overlapping generations economy. Quarterly Journal of Economics 107, 29–42.