Case 2. Non-rationing
5.3 Comparing the Welfare
32 Indeed, an increase in the inflation rate or an increase in the tax rate leads to an increase in .
29
The analysis proceeded so far indicates that ∗ ∗ ∗ ∗ in the rationing equilibrium and ∗ ∗ ∗ ∗ in the non-rationing
equilibrium. Given these results, we now compare the welfare of type-1 and type-2 agents under tax financing and money financing.
At the initial period 0 , there are type-1 old agents and each of them is endowed with units of capital. These agents utilize the capital and hire young type-1 agents to produce output, which is taxed at a rate of (under tax financing).
As a result, the after-tax wage income for each young type-1 agent is 1
while the after-tax capital income for each (initial) old type-1 agent is 1
.
Each young type-1 agent lends 1 / units to a type-2 young agent and each type-2 agent producesunits of time 1 capital.
As a result, each old type-1 agent at 1 receives 1 / units of capital. The old type-1 agent rents this capital to firms and obtains the after-tax return given as 1.
Since the loan rate is equal to ∗, it is clear that the old type-2 agent at
1 willreceive 1 1 ∗ / .
Denote ∗
as well as
∗ as the equilibrium growth rate and rate of returns from money. Then, the above scenario implies that the welfare of type-1 agents for all generations under the balanced growth path can be expressed as331 1 1 is less than to ensure the boundedness of the utility. Obviously, for given
33 We follow the standard practice by ignoring the initial old type-1 agents' utility. Note that each agent cares only old-age consumption. As a result, the utility of the first generation (born at time 0) is discounted at the rate , because the government evaluates the social welfare from time 0. Note that the population of type-1 agent is equal to .
30
parameters tax financing yields a higher (lower) welfare level for type-1 agents than money financing if
1 ∗
∗
∗
∗. 26 Recall that, under the rationing equilibrium, tax financing yields a higher rate of economic growth and a lower rate of inflation compared with money financing.
However, this does not guarantee that tax financing yields a higher level of welfare to type-1 agents than money financing, as the tax rate (which is equal to ) appears in the welfare function under tax financing. Note that ∗ . Hence, if 1 ∗ ∗ (a sufficient condition) for a given in the rationing equilibrium, we can be sure that tax financing yields a higher level of welfare to type-1 agents than money financing. To compare 1 ∗ with ∗, we depict
After some manipulations, the above equation reduces to .
1 1 ∗
31
which always holds for 0. Hence, under the rationing equilibrium tax financing yields a higher level of welfare to type-1 agents than money financing. Note that this result confirms that 1 ∗ ∗.
In the case of the non-rationing equilibrium, recall that ∗ ∗ ∗
∗. From this, we cannot directly infer the relative merits of government financing from the perspective of type-1 agents’ welfare. Nevertheless, we can derive a
condition showing that money financing may be better than tax financing under the non-rationing equilibrium. To see this, eq. (26) leads to the following sufficient condition under which money financing yields a higher level of welfare for type-1 agents:34 Rewrite the above equation as
∗
1 ∗
∗
1 ∗,
which, after some manipulations, is equivalent to
1 ∗
∗
∗ ∗ ≡ ∗. 27 Since ∗ ∗ ∗ ∗, ∗ 1. Eq. (27) is quite intuitive. From eqs. (25) and (26), the importance of economic growth in affecting the welfare of type-1 agents depends positively on the discount rate. In other words, if the discount rate is not too small, economic growth is more important on the welfare compared with the rate of returns from money. Since ∗ ∗ ∗ ∗, it is then clear that if the discount rate is not too small, money financing yields a higher level of welfare for type-1 agents.
The welfare of type-2 agents for all generations is given by35
34This is a sufficient condition because ∗
∗
∗
∗.
35 Note that and (as well as ) in all periods are in terms of per type-l agent. As a result, the utility of all type-2 agents (total population is equal to 1 is also expressed in terms of per type-1 agent, implying that the total utility of all type-2 agents should be divided by .
32
1 ∞
1 1 ∗
1 ∗ ⋯ 28 where
is equal to (
under the rationing (non-rationing) equilibrium. From eq. (11), the above equation can be rewritten as1
∗ 1 1
∗
∗ 1 1
∗ ⋯, where ∗ has been substituted. Under the rationing equilibrium,
1 1 ∗ / and ∗ ∗. Then, the above equation Since ∗ ∗ ∗ ∗ under the rationing equilibrium, tax financing yields a higher level of welfare to type-2 agents if
∗
33
Recall also that 1 ∗ ∗ under the rationing equilibrium. Thus, a
sufficient condition that leads to a better tax financing than money financing (from the perspective of type-2 agents’ welfare) is given as
∗ above equation implies that
1
∗ ∗ ≡ ∗.
Note that ∗ 1. Thus, if the discount rate is not too small, tax financing under the rationing equilibrium yields a higher level of welfare for type-2 agents than money financing.
The above equation shows that the welfare of type-2 agents is increasing in economic growth and decreasing in the rate of returns from money. Recall that ∗
∗ ∗ ∗ under the non-rationing equilibrium. Thus, in the non-rationing equilibrium money financing always leads to a higher level of welfare for type-2 agents than tax financing. We summarize our analysis in the following proposition:
The social welfare function for the economy as a whole is the summation of the welfare functions of type-1 and type-2 agents. By assuming that ∗, ∗ , we have the following result:
Proposition 10. Under the rationing equilibrium, tax financing yields a higher level
of the social welfare than money financing. On the contrary, under the non-rationing
equilibrium money financing yield a higher level of the social welfare than tax
financing.
34
It is interesting to note that economists are not always agreed with the effect of money and tax financing. McKinnon (1991) asserts that money financing usually leads to higher inflation and lower economic growth. Palivos and Yip (1995),
however, find that money financing does yield a higher inflation rate, but it also leads to a higher growth rate than tax financing. Interestingly, our model contribute to this debate by showing that asymmetric information may play an important role in determining the relative merits of government financing.
In terms of social welfare, there is controversy in the recent literature. While money financing leads to both higher inflation and economic growth, Palivos and Yip (1995) find that money financing is better than tax financing in terms of social welfare if the fraction of liquidity constrained investment purchases is relatively large. On the other hand, Gokan (2002) suggests that taxation is more desirable than seigniorage for the government to finance its expenditure. Our model concludes that which case is true depends on whether or not credit is rationing.
It is also worth noting that our results on the social welfare may provide an explanation to Mankiw's (1987) hypothesis of optimal seigniorage, which asserts that tax and inflation rates should co-vary positively. As is shown, the presence of
asymmetric information in this model gives rise to credit rationing (non-rationing) when the inflation rate is relatively high (low), and tax financing is better (worse) than money financing if credit is rationing (non-rationing). This implies that the
government should utilize taxation (i.e., raise the tax rate) when the inflation rate is relatively high. Therefore, we can observe a positive correlation between inflation and tax rates.
6 Conclusion
This paper incorporates asymmetric information into a simple model of endogenous growth to assess the relative merits of money and tax financing. As is well known, the presence of asymmetric information gives rise to the possibility of credit rationing. It is then found that whether or not credit is rationing plays a significant role in
determining the relative merits of money and tax financing.
35
Results demonstrate that money financing leads to higher inflation for the cases of credit rationing and non-rationing; nevertheless, the growth rate is higher under money (tax) financing if credit is non-rationing (rationing). In terms of social welfare, money (tax) financing is superior to tax (money financing) when credit is
non-rationing (rationing). These results reconcile the pre-existing literature as some studies suggest the government financing its expenditure via seigniorage while others via taxation. Moreover, our model may provide theoretical explanations to the
nonlinear correlation between inflation and economic growth as well as a positive correlation between inflation and tax rates.
36
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Figures
Figure 1. Credit Rationing and Non-Rationing
Figure 2. Equilibrium under Money Financing
40
Figure 3. Equilibrium under Tax Financing
Figure 4. Comparison of Money and Tax Financing: Rationing
41