# Money as Insurance

## Abstract

**:**

## 1. Introduction

## 2. Money and Rational Precaution

_{1}, q

_{2})

_{1}is consumption in the first period, and q

_{2}is consumption in the second period. The standard assumptions u

_{1}, u

_{2}> 0; u

_{11}, u

_{22}< 0; u

_{12}, u

_{21}> 0 hold, where the subscripts denote first and second order derivatives according to the two arguments of the utility function. Moreover, the utility function is assumed homothetic, meaning that plain income effects do not change the consumer’s time preference in the short term. This is a reasonable assumption in the static model.

_{1}= 1 − m − s

_{2}= 1 + αm + (1 + r)s

_{2}= 1 + αm + (1 + r)(1 − m) − (1 + r)q

_{1}

_{1}, q

_{2}) s.t. q

_{2}= 1 + αm + (1 + r)(1 − m) − (1 + r)q

_{1}

_{1}. This produces:

_{1}= 1 − m − s, and the expected consumption in period 2 is q

_{2}= 1 + αm + (1 + r)s.

_{1}, q

_{2}). For comparison, a dashed life-time budget constraint going through point E is drawn to illustrate the hypothetical case, where there would be no uncertainty about the yield of illiquid deposits and, thus, no need for insurance. In that case, the optimization would include only one analytical step according to rule (5). The optimum would occur at point ε, where the highest possible indifference curve from the set of the homothetic utility function touches the dashed life-time budget constraint. The consumer would then reach to the higher utility level u* by depositing s

^{ɛ}and receiving the warranted (1 + r)s

^{ε}in return.

^{ε}, the rational insurance motive to save money increases the amount of total savings compared to the case of perfect foresight. Therefore, secure cash saving replaces insecure deposit saving more than one-to-one. It can also be inferred from Figure 1 that the higher is the expected market interest rate r the larger is the share of deposit savings in total savings σ/s, and vice versa. The results are clear under the assumption of homothetic preferences saying that the consumer’s time preference is not affected by pure income effects.

_{1}from (3) in both cases. The point of intersection of the solid constraint with m > 0 along the horizontal axis is:

_{1}= 1 − m + (1 + αm)/(1 + r)

_{1}= (2 + r)/(1 + r)

## 3. Money, Education and Saving

_{2}= w(e) with w′ > 0, w″ < 0, where the sub-primes denote first and second order derivatives. With these amendments, the representative consumer’s periodic budget constraints read:

_{1}= 1 − e − m − s

_{2}= w

_{2}+ αm + (1 + r)s

_{2}= w(e), the optimum condition is:

_{1}produces:

_{1}and the expected labor income in period 2 is w

_{2}= ew′.

_{1}= (1 − e) − (m + s) and the expected consumption in period 2 is q

_{2}= αm + ew′ + (1 + r)s.

^{϶}= e and e

^{϶}w′ = ew′ = w

_{2}in Figure 2.

^{ε}. The effect remains qualitatively the same as in Figure 1 due to the neutrality of cash saving on education. However, Figure 2 also reveals that a fall in the expected market interest rate would make the investment in education e grow at the expense of deposit saving s. From the macroeconomic point of view, the partial replacement of s by e would mean replacement of physical capital accumulation by human capital accumulation. In the case of a rise in the expected interest rate, the effects would be reversed.1

_{1}= (1 − e − m) + (w

_{2}+ αm)/(1 + r)

_{1}= (1 − e) + w

_{2}/(1 + r)

## 4. Money Illusion

^{i}according to the point c

^{i}instead of choosing the rationally motivated m according to point c. This means that the difference m

^{i}− m > 0 is due to irrationality.

^{i}has made the yield curve of education start from point c

^{i}instead of point c. By (11), the choice of education is not distorted by money illusion, because it is based on the expected real interest rate that is externally given to the consumer. Thus, the consumer chooses education according to the point a

^{i}instead of choosing it according to point a. Nevertheless, e

^{i}= e and e

^{i}w′ = ew′ because the model omits the opportunity cost of leisure.

^{i}according to b

^{i}instead of choosing s according to b, which means that s

^{i}< s. While excess cash saving m

^{i}cuts available working time, thus placing point c

^{i}north-west from point c, homothetic preferences keep points b

^{i}and b on the dashed line starting from the origin.

^{i}against uncertainty. Moreover, albeit total savings are smaller under money illusion than under perfect foresight, that is, m

^{i}+ s

^{i}= σ

^{i}< σ = m + s, the share of cash of total savings is larger under money illusion, m

^{i}/σ

^{i}> m/σ. The qualitative effects of changes in the expected market interest rate remain the same as discussed in the previous sections.

_{1}= 1 − m − e + (w

_{2}+ αm)/(1 + r)

_{2}= w

_{2}+ m

^{i}+ (1 + r)(1 − m

^{i}− e − q

_{1}) where m

^{i}denotes the cash saved in period 1 and anticipated to be recollected as such in period 2. Calculation and manipulation produce the horizontal intersection point of the solid constraint:

_{1}= (1 − e) + (w

_{2}− rm

^{i})/(1 + r)

^{i}in (14) is calculated inversely to catch the fact that money illusion lures the consumer to reach beyond truly attainable consumption possibilities, thus motivating excessive saving of money. On the other hand, the CV in (6) measures what would be the cost of rational precaution to external uncertainty, that is, the horizontally measured inward shift of the dashed constraint caused by holding m. Therefore, the total cost of holding m

^{i}amounts to CV + CV

^{i}. Since the CV in (6) cancels out the first term of the CV

^{i}in (14), the remainder is:

^{i}. The result is economically intuitive, and the result holds qualitatively under partial money illusion.

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Note

1 | The comparative static properties of the model can be checked by totally differentiating (11) and (12) with recall of (13) and taking m fixed in (7) and (8). Denoting −u _{11} + 2(1 + r)u_{12} − (1 + r)^{2}u_{22} = A > 0 and u_{12} − (1 + r)u_{22} = B > 0 and using Cramer’s rule produces:
$$\frac{de}{d\alpha}=0$$
$$\frac{ds}{d\alpha}=-m\frac{B}{A}<0$$
$$\frac{de}{dr}=\frac{1}{w\u201d}0$$
$$\frac{ds}{dr}=-(\frac{1}{w\u201d}+s\frac{B}{A})+\frac{{u}_{2}}{A}>0$$
_{2}/A > 0. In the income effect, 1/w” < 0 and sB/A > 0. Their sum is negative, 1/w” + sB/A < 0, because homothetic preferences dictate that a marginal change in r induces a larger negative change in education 1/w” compared to the positive change in deposit saving sB/A. Noting the minus sign outside the parentheses, the income term is positive, and so also is the effect of a marginal change in the expected market interest rate r on illiquid saving s. |

## References

- Akerlof, George, and Robert Shiller. 2009. Animal Spirits. Princeton: Princeton University Press. [Google Scholar]
- Baiardi, Donatella, Marco Magnani, and Mario Menegatti. 2019. The theory of precautionary saving: An overview of recent developments. Review of Economics of the Household 18: 513–52. [Google Scholar] [CrossRef]
- Barr, Nicholas. 2001. Welfare State as Piggy Bank. Oxford: Oxford University Press. [Google Scholar]
- Baumol, William. 1952. The Transactions Demand for Cash: An Inventory Theoretical Approach. Quarterly Journal of Economics 66: 545–56. [Google Scholar] [CrossRef]
- Begg, David, Gianluigi Vernasca, Stanley Fischer, and Rudiger Dornbusch. 2014. Economics. New York: McGraw-Hill Education. [Google Scholar]
- Blanchard, Oliver, Alessia Amighini, and Francesco Giavazzi. 2010. Macroeconomics, a European Perspective. London: Pearson Education Ltd. [Google Scholar]
- Boar, Corina. 2021. Dynastic Precautionary Savings. The Review of Economic Studies 88: 2735–65. [Google Scholar] [CrossRef]
- Burda, Michael, and Charles Wyplosz. 2017. Macroeconomics—A European Text. Oxford: Oxford University Press. [Google Scholar]
- Clower, Robert. 1967. A Reconsideration of the Microfoundations of Monetary Theory. Western Economic Journal 6: 1–8. [Google Scholar] [CrossRef]
- Fisher, Irving. 1914. The Purchasing Power of Money. Bulletin of American Mathematical Society 7: 377–81. [Google Scholar]
- Fisher, Irving. 1928. The Money Illusion. New York: Adelphi. [Google Scholar]
- Fisher, Irving. 1930. The Theory of Interest. New York: Macmillan. [Google Scholar]
- Holman, Jill. 1998. GMM Estimation of a Money-in-the-Utility-Function Model: The Implications of Functional Forms. Journal of Money, Credit and Banking 30: 679–98. [Google Scholar] [CrossRef]
- Imrohoroglu, Ayse. 1992. The Welfare Costs of Inflation under Imperfect Insurance. Journal of Economic Dynamics and Control 16: 79–91. [Google Scholar] [CrossRef]
- Kam, Eric, and Paul Missios. 2003. Wealth effects in a cash-in-advance economy. Economics Bulletin 5: 1–7. [Google Scholar]
- Keynes, John Maynard. 1930. Treatise of Money. London: Macmillan. [Google Scholar]
- Marshall, Alfred. 1890. Principles of Economics. London: Macmillan. [Google Scholar]
- Orland, Andreas, and Davud Rostam-Afschar. 2021. Flexible Work Arrangements and Precautionary Behavior. Theory and experimental evidence. Journal of Economic Behavior & Organization 191: 442–81. [Google Scholar]
- Robinson, Joan. 1937. Introduction to the Theory of Employment. London: Macmillan. [Google Scholar]
- Sidrauski, Miguel. 1967. Rational Choice and Patterns of Growth in a Monetary Economy. American Economic Review 57: 534–44. [Google Scholar]
- Tobin, James. 1956. The Interest-Elasticity of Transactions Demand for Cash. The Review of Economics and Statistics 38: 241–47. [Google Scholar] [CrossRef]
- Tversky, Amos, and Daniel Kahneman. 1992. Rational Choice and the Framing of Decisions. The Journal of Business 59: 251–78. [Google Scholar] [CrossRef]
- Varian, Hal. 2019. Intermediate Microeconomics with Calculus. New York: Norton. [Google Scholar]
- Vasilev, Aleksandar. 2022. A Business-Cycle Model with Money-in-Utility (MIU) and Government Sector: The Case of Bulgaria (1999–2020). Journal of Economic and Administrative Sciences. [Google Scholar] [CrossRef]
- Walsh, Carl. 2013. Monetary Theory and Policy. Cambridge: MIT Press. [Google Scholar]
- Zhao, Qian, and Tak Kuen Siu. 2020. Consumption-leisure-investment strategies with time-inconsistent preference in a life-cycle model. Communications in Statistics—Theory and Methods 49: 6057–79. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Laurila, H.
Money as Insurance. *Risks* **2022**, *10*, 238.
https://doi.org/10.3390/risks10120238

**AMA Style**

Laurila H.
Money as Insurance. *Risks*. 2022; 10(12):238.
https://doi.org/10.3390/risks10120238

**Chicago/Turabian Style**

Laurila, Hannu.
2022. "Money as Insurance" *Risks* 10, no. 12: 238.
https://doi.org/10.3390/risks10120238