Euler, Transversality and Intertemporal Disequilbrium

This note provides a brief intuitive review of the Euler equation and the associated transversality/no-Ponzi-game constraints.  The Euler equation is frequently described as the workhorse or – as this writer prefers – the flux-capacitor that drives most modern macroeconomic models. Understanding the concept is necessary for fully appreciating a capital-focused framework. A mathematical proof of the equation is beyond the scope of this note but can be found in any graduate-level economics textbook.

The Intertemporal Budget Constraint

Diagrams A and B below provide an intuitive starting point for the concept of intertemporal distribution of income and consumption across time. In this example the consumer has a fixed annual income and the ability to alter their consumption profile by saving or borrowing. The two key rates guiding the consumer’s intertemporal choice are the market rate of interest and the rate of time preference (i.e. the “impatience” of the consumer). When the two rates are equal consumption equals income. When the rate of interest is lower than the rate of time preference the consumer will borrow in the present and pay it back in the future. As a result, the profile of consumption will be downward sloping relative to consumption. 

Assuming everything is kept in equilibrium, the consumer must respect an intertemporal budget constraint. In other words, the present value of their consumption must equal the present value of their income. The equation below uses discrete time for ease of understanding; more complex versions use continuous time. For our purposes dividing time into “today” and “future” gets the job done. Diagram C provides a graphical depiction of the budget constraint line. Intuitively, if today’s consumption is to the left of (i.e. less than) today’s income, then future consumption will be greater than future income.   

The Euler Equation

The intertemporal budget constraint tells us the choices available to the consumer, but not how they will actually choose to allocate consumption between today and the future. To do that we turn to the Euler equation and the indifference curve. 

The Euler equation determines the marginal rate of substitution of consumption between today and the future. Diagram D shows a visualization of the indifference curve formed by the marginal rate of substitution. When the rate of interest and the rate of time preference are equal the slope of the tangent is one and the trade-off between consumption today and consumption in the future is equal. The curve is concave because each additional unit of consumption in either direction provides diminishing returns.

Putting the Equations Together

Putting together the budget constraint with the indifference curve gives us the time path of consumption. In Diagram E the rate of interest is equal to the rate of time preference. As a result, consumption in each period is equal to income in each period and there is no desire to trade consumption across time because the utility of consumption today is equal to the utility of consumption in the future.

In Diagram F the interest rate falls below the rate of time preference. As a result, the slope of the budget constraint becomes shallower. The shift of the budget line’s slope represents the fact that borrowing consumption from the future has become less expensive. The point of tangency between the budget line and the indifference curve shifts down and to the right. That means consumption today is higher than income today and consumption tomorrow is lower than income tomorrow.

Implications for Intertemporal Disequilibrium

To understand the implications of the changing time path of consumption for intertemporal disequilibrium We shift back to an intuitive view of the situation. Diagram G shows a time series of the rate of time preference and the market rate of interest. Diagram H shows a time series of income and consumption.At time t the rate of interest drops below the rate of time preference so consumption moves above the trend growth of income. Once we get to t+1 the trend rate of consumption growth must fall relative to the trend rate of income growth. At t+2 a negative output gap has emerged[1]  and the market rate of interest must fall again to close the gap. This process will continue over and over, but because the utility of bringing forward consumption is diminishing the interest rate must fall fast and faster to prevent the negative gap from appearing. This is the “conveyor belt” that is referenced frequently in the work of renowned economist Bernard Connolly.

Transversality/No Ponzi-Game Constraint

The Euler equation and the intertemporal budget constraint can be extended to show in theory why asset price bubbles should not form. The strict version of the transversality condition says that nothing should be saved or owed in the final period. A less restrictive version of the constraint, the no-Ponzi-game condition, is that the present value of wealth at infinity must be non-negative.      

The equation below is a version of the Euler equation that balances the utility of transforming a unit of an asset into consumption today with the utility lost from – using Obstfeld and Rogoff’s terminology –  “dividends” forgone over the next T periods and the need to repurchase the asset at t+T. Unless the asset is repurchased, the no-Ponzi-game constraint does not hold.      

The value of an asset at infinity is just the present value of all of its cash flows so the final term in the Euler equation should always equal zero. If the value at infinity is greater than zero, there is net gain to consuming the asset today despite forgoing future dividends from asset.

The implication is that an individual can increase the present value of their lifetime consumption above lifetime resources by borrowing and then maintaining a negative net asset position indefinitely, in other words, by borrowing money they could never pay back. By lending to a borrower who will forever maintain a negative net asset position the lender is falling victim to a Ponzi game.  

As pointed out frequently by Bernard Connolly, the Euler equation can be “turned off” for a time via a credit bubble (a “bankers bubble”, or Ponzi game), in which borrowers and lenders behave as though the transversality constraint does not hold, and the equation’s macro implications can be distorted, for a time, by over-optimistic assessments of wealth. Such assessments can refer to human wealth (future labor income) or from non-human wealth (financial and real assets). If expectations of wealth are too optimistic, however, a correction of such assessments will accentuate any downturn in future consumption. A downward revision of productivity expectations, for instance, is likely to provoke a “Super-Say’s-Law-in-reverse” effect. And if asset prices are inflated by sub-“normal” interest rates (a “traders’ bubble”), any attempt to return to “normal” interest rates will have a significant negative effect on consumption.

References

Jones, Charles I. Macroeconomics. New York: W.W. NORTON, 2014. Print.

Kamihigashi, Takashi. 2006. "Transversality Conditions and Dynamic Economic Behavior,"

Discussion Paper Series 180, Research Institute for Economics & Business Administration, Kobe University.

Parker, Jonathan. December 2007. “Euler Equations”,  New Palgrave Dictionary of Economics.

Rogoff, Kenneth, and Maurice Obstfeld. 1996. Foundations of International Macroeconomics. Cambridge, Massachusetts: MIT Press.

[1] It is important to note that these diagrams depict the position of an individual. Many important issues arise in transposing them to the macro economy. Such issues have frequently been discussed by Bernard in his work over the years. The present note is intended only to give a simple, intuitive exposition of the individual behaviour underlying our analysis of the macro economy.