Why is there a positive risk-free interest rate?

Question

I've been told that the interest rate is the price needed to be willing to take on the risk associated with lending. This is why personal loans have higher interest rates than bank deposits, for example. So why is there a positive (as opposed to zero) risk-free rate? It is usually assumed that government bonds are risk-free, yet they have a positive return. If there is no risk, how can there be a positive rate?

Answer 1

The interest rate is (1) the price needed to take on risk and (2) the price needed to delay consumption.

The reason there is a positive risk free rate, even though there is no risk, is because of the time preference typical of any economic agent. It is preferable to consume today, than to consume tomorrow. To put off consumption today and invest in the risk free asset instead, the agents want to earn interest, otherwise they will consume instead of saving.

Normally, if the risk free rate was zero, nobody would buy the risk free asset, but spend everything on consumption. E.g. nobody would buy government bonds. The government has to offer a positive interest rate to find buyers.

Current negative interest rates are actually a reflection of how our classical assumption, that agents prefer to consume today rather than tomorrow, has been turned on its head. Nobody wants to consume today despite zero or negative interest rates contrary to the classic theory above (and people buy government bonds with negative interest rates). This can happen if agents are apprehensive enough of the future to want to increase their precautionary savings instead of consuming. If they are worried enough, they will do this even if they lose money on their savings (due to the negative interest rate).

Answer 2

Risk is not the same than the interest rate an asset pays. In a world without risk, there is still room for positive interest rates. Why? Because of the opportunity cost of money. Instead of lending that money to the government, a company or an individual, the owner of the money could investing in some profit making business. In a stylised Econ 101 neoclassical economy, decreasing marginal product and the Inada conditions imply that the equilibrium interest rate is only zero when the capital in the economy is infinity. As this is never the case, the equilibrium interest rate is necessarily positive. Its level is given among other things by technology and stocks of factors of production.

To see an example, consider an aggregate production function

$$ Y = AK^{\alpha}L^{1-\alpha} $$

where $\alpha \in (0,1)$.

Optimal choice of factors is given by

$$ \frac{r}{w} = \frac{MP_K}{MP_L} $$

which is equivalent to

$$ \frac{r}{w} = \frac{1-\alpha}{\alpha}\frac{L}{K} $$

All the right hand side terms (technology parameters and stock of factors, including employment/population) are positive. Therefore, the left hand side is too. Notice that $K\rightarrow \infty$ means, ceteris paribus, $r \rightarrow 0$.

This post and answer provide more insights about profits and interest rates in a neoclassical economy.

Answer 3

In a standard neoclassical framework like the Ramsey-Cass-Koopmans model, the interest rate along the balanced growth path is a function of preferences and technology. What matters is essentially the growth-adjusted discount factor of households appearing in the Euler equation. This points to two sources for positive risk-free rates:

1. There is the pure rate of time preference pointed out in the answer by M3RS. Economic agents tend to prefer consumption today relative to tomorrow and need to be compensated for delaying consumption. This introduces a substitution effect from today's consumption to the future.
2. There is steady economic growth over time, implying that the available consumption path over time is increasing. But reasonable preference specifications imply that households prefer a smooth consumption path, i.e. are not arbitrarily willing to substitute consumption between periods. Typically, households would prefer a perfectly flat consumption path over time out of their permanent income, i.e. would like to pull consumption forward from the future when they are richer (and marginal utility is lower). So the real interest rate also needs to compensate them for their low intertemporal elasticity of substitution that punishes having a non-flat consumption profile: delayed consumption today earns a non-zero return in the form of the risk-free interest rate.

In point 2 there is an additional complication: changes in the real risk-free interest rate have both income and substitution effects, because they will affect the present discounted value of life-time income as well as the relative price of consumption today versus tomorrow. With log utility, which implies a unit intertemporal elasticity of substitution, the two effects exactly cancel. There is no additional compensation required beyond the one for the pure rate of time preference. In contrast, with an intertemporal elasticity of substitution below 1, the real risk-free interest rate carries an additional compensation to induce people to have an increasing consumption profile over time.

See for example page 947 of King/Rebelo 1999: Resuscitating Real Business Cycles