<p>The interest rate is (1) the price needed to take on risk <em>and</em> (2) the price needed to delay consumption.</p>

<p>The reason there is a positive risk free rate, even though there is no risk, is because of the <strong>time preference</strong> 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.</p>

<p>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.</p>

<p>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).</p>

CC BY-SA 3.0
2017-09-10T19:34:41.213Z
If people have preference for consumption smoothing, or value lifetime consumption, then they will spread consumption across periods, regardless of whether $r>0$ or not. Think of the trivial [Fisher's model](https://en.wikipedia.org/wiki/Intertemporal_choice#Fisher.27s_model_of_intertemporal_consumption). Consumption spreading does not hinge on positive $r$. Neither does Friedman's permanent income hypothesis, not life cycle models.
luchonacho
CC BY-SA 3.0
2017-09-11T07:08:04.370Z
These models provide support to the argument above, as they predict consumption will be higher today if interest rates are lower. I wouldn't argue that consumption in Period 2 would be ZERO if interest rates were not positive. That would be clearly unrealistic. The argument is simply that positive interest rates induce people to save more (lend to the government). The government can then invest in infrastructure and improve long run growth. Interest rates are positive to induce saving.
M3RS
CC BY-SA 3.0
2017-09-11T08:11:44.523Z
Your answer says: " It is preferable to consume today, than to consume tomorrow." "Normally, if the risk free rate was zero, nobody would buy the risk free asset, but spend everything on consumption.". That is not necessarily true. **Savings are possible even with $r=0$**. If I earn 100 today and nothing tomorrow, and value consumption tomorrow, then I will save regardless of the value of $r$.
luchonacho
CC BY-SA 3.0
2017-09-11T08:25:30.133Z
I agree with you. These kind of discussions are a perfect demonstration of why economics is a dying profession :)
M3RS
CC BY-SA 3.0
2017-09-11T08:27:29.457Z
Not sure what you mean. :/
luchonacho
CC BY-SA 3.0
2017-08-22T12:16:53.013Z
2017-08-22T18:44:28.773Z
2017-08-22T18:44:28.773Z
M3RS
<p>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? <strong>Because of the opportunity cost of money.</strong> 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 <a href="https://en.wikipedia.org/wiki/Inada_conditions" rel="nofollow noreferrer">Inada conditions</a> 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.</p>

<p>To see an example, consider an aggregate production function</p>

<p>$$ Y = AK^{\alpha}L^{1-\alpha} $$</p>

<p>where $\alpha \in (0,1)$.</p>

<p>Optimal choice of factors is given by</p>

<p>$$ \frac{r}{w} = \frac{MP_K}{MP_L} $$</p>

<p>which is equivalent to</p>

<p>$$ \frac{r}{w} = \frac{1-\alpha}{\alpha}\frac{L}{K} $$</p>

<p>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, <em>ceteris paribus</em>, $r \rightarrow 0$.</p>

<p><a href="https://economics.stackexchange.com/a/16906/2776">This post and answer</a> provide more insights about profits and interest rates in a neoclassical economy.</p>

CC BY-SA 3.0
2017-08-22T12:41:25.497Z
Can you be more specific about what determines the equilibrium interest rate (in a stylised Econ101 neoclassical economy)? Is there an equation which gives the interest rate in terms of technology, stocks of factors of production, and other things? Is the intuition that the risk-free interest rate adjusts the value of your *absolute* savings in line with economic growth, but not your *relative* savings (your proportion of the economy's total wealth remains the same if you invest in risk-free assets)?
Peter
CC BY-SA 3.0
2017-08-22T12:00:54.517Z
2017-08-22T13:50:25.927Z
2017-08-22T13:50:25.927Z
luchonacho
<p>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:</p>
<ol>
<li>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.</li>
<li>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.</li>
</ol>
<p>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.</p>
<p>See for example page 947 of <a href="https://people.bu.edu/rking/EC702/KR99hom.pdf" rel="nofollow noreferrer">King/Rebelo 1999: Resuscitating Real Business Cycles</a></p>

CC BY-SA 4.0
2022-03-07T13:47:37.790Z
2022-03-07T13:47:37.790Z
jpfeifer
<p>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?</p>

<interest-rate>
Why is there a positive risk-free interest rate?
Peter
# Why is there a positive risk-free interest rate?

(Asked by: username_0 on 2017-08-22)

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?

**Tags:** interest-rate

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## 3 answers

### Answer by username_1 on 2017-08-22. Score: 7

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).

> **Comments:**
> **username_2**: If people have preference for consumption smoothing, or value lifetime consumption, then they will spread consumption across periods, regardless of whether $r>0$ or not. Think of the trivial [Fisher's model](https://en.wikipedia.org/wiki/Intertemporal_choice#Fisher.27s_model_of_intertemporal_consumption). Consumption spreading does not hinge on positive $r$. Neither does Friedman's permanent income hypothesis, not life cycle models.  
> **username_1**: These models provide support to the argument above, as they predict consumption will be higher today if interest rates are lower. I wouldn't argue that consumption in Period 2 would be ZERO if interest rates were not positive. That would be clearly unrealistic. The argument is simply that positive interest rates induce people to save more (lend to the government). The government can then invest in infrastructure and improve long run growth. Interest rates are positive to induce saving.  
> **username_2**: Your answer says: " It is preferable to consume today, than to consume tomorrow." "Normally, if the risk free rate was zero, nobody would buy the risk free asset, but spend everything on consumption.". That is not necessarily true. **Savings are possible even with $r=0$**. If I earn 100 today and nothing tomorrow, and value consumption tomorrow, then I will save regardless of the value of $r$.  
> **username_1**: I agree with you. These kind of discussions are a perfect demonstration of why economics is a dying profession :)  
> **username_2**: Not sure what you mean. :/  

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### Answer by username_2 on 2017-08-22. Score: 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](https://en.wikipedia.org/wiki/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](https://economics.stackexchange.com/a/16906/2776) provide more insights about profits and interest rates in a neoclassical economy.

> **Comments:**
> **username_0**: Can you be more specific about what determines the equilibrium interest rate (in a stylised Econ101 neoclassical economy)? Is there an equation which gives the interest rate in terms of technology, stocks of factors of production, and other things? Is the intuition that the risk-free interest rate adjusts the value of your *absolute* savings in line with economic growth, but not your *relative* savings (your proportion of the economy's total wealth remains the same if you invest in risk-free assets)?  

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### Answer by username_3 on 2022-03-07. Score: 0

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](https://people.bu.edu/rking/EC702/KR99hom.pdf)


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