<p>Consider the life-time utility maximization problem of the individual. At this stage of the paper, while the utility function &quot;has the usual regular indifference-curve
concavities&quot; as Samuelson writes, nothing has been assumed about the form of the intertemporal utility function, so the problem is stated as</p>
<p><span class="math-container">$$\max_{C_1,C_2,C_3}U(C_1,C_2,C_3),\;\;\; s.t.\;\;\;  C_3 = \frac {1+ R_t}{R_tR_{t+1}} - \frac{C_1}{R_tR_{t+1}} - \frac{C_2}{R_{t+1}} \tag{1}$$</span></p>
<p>where the constraint comes from re-arranging the intertemporal budget expression (eq. 1 of the paper), and interest rates/discount factors are treated as exogenous. Since this is an equality constraint (implying no bequest motive and no debts at the end of life), we can incorporate it into the utility function and solve with respect to <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span> only. The first-order conditions are then</p>
<p><span class="math-container">$$\frac{d U}{d C_k} = \frac{\partial U}{\partial C_k} + \frac{\partial U}{\partial C_3}\cdot \frac{\partial C_3}{\partial C_k} =0, \;\;\; k=1,2 \tag{2} $$</span></p>
<p>Because consumption is a good, marginal utility is positive. This implies that at the solution also,</p>
<p><span class="math-container">$$\frac{\partial C_3}{\partial C_k} &lt; 0,\;\;\; k=1,2 \tag{3}$$</span></p>
<p>By the chain rule we can write</p>
<p><span class="math-container">$$\frac{\partial C_3}{\partial R_{t+1}} = \frac{\partial C_3}{\partial C_k} \cdot \frac{\partial C_k}{\partial R_{t+1}} , \;\;\; k=1,2 \tag{4}$$</span>.</p>
<p>Due to <span class="math-container">$(3)$</span> we obtain the fundamental result that, at the solution, the <span class="math-container">$t+1$</span> discount factor (or equivalently, the <span class="math-container">$t+1$</span> interest rate) will tend to affect <span class="math-container">$C_3$</span> in the opposite direction than <span class="math-container">$C_1, C_2$</span>.</p>
<p>This is crucial, because it says that changes in the exogenous interest rates will  lead to solutions where we will observe higher (lower) <span class="math-container">$C_1,C_2$</span> and lower (higher) <span class="math-container">$C_3$</span>.</p>
<p>Samuelson asserts that  <span class="math-container">$\partial C_3/\partial R_{t+1} &gt;0$</span>, which is equivalent to assert that</p>
<p><span class="math-container">$$\frac{\partial C_k}{\partial R_{t+1}} &lt; 0 , \;\;\; k=1,2$$</span></p>
<p>or equivalently, that</p>
<p><span class="math-container">$$\frac{\partial C_3}{\partial i_{t+1}} &lt; 0,\qquad \frac{\partial C_k}{\partial i_{t+1}} &gt; 0 , \;\;\; k=1,2$$</span></p>
<p>Is this the case?</p>
<p>For given interest rates we obtain a solution for the consumption vector that implies certain savings and a certain interest income. If we contemplate an increase in <span class="math-container">$i_{t+1}$</span>, total income, <span class="math-container">$I$</span>, will tend to increase, <span class="math-container">$\partial I/\partial i_{t+1}&gt;0$</span> . Then, because consumption is assumed to be a &quot;superior&quot; good, <span class="math-container">$C_k,\; k=1,2$</span> are certainly normal goods, <span class="math-container">$\partial C_k/\partial I &gt;0$</span>, and so</p>
<p><span class="math-container">$$\frac{\partial C_k}{\partial i_{t+1}} = \frac{\partial C_k}{\partial I}\cdot \frac{\partial I}{\partial i_{t+1}} &gt;0$$</span></p>
<p>Then for <span class="math-container">$C_3$</span> the same relation implies that <span class="math-container">$\partial C_3/\partial I &lt;0$</span>, <em>at the solution</em>. This does not make <span class="math-container">$C_3$</span> an inferior good, when viewed on its own, but it is the result of it being determined residually, as the last element in a fixed-horizon intertemporal problem.</p>

CC BY-SA 4.0
2020-08-16T20:33:09.040Z
Hi Alecos, thanks for your answer, I can follow most of it. However, I don't understand 2 things: How do you define total income? And second: how did you derive equation 4?
user18214
CC BY-SA 4.0
2020-08-17T01:42:59.423Z
@MiltonKeynes  Total Income is life-time income: the two units that are produced in periods 1 and 2, plus any interest income from saving. Equation 4 is just the chain rule for differentation.
Alecos Papadopoulos
CC BY-SA 4.0
2020-08-16T18:02:44.460Z
2020-08-16T18:02:44.460Z
Alecos Papadopoulos
<p>I was reading <a href="http://public.econ.duke.edu/%7Ehf14/teaching/socialinsurance/readings/Samuelson58(6.3).pdf" rel="noreferrer">this paper</a> by Paul Samuelson. I am puzzled by a bit on page 470, left column. Here he claims that <span class="math-container">$\frac{\partial C_3}{\partial R_{t+1}}&gt;0$</span>. Let me give a short description of what exactly this means and then I'll explain why I disagree.</p>
<p>In this model every individual lives 3 periods. Every person produces 1 unit of output in period 1, one unit in period 2 and no output in period 3. He can however trade some of his output in the first 2 periods for consumption in the last period and this gives rise to his consumption pattern <span class="math-container">$(C_1, C_2, C_3).$</span></p>
<p>Suppose a person is in period 1 at time <span class="math-container">$t$</span>. So he is in period 2 at time <span class="math-container">$t+1$</span>, and in period 3 at time <span class="math-container">$t+2$</span>. He can earn an interest rate of <span class="math-container">$i_t$</span> on savings at time <span class="math-container">$t$</span> for all <span class="math-container">$t$</span> (that is <span class="math-container">$i_t$</span> is the interest rate earned on savings you carry from time <span class="math-container">$t$</span> to period <span class="math-container">$t+1$</span>). Now define <span class="math-container">$R_t=1/(1+i_t)$</span> for all <span class="math-container">$t$</span>.</p>
<p>The claim by Samuelson is that <span class="math-container">$\frac{\partial C_3}{\partial R_{t+1}}&gt;0$</span>. Or equivalently: <span class="math-container">$\frac{\partial C_3}{\partial i_{t+1}}&lt;0$</span>. How can this be true? If <span class="math-container">$i_{t+1}$</span> goes up (down), sacrificing a fraction of <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span> to increase <span class="math-container">$C_3$</span> becomes more (less) profitable. So common sense would suggest: <span class="math-container">$\frac{\partial C_3}{\partial R_{t+1}}&lt;0 \Leftrightarrow \frac{\partial C_3}{\partial i_{t+1}}&gt;0$</span>.</p>
<p>Samuelson gives no proof of his claim but he does claim it can be inferred from the following: &quot;Of course, these functions are subject
to all the restrictions of modern consumption theory of the ordinal utility or
revealed preference type. Thus, with
consumption in every period being a
&quot;superior good,&quot; &quot;</p>
<p>Please enlighten me. Is Samuelson wrong/am I wrong?</p>

CC BY-SA 4.0
2020-08-14T19:14:46.873Z
Naive questions: How crucial is the sign of $\frac{\partial C_3}{\partial R_{t+1}}$ for the results? Do the results change dramatically if we assume this expression is negative or positive?
emeryville
CC BY-SA 4.0
2020-08-14T21:21:44.450Z
Well, I haven't read the entire paper yet, so I don't know. However, I believe the question is worth asking in its own right. The question shouldn't be seen as a challenge to of the paper its thesis, but as an effort to better my (and hopefully others) understanding of economics.
user18214
CC BY-SA 4.0
2020-08-14T21:23:35.670Z
That being said, Samuelson does follow the statement of $\frac{\partial C_3}{\partial R_{t+1}}<0$ up with the comment  "(This
says that lowering the interest rate
earned on savings carried over into retirement must increase retirement consumption.) "
user18214
CC BY-SA 4.0
2020-08-14T21:24:12.270Z
Thus it isn't a simple accident of switching the less than and greater than sign
user18214
CC BY-SA 4.0
2020-08-14T21:25:49.017Z
If Samuelson is right than it must be a counter-intuitive result of the economics of intertemporal decision making more generally and it seems to me to be an interesting result
user18214
<interest-rate><errata>
Effect of change in discount rate on consumption -- (Samuelson, 1958)
user18214
# Effect of change in discount rate on consumption -- (Samuelson, 1958)

(Asked by: username_0 on 2020-08-14)

I was reading [this paper](http://public.econ.duke.edu/%7Ehf14/teaching/socialinsurance/readings/Samuelson58(6.3).pdf) by Paul Samuelson. I am puzzled by a bit on page 470, left column. Here he claims that $\frac{\partial C\_3}{\partial R\_{t+1}}>0$. Let me give a short description of what exactly this means and then I'll explain why I disagree.

In this model every individual lives 3 periods. Every person produces 1 unit of output in period 1, one unit in period 2 and no output in period 3. He can however trade some of his output in the first 2 periods for consumption in the last period and this gives rise to his consumption pattern $(C\_1, C\_2, C\_3).$

Suppose a person is in period 1 at time $t$. So he is in period 2 at time $t+1$, and in period 3 at time $t+2$. He can earn an interest rate of $i\_t$ on savings at time $t$ for all $t$ (that is $i\_t$ is the interest rate earned on savings you carry from time $t$ to period $t+1$). Now define $R\_t=1/(1+i\_t)$ for all $t$.

The claim by Samuelson is that $\frac{\partial C\_3}{\partial R\_{t+1}}>0$. Or equivalently: $\frac{\partial C\_3}{\partial i\_{t+1}}<0$. How can this be true? If $i\_{t+1}$ goes up (down), sacrificing a fraction of $C\_1$ and $C\_2$ to increase $C\_3$ becomes more (less) profitable. So common sense would suggest: $\frac{\partial C\_3}{\partial R\_{t+1}}<0 \Leftrightarrow \frac{\partial C\_3}{\partial i\_{t+1}}>0$.

Samuelson gives no proof of his claim but he does claim it can be inferred from the following: "Of course, these functions are subject
to all the restrictions of modern consumption theory of the ordinal utility or
revealed preference type. Thus, with
consumption in every period being a
"superior good," "

Please enlighten me. Is Samuelson wrong/am I wrong?

**Tags:** interest-rate, errata

> **Comments:**
> **username_1**: Naive questions: How crucial is the sign of $\frac{\partial C_3}{\partial R_{t+1}}$ for the results? Do the results change dramatically if we assume this expression is negative or positive?  
> **username_0**: Well, I haven't read the entire paper yet, so I don't know. However, I believe the question is worth asking in its own right. The question shouldn't be seen as a challenge to of the paper its thesis, but as an effort to better my (and hopefully others) understanding of economics.  
> **username_0**: That being said, Samuelson does follow the statement of $\frac{\partial C_3}{\partial R_{t+1}}<0$ up with the comment  "(This says that lowering the interest rate earned on savings carried over into retirement must increase retirement consumption.) "  
> **username_0**: Thus it isn't a simple accident of switching the less than and greater than sign  
> **username_0**: If Samuelson is right than it must be a counter-intuitive result of the economics of intertemporal decision making more generally and it seems to me to be an interesting result  

---

## 1 answers

### Answer by username_2 on 2020-08-16. Score: 4

Consider the life-time utility maximization problem of the individual. At this stage of the paper, while the utility function "has the usual regular indifference-curve
concavities" as Samuelson writes, nothing has been assumed about the form of the intertemporal utility function, so the problem is stated as

$$\max\_{C\_1,C\_2,C\_3}U(C\_1,C\_2,C\_3),\;\;\; s.t.\;\;\; C\_3 = \frac {1+ R\_t}{R\_tR\_{t+1}} - \frac{C\_1}{R\_tR\_{t+1}} - \frac{C\_2}{R\_{t+1}} \tag{1}$$

where the constraint comes from re-arranging the intertemporal budget expression (eq. 1 of the paper), and interest rates/discount factors are treated as exogenous. Since this is an equality constraint (implying no bequest motive and no debts at the end of life), we can incorporate it into the utility function and solve with respect to $C\_1$ and $C\_2$ only. The first-order conditions are then

$$\frac{d U}{d C\_k} = \frac{\partial U}{\partial C\_k} + \frac{\partial U}{\partial C\_3}\cdot \frac{\partial C\_3}{\partial C\_k} =0, \;\;\; k=1,2 \tag{2} $$

Because consumption is a good, marginal utility is positive. This implies that at the solution also,

$$\frac{\partial C\_3}{\partial C\_k} < 0,\;\;\; k=1,2 \tag{3}$$

By the chain rule we can write

$$\frac{\partial C\_3}{\partial R\_{t+1}} = \frac{\partial C\_3}{\partial C\_k} \cdot \frac{\partial C\_k}{\partial R\_{t+1}} , \;\;\; k=1,2 \tag{4}$$.

Due to $(3)$ we obtain the fundamental result that, at the solution, the $t+1$ discount factor (or equivalently, the $t+1$ interest rate) will tend to affect $C\_3$ in the opposite direction than $C\_1, C\_2$.

This is crucial, because it says that changes in the exogenous interest rates will lead to solutions where we will observe higher (lower) $C\_1,C\_2$ and lower (higher) $C\_3$.

Samuelson asserts that $\partial C\_3/\partial R\_{t+1} >0$, which is equivalent to assert that

$$\frac{\partial C\_k}{\partial R\_{t+1}} < 0 , \;\;\; k=1,2$$

or equivalently, that

$$\frac{\partial C\_3}{\partial i\_{t+1}} < 0,\qquad \frac{\partial C\_k}{\partial i\_{t+1}} > 0 , \;\;\; k=1,2$$

Is this the case?

For given interest rates we obtain a solution for the consumption vector that implies certain savings and a certain interest income. If we contemplate an increase in $i\_{t+1}$, total income, $I$, will tend to increase, $\partial I/\partial i\_{t+1}>0$ . Then, because consumption is assumed to be a "superior" good, $C\_k,\; k=1,2$ are certainly normal goods, $\partial C\_k/\partial I >0$, and so

$$\frac{\partial C\_k}{\partial i\_{t+1}} = \frac{\partial C\_k}{\partial I}\cdot \frac{\partial I}{\partial i\_{t+1}} >0$$

Then for $C\_3$ the same relation implies that $\partial C\_3/\partial I <0$, *at the solution*. This does not make $C\_3$ an inferior good, when viewed on its own, but it is the result of it being determined residually, as the last element in a fixed-horizon intertemporal problem.

> **Comments:**
> **username_0**: Hi username_2, thanks for your answer, I can follow most of it. However, I don't understand 2 things: How do you define total income? And second: how did you derive equation 4?  
> **username_2**: @MiltonKeynes  Total Income is life-time income: the two units that are produced in periods 1 and 2, plus any interest income from saving. Equation 4 is just the chain rule for differentation.  

---

