# Effect of change in discount rate on consumption -- (Samuelson, 1958)

I was reading this paper 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?

• 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? Aug 14 '20 at 19:14
• 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. Aug 14 '20 at 21:21
• 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.) " Aug 14 '20 at 21:23
• Thus it isn't a simple accident of switching the less than and greater than sign Aug 14 '20 at 21:24
• 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 Aug 14 '20 at 21:25

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.

• 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? Aug 16 '20 at 20:33
• @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. Aug 17 '20 at 1:42