# Why is instantaneous utility of current period discounted?

Consider a two period model of consumption.

I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the next period.

Could someone give me an intuition behind this result?

Below, the derivation is presented.

$$\max_{c_{t},c_{t+1}}u_{t}\left(c_{t},c_{t+1}\right)$$

such that

$$c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1}$$

FOC: (MU denotes marginal (aka instantaneous) utility)

$$\begin{cases} MU_{t}=\lambda\\ MU_{t+1}=\lambda\frac{1}{1+r_{t+1}}\\ c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1} \end{cases}\Rightarrow\frac{1}{1+r_{t+1}}MU_{t}=MU_{t+1}$$

What you see as $$\frac{1}{1+r_{t+1}}MU_t = MU_{t+1}$$ can also be seen as $$\frac{1}{1+r_{t+1}}MU_t = \frac{1}{1}MU_{t+1}$$ where $1+r_{t+1}$ and $1$ are the respective prices of current and future money measured in future money. So basically what is happening is that you calculate how much utility an additional unit of future money would buy you in the current and in the future period. In optimum these quantities are equal. You can also rewrite the equation into the form of the familiar MRS condition: $$\frac{MU_t}{MU_{t+1} } = \frac{1}{\frac{1}{1+r_{t+1}}}.$$

• 1. You suggested (linear) utility function that lead to consumption only at 1 period and starvation in the other. Kind of an extreme example, don't you think? 2. I was talking about discounting by using interest rate, while you are talking about time preference rate. Aug 3 '15 at 8:55
• @lyuboslaw 1. Yes, it's an extreme example, but it did not matter in this case. 2. I see, this was not clear to me. Based on your comment I edited your question. If you do not agree with the edit change it back. Aug 3 '15 at 9:09

It is important to note that the OP uses a general form of a utility function, where utility is not necessarily separable per period. But in such a general setting, $MU_t$ is not "marginal utility", but only the partial derivative of $u(c_t, c_{t+1})$ with respect to $c_t$. "Marginal utility" is a concept suited for a uni-variate utility function.

So it is more accurate to write the optimal relationship as

$$\frac{1}{1+r_{t+1}}\frac {\partial u(c_t, c_{t+1})}{\partial c_t} = \frac {\partial u(c_t, c_{t+1})}{\partial c_{t+1}}$$

Indeed, it appears that we are "discounting" the effect of the present, rather than that of the future. But it perhaps becomes more clear if we view it not as dividing the "present" by a discount factor, but rather as multiplying it by the relative price between present and future:

We can "buy" one unit of future consumption by trading only $1/(1+r_{t+1})$ units of current consumption (i.e. by saving $1/(1+r_{t+1})$ amount now and obtain $1$ unit tomorrow). So by multiplying the "effect of the present on utility" by $[1/(1+r_{t+1})]$, we express it in terms of the future.

• "Marginal utility is a concept suited for a uni-variate utility function." Really? I understand the intuition of what you are saying but I see this used quite often (Varian, Mas-Colell) with non-separable utility functions. By used I mean that define $MU_x(x,y) = \frac{\partial U(x,y)}{\partial x}$. Perhaps the convention is not ideologically tidy? Oct 8 '15 at 9:49
• @denesp this is mainly a nomenclature issue, not of mathematical operations. I would not call "marginal output" the partial derivative of the production function with respect to capital holding labor fixed, for example. We call it "marginal product of capital", so I guess it is consistent to say "marginal utility of good A" (holding good B constant). But not "marginal utility" alone. Oct 8 '15 at 11:45