## Utility and Time – Statement of the Problem

Ultimately our goal is to get some description of price evolution derived from first (economic) principles. In earlier posts (1, 2, 3) I have shown what can be deduced from ‘demand invariance under price-scaling’. As described there we still assume ${n}$ goods being traded in a market, hence there are prices ${p_i}$ and demands ${d_i}$ for ${1\leq i\leq n}$ attributed to these goods. That was the setting so far and now we are going to take the first steps into time.

We assume that good ${i}$ is consumed over time and describe consumption ${c_i(\cdot)}$ as a positive real function. Consumption of good ${i}$ from time ${a}$ to time ${b}$ is measured by ${\int_a^b c(s)ds}$. The participants in the market we call agents. An agent attributes to each consumption vector ${c}$ a utility ${u}$. Technically this is a positive, increasing and concave function. In all our examples ${u}$ and ${c}$ will be sufficiently differentiable. Utility is increasing since more consumption is considered better and it is concave since we assume ‘diminishing marginal utility‘. The latter does not always hold in economic situations. However, most introductory examples are concave and as a start this seems safe.
I assume ‘time impatience‘, that means, consumption now is better than consumption in the future. That assumption is not undisputed, but, as a model for the finite life span of the agents, this too seems safe for such an introductory text. Overall utility from time ${a}$ to time ${b}$ is measured by ${\int_a^b r^s u(c(s)) ds}$ for some discount rate ${0.

Agents have attached a wealth level ${w(\cdot)}$, that means for all time holds ${\sum_i^n p_i(t) c_i(t) = w_i(t)}$. They consume according to their prescribed wealth and they consume according to their demand (${c_i=d_i}$). The last assumption closes the gap between ${p}$ and ${c}$. We assume that demand is given as ${d_i=\dot{p_i}}$ and thus we obtain in summary ${c_i = \dot{p}_i}$.

Now we are in the shape to state the problem: agents in a market maximize utility

$\displaystyle \int_0^T r^s u(c(s)) ds$

according to the constraint

$\displaystyle \sum_{i=1}^n p_i(t) c_i(t) - w(t) = 0$

for given time ${0, discount rate ${0, wealth function ${w}$ and utility function ${u}$.

Voilà, we end up with a constrained Euler-Lagrange equation. But beware! There are a couple of traps jamming all intuition we might have from mechanics or similar theories with conserved energy (understood as the Legendre transform of the Lagrangian). I certainly elaborate on this in one of the next entries.
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