Intertemporal budget constraint

Key takeaways

Intertemporal budget constraint belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Intertemporal budget constraint to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Intertemporal budget constraint from memory before moving on to harder problems.

Reference excerpt

In economics and finance, an intertemporal budget constraint is a constraint faced by a decision maker who is making choices for both the present and the future. The term intertemporal is used to describe any relationship between past, present and future events or conditions. In its general form, the intertemporal budget constraint says that the present value of current and future cash outflows cannot exceed the present value of currently available funds and future cash inflows. Typically this is expressed as

∑ t = 0 T x t ( 1 + r ) t ≤ ∑ t = 0 T w t ( 1 + r ) t , {\displaystyle \sum _{t=0}^{T}{\frac {x_{t}}{(1+r)^{t}}}\leq \sum _{t=0}^{T}{\frac {w_{t}}{(1+r)^{t}}},}

where x t {\displaystyle x_{t}} is expenditure at time t, w t {\displaystyle w_{t}} is the cash that becomes available at time t, T is the most distant relevant time period, 0 is the current period, and 1 1 + r {\displaystyle {\frac {1}{1+r}}} is the discount factor computed from the interest rate r. Complications are possible in various circumstances. For example, the interest rate for discounting cash receipts might be greater than the interest rate for discounting expenditures, because future inflows may be borrowed against while currently available funds may be invested temporarily pending use for future expenditures, and borrowing rates may exceed investment returns.

Applications In most applications, the entire budget would be used up, because any unspent funds would represent unobtained potential utility. In these situations, the intertemporal budget constraint is effectively an equality constraint. In an intertemporal consumption model, the sum of utilities from expenditures made at various times in the future, these utilities discounted back to the present at the consumer's rate of time preference, would be maximized with respect to the amounts xt consumed in each period, subject to an intertemporal budget constraint. In a model of intertemporal portfolio choice, the objective would be to maximize the expected value or expected utility of final period wealth. Since investment returns in each period generally would not be known in advance, the constraint effectively imposes a limit on the amount that can be invested in the final period—namely, whatever the wealth accumulated as of the end of the next-to-last period is.

Worked examples

Example 1 — a first encounter with Intertemporal budget constraint

Start with the simplest possible case. Write down what Intertemporal budget constraint claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Intertemporal budget constraint before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Intertemporal budget constraint ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Intertemporal budget constraint

In research: Intertemporal budget constraint appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Intertemporal budget constraint in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Intertemporal budget constraint is common in secondary-school and first-year university syllabi. It links to neighbouring topics Constraint programming, Intertemporal economics, Mathematical finance, so understanding it makes those chapters shorter.
In everyday life: Look for Intertemporal budget constraint outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Intertemporal budget constraint in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Intertemporal budget constraint means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Intertemporal budget constraint out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Intertemporal budget constraint in simple terms?

In economics and finance, an intertemporal budget constraint is a constraint faced by a decision maker who is making choices for both the present and the future. The term intertemporal is used to describe any relationship between past, present and future events or conditions.

Why does Intertemporal budget constraint matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Intertemporal budget constraint?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Intertemporal budget constraint.