Macroeconomics For Dummies®, uk edition Published by: John Wiley & Sons, Ltd

Looking to the household’s future

One way of discovering the NPV of £100 next year is to ask another

question: how much money would you need today in order to turn it into

£100 next year? Well, if you have £X today and you save it at interest rate r,

next year you have £(1 + r)X. Setting this expression equal to £100 gives you

the following:

That is, you need exactly £100/(1 + r) today in order to turn it into £100 next year, which is why the NPV of £100 next year is £100/(1 + r).

Applying this logic more generally, the NPV of £X in t years’ time is equal to:

Notice that the farther away in the future something is, the more heavily it’s discounted. That is, the NPV of £100 in two years’ time is

less than the NPV of £100 in one year’s time. But they’re both much larger than the NPV of £100 in 100 years’ time!

So what the earlier equation

is really saying is that the NPV of all the household consumption better be equal to the NPV of all your income, otherwise:

If the NPV of consumption is greater than the NPV of income: The consumption stream (the consumption over time taken as a whole) is unaffordable.


If the NPV of consumption is less than the NPV of income: The

household could increase its consumption.

Moving from a household to the government

The government is much like a household in that every year it gets some income in the form of taxes (T ) and every year it has to decide how much to spend (G ). Like a household, the government can choose to spend less than it earns and save the difference or (more commonly) spend more than it earns and borrow the difference. So, from year-to-year, no reason exists for T = G.

But the government’s intertemporal budget constraint does put a constraint on what kind of fiscal policy is feasible. It says that the NPV of all the tax revenue better equal the NPV of all its spending:

On the left-hand side of this equation is the NPV of an infinite stream of government purchases and on the right-hand side is the NPV of an infinite stream of taxes. (You may think that assuming that the government is going to be around forever is a bit far-fetched, but because knowing how long is impossible and because cash flows very far in the future have a negligible NPV, the simplification is useful.)

Therefore, looking at this equation you can clearly see that increasing

government spending today (G₁) must either coincide with a fall in future government spending or an increase in taxes at some point. Equally, a fall in taxes today (T₁) must be made up for either by an increase in taxes in the future or a fall in government spending at some point. In short, there’s no free lunch (though we hear the MPs’ restaurants in the Houses of Parliament come close!).