The “margin” refers to the consequences of a small change (often a one unit change) in a key variable.
For instance, suppose you consider whether to put more effort into your work.
There may be a marginal benefit of an extra hour of work. This can be better quality of work or higher wages depending on the work.
Yet there may be a marginal cost. For example, the extra hour of work requires more effort. It also takes time away from other activities.
The “marginal” equations relate to links between the marginal benefit and marginal cost of actions.
So, here are six different marginal equations and their real world uses:
Marginal benefit and marginal cost
One fundamental equation is:
Marginal benefit (MB) = marginal cost (MC)
Here are some specific examples of how it works:
1) Firms
- If a firm increases its output by one unit, it receives extra revenue. This call this marginal revenue or MR. This is the marginal benefit from increasing output.
- But to increase output, a firm must pay its workers more, or keep its machines running longer, or provide raw materials. To increase output by one unit, the firm thus incurs a cost, the “marginal cost” or MC.
- Under assumptions on the behaviour of MR and MC, firms maximise profits when MR=MC.
- If MR > MC, the firm can increase its production to increase profits.
- If MR < MC, the firm can lower production to decrease profits.
2) Consumers
- A consumer may get satisfaction (“utility”) from consuming one more chocolate bar. This is the marginal benefit or marginal utility here.
- But the consumer must pay the price of the chocolate bar. So the marginal “cost” here is the price.
- Thus the consumer consumes until marginal utility = price.
- Note this assumes diminishing marginal utility. As you consume more, each extra unit of a good delivers less and less extra satisfaction.
- Also this needs to be adjusted for income. Income may be the limiting factor in a consumer’s decision.
- A more complex version of this idea is the equi-marginal principle. This takes into account multiple goods.
3) Policy, externalities and social welfare
- A policy maker may want to maximise “social welfare”.
- One way to define social welfare is total social benefit – total social cost.
- Social benefits include private benefits and external benefits to third parties not involved in a transaction. Similarly for social costs.
- When is social welfare maximised? When the marginal social benefit equals the marginal social cost (MSB=MSC).
- If MSB>MSC, we can increase social welfare by increasing the quantity of the good. This often occurs in the free market when there are positive externalities, such as with education (consumption externalities) or with firms training workers (production externalities).
- If MSB<MSC, we can increase social welfare by reducing the quantity of the good. This can occur in the free market with negative externalities, such as with car traffic (consumption externalities) or factory pollution (production externalities).
- This can be used to analyse market failures and government policy. For example, what taxes should firms face, so that they take into account the external costs of pollution?
- This approach does make assumptions about welfare. For instance, it is likely to involve a “utilitarian” approach to defining social welfare. We may want to define social welfare in an entirely different way. What is social welfare?
4) Equi-marginal principle (the return on investment)
Marginal benefit = marginal cost.
This assumed one choice variable. For example, how many chocolate bars to consume to maximise utility? Or, how many cars to produce to maximise profits?
How about when there are two or more choices?
The example of consumer utility – the equi-marginal principle
We have a second economics equation for this.
Suppose a consumer can consume two goods, apples or bananas.
Then, the consumer maximises their utility (under some conditions) when:
What does this equation mean?
Marginal utility = extra utility (satisfaction) from consuming one more unit of the good.
Each side of the equation describes the “value for money“, a.k.a. the “return on investment (ROI)” or “bang for your buck”.
In other words, the extra utility gained from spending one more pound on apples.
The consumer maximises their utility when they get the same “bang for their buck” or ROI on spending on each good.
Suppose there is higher ROI on apple spending.
Then the consumer should spend more on apples. Less on bananas. Until the equation above holds again.
Note this assumes diminishing marginal utility.
Another application – effort allocation for a firm and consulting
When managing a business, you may have to choose where to focus your workers or effort.
Should your workers focus on a) product development or b) branding?
That’s just one example of the many choices firms and managers face.
There are many other possible tasks too, such as sales, accounting or human resources.
But let’s run with the product development versus branding example.
Suppose the ROI of having an extra worker on product development is currently higher than the ROI for an extra worker for branding.
Then the manager may want to move more workers toward product development.
This kind of thinking can be critical to consulting.
For example, consulting firms may advise managers how to prioritise work based on its ROI.
Note this is not just which activity delivers greatest total benefit.
Instead it takes into account A) the amount of resources already deployed to that activity and B) the effects of small changes, in other words marginal benefits.
Key takeaways from the equi-marginal principle
- Put more effort into tasks which generate greater marginal benefit for the same extra time, effort or monetary cost. Less effort into tasks with lower marginal benefit for the same extra cost.
- Put more effort into tasks which generate the same marginal benefit at lower time, effort or monetary cost.
- More generally, put more effort into tasks that generate greatest “return on investment“.
- Bear in mind the limitations of the equi-marginal principle. Other factors will matter for effort / allocation decisions.
5) Marginal equations over time – the intertemporal Euler equation
Suppose a consumer has an amount of income. They can choose to spend this income today or in the future. How do they spread their consumption over time?
The key result here is the “intertemporal Euler equation”. Intertemporal meaning over time.
Namely, the marginal cost of saving today (foregone consumption) equals the discounted marginal benefit of saving today. The latter is the interest earnt on saving, converted into consumption utility.
This is yet another application of marginal benefit equals marginal cost.
This leads to the idea that consumers “smooth their consumption” over time. The prediction here is that consumption will stay fairly constant in the face of temporary income changes. In the face of permanent income changes, consumption levels will adapt over time.
Indeed consumption tends to be much less volatile compared with investment (as a % of GDP).
This may relate to other intertemporal decisions. For instance, deciding how much to draw down from a retirement fund.
6) How margins relate to taxes – the Saez tax formula and the Laffer curve
What level of income taxes should the government set?
Well, what are the effects of a small rise in income tax rates?
There are three key effects:
- The “mechanical” effect (ME). Provided people work the same hours for the same pay, a higher tax rate “mechanically” increases tax revenue.
- The “behavioural effect” (BE). A higher income tax rate reduces the incentive to work. This reduces pay and has a negative effect on tax revenue.
- Finally the “welfare effect” (WE). Consider a government who wants to lower income inequality. Higher income taxes, particularly higher up in the income distribution, may reduce income inequality.
Suppose a government only cares about maximising revenue. Then the revenue-maximising tax rate occurs when ME+BE=0.
If ME+BE>0, then raising the tax rate increases revenue. But if ME+BE<0, the government should decrease the tax rate to raise revenue.
Suppose the government considers inequality. Then the welfare-maximising tax rate occurs when ME+BE+WE=0.
There will be similar tradeoffs in other models with behavioural and inequality effects.
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