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What would happen to nominal income and wages if the money supply were fixed (100% reserve banking). Would real wages increase, even though nominal wages would stay the same or even decrease?

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    $\begingroup$ What would most likely happen is that new private short-term debt instruments would pop up, and become the de facto money supply, which can grow. How do the authorities react to this? $\endgroup$ – Brian Romanchuk Mar 22 '19 at 1:19
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Roughly speaking nominal wages should remain fixed but things would still get invented and production would become ever ore efficient. So real wages would increase as deflation boosted the purchasing power of your wages.

However, it may not be such a good idea to have a completely fixed money supply and deflation - it may become unstable if people noticed that storing money under your mattress was a perfectly effective way to save. If many savers started doing this then it could induce cycles of bubbles and crashes in the value of money. It would probably be better to deliberately increase the money supply by a few percent each year - just enough so that people were not inclined to save simply by storing money.

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As Mick explains in the first paragraph of his answer, real wages would gradually increase along with productivity.

However, the fear that this effect would destabilize the economy by causing everyone to stuff their matterses with money and never buy anything is unfounded. The thought is that if goods are constantly becoming cheaper and better, nobody would anyone spend any money now if they can get more in the future.

This overlooks the fact that people by default have a certain time preference - why would I care that bread becomes cheaper 1 year from now if I am hungry now? In cases of decreasing prices, people will save until the good is cheap enough to be more immediately valuable than whatever else could have been bought in the future.

A good example is personal computers/smartphones, which have become exponentially better with time. People who want the latest model for important work or personal prestige have a high time preference and buy even though a better version comes out next year. Students tend to wait for sales where they can buy last year's model at a discount.

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