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Does decision of Individuals to save money affect the economy of a country negatively? I think so yes, because if a high earning individual refuses to spend money then it stop the 'gears' of the economy from turning.

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The answer is that it depends how the money is saved and whether we talk about short-run or long-run.

in the short-run

Saving in form of investing in stocks and bonds or in other form that increases investment spending on productive activities does not hurt economy.

However, saving by hoarding cash under the mattress, or saving by firms in form of increasing their inventory of goods, or investing into companies that will just ‘sit on the cash’ is bad for the economy.

This is because investment spending has the same multiplier as consumer spending:

$$Y= \frac{1}{1-c_1}\left(c_0 + I + G \right) \tag{*}$$

Where Y is output, $c_1$ marginal propensity to consume, $c_0$ autonomous consumption, $I$ investment and $G$ government spending.

Regular saving has no effect on $Y$ because decreasing $c_0$ will be offset by increase in $I$. However, in case of saving in form of inventory or cash under mattress $Y$ contracts and so the increase in savings is offset by fall in $Y$.

What matters more than the total saving is the fraction of income saved as it affects the size of the multiplier.

in the long-run

In the long-run the equation * which describes aggregate demand does not matter. No matter what aggregate demand is the aggregate supply is perfectly vertical so the output is fully determined just by productive capacity of the economy. That is in long run $Y=f(K,L)$ where $f()$ is production function that depends on inputs of capital and labor.

In the long run saving has either no negative effect but no positive effect either or positive effect. This depends on whether you believe growth is exogenous or endogenous (i.e. whether saving has som effects on the production technology or not). This is not settled question yet (see Romer Advanced Macroeconomics ch 3).

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PS: If we just talk about single individual and not extremely large sums of money not group of individuals doing the same thing the effect, positive or negative, will be negligible.

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