Do Keyensians believe that it's wrong for households to aggresively save money? Many Austrian/Neoclassical websites argue that Keynes and Keynesians demonise saving and encourage individual people to spend rather than save.  Someone (not an economist) recently told me that I need to feel guilty for my frugal lifestyle and that I'm being unpatriotic and should spend more money to help the economy.  But I'm not convinced that this is what Keyensians actually believe, as opposed to a straw-man used by the anti-Keynesians.  I thought the point was not that it's damaging to the economy for individual people to save their money as aggressively as they want to, but rather that if there is a paradox-of-thrift type situation during a recession, the government should take corrective action to reflate the economy.  And I also thought that Keyensians believed that a higher household savings rate is always good in the long run, and that the debate was specifically about what happens during a recession.  

Is my friend right or am I?  Is it really true that there is any significant school of economists who would think that individual people saving their money prudently was bad for the economy and that such people ought to feel guilty for being savers?  Or can I carry on saving aggressively guilt-free?
The issue is more complex than your friend suggest and he is not completely right but at the same time there is a kernel of truth to what he is claiming. There are differences in the effect of savings short and medium and long run. However, to fully explain this I will have to use some math to ground the reasoning and make everything consistent. 

Consider simple closed economy. By definition in any closed economy the total output/income $Y$ must be equal to consumption $C$, investment $I$ and government spending $G$. This is just accounting and will always hold so we have: 

$$Y = C+I+G$$

Now we know from basic economics that consumption must be some positive function of disposable income $C=C(Y_d)$ with $C'&gt;0$. This is just common sense if you have more money avaiable you will consume more goods and services in general. This is consequence of the basic non-satiation assumption - more is always better on which most economics builds on. To make things simple let us assume that the relationship is linear so we have:

$$C=c_0 + c_1Y_d$$

where $c_0$ is some basic consumption that you have to undergo irrespective of income to just stay alive and $c_1$ the marginal propensity to consume, i.e. the share of income you use for consumption as opposed to saving. Now by definition disposable income is the income after taxes $Y_d = Y-T$. So in fact our consumption function turns into:

$$C= c_0 + c_1 (Y-T)$$

Second we will hold investment constant which reasonably holds in the short run (but not necessary in the medium long run) as most investments are not very liquid so we have $I= \bar{I}$. Government spending will be treated as completely exogenous (determined outside this model) so it stays $G$.

Now to solve this model we can just substitute consumption function $C$ into the first equation which gives us:

$$Y = c_0 + c_1 (Y-T)+\bar{I}+G$$

Now solving for $Y$ using basic algebra we get the following:

$$Y = \frac{1}{1-c_1} \left(c_0  +\bar{I}+G-c_1T\right)$$

Now this last equation is all you need to understand the paradox of thrift. First consider the second part of the equation $\left(c_0  +\bar{I}+G-c_1T\right)$. This part is called autonomous spending because it does not depend on income as $c_0$ is basic consumption of necessities  $\bar{I}$ is fixed investment and $G$ and $T$ is government spending and taxation which is exogenously given so in this model it wont change. 

So now you must be wondering where is the effect of saving on income? Well it hides in the first term:

$$\frac{1}{1-c_1} $$ 

recall $c_1$ is the share of income you spend on consumption - since you can only consume or save $(1-c_1)$ is the share of income saved. 

Now its trivial to see that as $(1-c_1)$ increases the $Y$ must fall holding everything else constant. So your friend is definitely right that in the short-run saving more - meaning saving larger proportion of your income not just saving more in dollar terms - leads to lower income.

However, here we set investment constant, which is reasonable in the short run, but not in the long run. Just as a matter of accounting identity the investment must be equal private $S$ and public $T-G$ savings so we have:

$$I=S +T-G$$

So increasing saving also increases income. Furthermore, saving is also necessary for economic growth. The reason for this is that economic output depends on factors of production like capital and labor. For simplicity lets focus just on capital. We know that the more capital economy has the more it can produce so we know that $Y = F(K)$ with $Y'_k&gt;0$. So the national income increases with amount of capital avaiable.

Now how do you get more capital? By saving! Again just by pure accounting identity the stock of capital tomorrow $K_{t+1}$ must be equal to capital today net of depreciation plus investment:

$$K_{t+1} = (1-\delta)K_t + I_t$$

We already know $I=S+T-G$ lets set taxes equal to gov. spending to simplify we get $I=S$. Now going back to the original equation on consumption since $c_1$ is marginal propensity to consume $1-c_1$ must be marginal propensity to save so we can replace $I$ by savings which are $S=(1-c_1)Y$ to get

$$K_{t+1} = (1-\delta)K_t + (1-c_1)Y$$

So in the long run you get more capital accumulation and economic growth when propensity to consume $c_1$ is small or consequently when share of income saved $1-c_1$ is large. So your friend is definitely not correct if he claims that being thrifty is always bad for the economy or that (New) Keynesians claim that. 

PS: This was not part of your question but note that while some pundits still use words like Neoclassical and Keynesian as synonyms for free market pro saving economists and pro more government regulation spending economists respectively, this distinctions no longer hold. The current mainstream in economics is  Neoclassical synthesis which is combination of Neoclassical micro models and Keynesian macro models as well as some other ideas. So nowadays words like Keynesian or Neoclassical dont carry the same meaning. For example, Mankiw is a New Keynesian but he is small government, pro saving, conservative/libertarian while Stiglitz work is neoclassical and he is pro big government, pro spending, liberal. The reason for this is that both the neoclassical and keynesian models are highly nuanced and the pro-spending vs pro-saving debate depends heavily not just on science but also on personal values, the ways how a person discounts future benefits of economic growth and all this discussion gets more complex once you include inequality and philosophical discussions around that.