I was wondering if I put $1000 in the piggy bank for a year and the inflation being 10%, and everything else being constant, 1 year later that $1000 in cash money can only buy goods that today would cost me $900. So my question is: where did that $100 in value go?
The core understanding of economics that you seem to be missing is that there is no value inherent in money. It has exactly as much value as people decide it is worth, no more, and no less; that is, money has only perceived value, and that value is therefore subject to the perceptions of the people who exchange goods and services for that money.
Inflation, then, is what happens when people collectively decide that money is no longer worth as much as it was yesterday. It's important to note that there is no one person who can make that determination; if one store owner decides that money is worth less, they can raise prices at their store to compensate for that perceived devaluation, but the implications of this decision are minimal, as customers can go to other stores whose owners value their money more. However, when many store owners have the same perception and raise prices in concert with each other, then you have inflation.
There are many factors that people take into consideration when deciding how much money is worth, including supply and demand (the idea that things are worth more when they are scarce and people want them), interest rates (the price of money, or how much the banks charge for lending money or pay for holding deposits), government fiscal policy (including tax rates, government spending, and financial responsibility — see also Liz Truss's disastrous mini-budget in the UK on that last), and the current state of the economy.
In summary, the value of money is subject to change in the same way the value of anything else is. There is nothing magical about money that exempts it from market forces, and there is no "universal law of conservation of value" that must be obeyed by money nor for any other commodity.
Pause on thinking about the concept of inflation as a whole, because I think you are considering multiple different concepts at once. Instead, think about your personal life, with your $1,000 piggy bank - let's plan out the cost of a hypothetical weekend vacation you might be planning to go on next month.
You buy your flight today for $300. You look at the cost of hotels in the area (looks like you can get a room for both nights at $250 total), and decide that you want to be flexible on where you stay, so you don't book anything in advance. You see there's a sightseeing tour there, and reserve a slot for yourself - looks like tickets will probably cost you $200 once you get there. That leaves you with $250 for food / entertainment costs - after doing some budgeting, it seems that should be perfect.
Then once you get there, what happens if things are more expensive than when you planned your trip? Perhaps fuel costs increased over the month, but you already bought your plane ticket, so no impact to you there. Maybe a bunch of hotels are all booked up, and the remaining ones bumped up their prices - now it will cost you $300 for hotels (not a lot you can do about this, as you didn't pay in advance, and you have to sleep somewhere). And the sightseeing tour became very popular, so it won't cost you $200, it would cost you $300 - you decide to cancel your reservation as it's too expensive. Restaurants in the area you're staying have also increased their prices - so you decide to buy food from a grocery store and eat some of your meals in your hotel room, instead.
'Inflation' is basically an estimate to show how much more expensive things become over time. Some things might impact you, some might not. Sometimes you can adjust your spending habits to compensate, and sometimes you can't.
When most people talk about 'inflation', they are basically talking about a country's 'Consumer Price Index', which is a specific way to calculate what a 'basket of goods' cost yesterday vs today. So like a liter of gas, a loaf of bread, a pair of shoes, etc. - add up an estimated mixture of what you might spend money on, and show what it cost yesterday vs today.
That money didn't "go" anywhere - it just can't buy what you thought it could previously. Below are a few reasons why this might be the case.
Individually, some businesses that now charge you more might simply be increasing their profits (https://globalnews.ca/news/9098447/canada-inflation-grocers-profits/ is an article describing how Canadian's are paying significantly higher food costs over the past 2 years, and the large grocery chains are raking in record profit). Some businesses might be directly covering their own increased costs. Maybe consumers are starting to demand higher quality goods, and the cheapest possible methods of manufacturing need to get replaced by more expensive methods.
Simplistically, the money in circulation in a country represents the value of all 'productive assets' within that country. If a country's real economic output drops, but the amount of money available stays the same, then each $ won't buy as much. In that case, 'where the money went' is the idea that there are simply fewer real goods to go around, being chased by the same amount of $ in circulation. This is a decrease in supply of goods.
As a final alternative, maybe your government just printed $1,000,000,000,000 - now the number of goods being produced is the same, but there are more $'s chasing those goods. This is an increase in the supply of money.
Remember that money is just a convenience. Instead of money, everyone could trade goods and service -- this is a barter-based economy.
So imagine instead of having $1,000 dollars in the piggy-bank, you're a farmer with 1,000 lbs of wheat in your silo. And when you want to buy something, you give them some of your wheat.
Suppose last year, you could buy a dozen eggs for a pound of wheat. But then there's a disease that kills off lots of chickens, so there aren't as many eggs to go around. So the chicken farmers raise the price of eggs. And the people who sell products made with eggs (e.g. bakers) also have to raise their prices. Now that same pound of wheat can only buy 11 eggs, and buying a cake will cost more wheat.
That's inflation. The 1,000 lbs now buy what would have cost 920 lbs a year ago. But those 80 lbs of wheat didn't go anywhere -- they're still in the silo, but you can't do as much with them.
The value of money lost to inflation doesn't really "go" anywhere. This is a misconception.
But there is something called seigniorage, that is almost as you meant in the question. That is, the very act of creating (minting, printing) money somewhat transfers purchasing power from all money that already exists to the newly minted/printed money, akin of debasement of coinage.
As Milton Friedman famously said:
Inflation is always and everywhere a monetary phenomenon, in the sense that it is and can be produced only by a more rapid increase in the quantity of money than in output.”
According to the so-called Cantillon Effect the missing value of the money is transferred in the course of the inflation to the institutions which are closest to the newly created money.
That means Banks, Government, and also e.g. people taking out a mortgage. Since new money is generated using Loans and Debt.
: [Milton Friedman, Inflation Causes and Consequences, Asian Publishing House, 1963.]
Provided the amount of money in circulation is static, there would be deflation since global wealth is rising every minute (through discoveries, improvements, trade, etc.). Also, some actors are adapting to the decreasing value of money through so-called shitflation and shrinkflation. That means the institutions controlling the amount of money in circulation (Governments, Banks etc.) are actually extracting much more from the economy than is the perceived inflation.
shitflation - lower quality goods for the same money
shrinkflation - lower quantity of goods for the same money
As others have said, after one year, the $1000 in your piggy bank will have $909.09 worth of spending power. On the flip side $1000 owed to someone will also be worth $909.09 after one year (interest considerations aside). So one way to think about where the value goes, is from savings to debts.
The same is true when governments print money (quantative easing). The value of a dollar, say, goes down, meaning that the value of savings, and debt, goes down. It's as if the government took money from your savings account to pay their debts (and anyone else who owes more than they have in savings). See also @Kevin's comment.
If you could buy something for $1000 and its value rose in line with inflation you could avoid the loss. I think this is why gold is attractive at during periods of high inflation.
Inflation happens when the government just prints more money so they are now worth less. This explanation is called monetarism. This may look like a nasty thing to do but there are arguments that small controlled inflation encourages people to buy and consume more rather than save, or to invest money somewhere. Both are useful for economy. Hence slow and steady rate of inflation is actually favored by the most of economists.
At the times of crisis, it is not unusual for the government to print enough money that all savings become worthless, essentially nationalizing them. It was one of the reasons why keeping dollars at home was illegal in the former Soviet Union: it was required to keep savings in roubles, so that the government could easily take them away without even visiting you, by just printing more roubles. Eventually they did. They also did with Czar roubles during revolution. Young post soviet states did that with they first iteration of currencies (not the second, and not with Euro that followed).
Should I think this happened because everybody stopped valuing roubles without any obvious reason or there were other very complex reasons beyond my understanding that turned all my money into rather bad toilet paper?
From Wikipedia, it looks like there are other explanations for inflation as well (Keynesian, etc), but after reading through them I do not really get how do they exactly work without printing more money, and why do we need other explanations after more money have eventually been printed.
I belong to the family where three generations in a row have lost all savings due inflation, hence would afford to consider these other explanations just PR.
Inflation has multiple causes, some are short-term and some are long-term. Currently we have a big inflation because of yet another fuel shock - multiple countries sanctioned Russia because we (temporarily) don't want to buy fossil fuels from war criminals. In that case the value of your money didn't evaporate - you just need to spend more money on energy, which became more valuable now... That is unfortunate for most people but if you own solar panels, windmills or a coal-based power station, that's good for you. Of course we wouldn't have any fuel crisis if we actually cared about environmental sustainability and energy security, but seems that we don't.
There are cases where natural resources are depleted or not usable (i.e. water shortage, soil salinity, storms) and in that case the increase of food price is higher because again, raw materials got more valuable or products are destroyed.
There are cases like COVID where people are unable to work, while still getting (government) money, and in that case there's a lot of money, and a lot of demand for goods, services and investments, while not enough work is done, which reduces supply of good and services... and the prices must go up. In that case, if your model of work wasn't affected by COVID (i.e. you work from home on your computer and you're more productive than ever because you are happier and you don't need to commute), the value of your money gets diminished, because now in the grocery stores or electronics shops you compete not only with other workers, but also with non-workers (people and corporations) who receive handouts while you are taxed.
And speaking of taxes, since gold has been confiscated from the population in 1933 in USA, and gold standard lifted in 1971, there is another long-term driver of inflation, which are the central banks. Using government money is mandatory. Central banks around the world mostly have a yearly inflation target of 2%, which means that on average peaceful crisis-free year they intend to decrease the value of your piggy bank by 2%, by printing money and buying assets with them, or by printing money and handing it out to poor people and rich people alike. Of course, it's under the pretense of stimulating the economy by making cash-hoarding cost 2% yearly, while strengthening demand, and sure it does increase demand, but that means that prices go up - and value of your cash goes towards government programs, not toward your personal goals.
Governments try to protect people from price increases by imposing minimum wages on business and it does work to some extend (workers in rich countries are as rich, as specialists in poor countries), but also it has some side-effects, like outsourcing work to far-away countries, or stimulating automation and robotization. Which is good if you ask me, but will eliminate unqualified work, which again forces government to issue handouts, or pay for education, or allow masses of people to go poor.
Inflation is defined as the rate at which the prices of goods and services rise over time. The main reason for inflation is the increase in the money supply. When the money supply increases, each dollar becomes worth less because there are more dollars chasing the same number of goods.
In your example, if the inflation rate is 10% and you put $1000 in the piggy bank for a year, then one year later that $1000 will only be able to buy goods that cost $900 today. The $100 in value has been lost due to inflation.
In short it is partially a tax. https://en.wikipedia.org/wiki/Seigniorage#Seigniorage_as_a_tax And it can partially happen due to market shortages.
One of the tools used by Federal Reserve to manipulate money supply are Open Market Operations. In short, central bank can create new money, and buy securities with them.
What does that mean for you? It means that price of securities is inflated, while the price of dollars in your pockets is decreased.
In essence, they can print money, and buy stuff with them, while money in your pocket loses value. The value of your money goes towards stocks or bonds owned by Federal Reserve / Central Bank.
TLDR: it goes into government programs
If you were free to use gold in everyday transactions, there would be deflation. Each unit of gold earned earlier in the pioneer times becomes more valuable each decade, because creating value is infinitely easier now with developed economy, to which the ones working years ago contributed.
Sadly we're forced to use fiat currency now. And thus the value goes to the government. Government can take out debts, and them pay them back by printing money. So it seems like they provide more public services than they charge in taxes. The difference is inflation.