"How can we guarantee that savings must be equal to investment?" In loanable funds market, demand comes from investment and supply comes from savings, so when the market reaches equilibrium, savings = = investment.
"If I use half of my money as savings and the other half as my own investment, that part of my investment is also state investment." In macroeconomics, investment is a proper term, which means "buying machines and related buildings for production, that is, production machines and buildings as production factors to expand production". When you say "invest", you mean what we usually mean, such as buying stocks or bonds yourself.
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