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Step-by-step mathematical derivation with typeset equations and expandable detail.
Sections
Saving supply and investment demand
The loanable-funds model is the classical theory of the real interest rate in a closed economy. It treats the bond market (equivalently, the market for loanable funds) as the arena where savers supply funds and investors demand them. The real interest rate adjusts to clear this market. National saving is the supply of loanable funds: the portion of income not consumed by households or the government. It is an increasing function of because a higher real return makes current consumption more expensive relative to future consumption, encouraging households to save more.
Investment is the demand for loanable funds. Firms borrow to finance capital projects (factories, equipment, R&D) whose returns must exceed the cost of borrowing. As rises, fewer projects clear that hurdle, so investment demand slopes downward. The upward-sloping saving supply and downward-sloping investment demand intersect to determine the equilibrium real interest rate and the equilibrium quantity of saving and investment .
National saving is an increasing function of the real interest rate.
Investment demand is a decreasing function of the real interest rate.
Market clearing and the equilibrium rate
Equilibrium in the loanable-funds market requires that the quantity of saving supplied equal the quantity of investment demanded: . With a linear specification and , setting supply equal to demand and solving for yields . The equilibrium rate rises when autonomous investment demand () increases or when autonomous saving () falls.
The equilibrium quantity of loanable funds is found by substituting back into either the saving or investment equation. At , every dollar saved finds a borrower, and every dollar of desired investment finds a lender. If the interest rate is above , saving exceeds investment (excess supply of funds), and competition among lenders pushes down. If below , investment exceeds saving (excess demand for funds), and competition among borrowers pushes up. The market thus gravitates toward the equilibrium rate.
Loanable-funds market equilibrium: saving equals investment.
Equilibrium real interest rate with linear saving and investment schedules.
Equilibrium quantity of loanable funds, obtained by evaluating saving at the equilibrium rate.
Government deficits and crowding out
National saving equals private saving plus public saving: . When the government runs a deficit (), public saving is negative, reducing the total supply of loanable funds. In the diagram, the saving supply curve shifts leftward. At the original interest rate, investment demand now exceeds the diminished supply of funds, creating excess demand that bids up .
The higher equilibrium interest rate discourages some private investment projects that were previously viable. This is crowding out: government borrowing absorbs funds that would otherwise have financed private capital formation, raising the cost of borrowing and reducing investment. The magnitude of crowding out depends on the slopes of the saving and investment schedules. If investment is highly sensitive to the interest rate (flat curve), the crowding-out effect on investment is large. If saving is highly elastic (flat curve), the interest rate rises little and crowding out is mild. In the extreme classical case where saving is perfectly inelastic, the deficit raises by the full amount needed to crowd out private investment dollar-for-dollar.
National saving: private saving plus the government budget surplus (or minus the deficit).
A larger deficit shifts saving supply left, raising the equilibrium interest rate by an amount that depends on the sensitivity of saving and investment to the rate.