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Are the two models compatible?

Consider the situation: consumers lost their confidence in the economy so that they consume less and save more, hence the IS curve shift down. Since the consumers save more, there should be more loanable money in the bank.

So, why doesn't the LM curve as well as the interest rate shift down when there is more lonable money?

(I can tell the two models apart.But I think it is natural for the interest rates to go down when people save more/consume less)

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The LM-curve is completely independent from the loanable funds market. This is because the loanable funds market is entirely determined by the real side of the economy. Therefore, it is linked to the IS curve, not the LM curve. Even more, equilibrium in the loanable funds market for given $Y$ allows you to derive the IS curve, just as the keynesian cross allows you to do.

To see this, consider both relationships in the loanable funds market. The demand side of loanable funds is given by the demand for investment (negatively related to the real interest rate $r$) and the supply side is given by all savings available (positiveley related to $r$). For equilibrium it is required that $S=I$ holds. Which is exactly the equilibrium condition represented by the IS curve: what are the combinations of $r$ and $Y$ for which savings equals investment.

To derive the negative IS-relationship from the loanable funds market, just go through the following reasoning. Suppose that the loanable funds market is in equilibrium. This equilibrium holds for a given $Y$. If $Y$ increases, and the marginal propensity to consume is smaller than unity, the level of savings will increase for each $r$, so that the savings curve shifts to the right. More income does not affect investment, so that a new equilibrium requires a lower $r$. Conclusion: if $Y$ increases, a lower $r$ is needed for $S=I$ to hold. This is the IS-curve.

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To my understanding, in the IS-LM model you have nominal interest rates, whereas in loanable funds model you have real interest rates.

The LM in the IS-LM model is the money supply, rather than money available to be loaned out. It is determined by the monetary authority and not by the economic conditions, hence it doesn't shift when there is an increase in savings.

Also, intuitively, the monetary authority should not be able to affect the real interest rates (the fact that there is more money in the economy, doesn't mean that there is more value created by capital).

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