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I have in a problem shown that for the IS curve (Y) contra the GDP in equilibrium ($Y^*$) it applies that: $\frac{\partial Y}{\partial G}>\frac{\partial Y^*}{\partial G}$. Where G is public consumption. I think my calculations make sense. But what will the economic interpretation of this result be? Can anyone help me?

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So graphically this is pretty straight forward, as you may have already understood: Since the LM curve has a positive slope the entire shift of IS curve ($\frac{\partial Y}{\partial G})$ is not fully translated into final output. It would happen when LM curve is flat (which is usually the case at very low interest rates in liquidity trap situation).

Interpretation: As govt's consumption/expenditure increases at a given interest rate, it increases output. But higher output requires more money circulating in the economy to sustain the higher output. Since no additional money is injected into the economy this additional money demand is met from the speculative demand for money. This means that speculative demand should come down.

The decrease in speculative demand comes from increase in interest rates. (Mechanically, this happens because as govt floats more bonds into the market (for expanding its borrowing), it triggers a speculation of increase in interest rates which, in turn, triggers the money held by speculators to invest back in the economy - increasing the money available for transactions). Further there are more bonds in the market too causing the bond prices to fall and interest rates to increase (wealth effect channel).

But as the interest rates goes up, the private investment demand comes down (called - crowding out of private investment) ultimately making final increase in demand ($\frac{\partial Y^*}{\partial G}$) to be lower at equilibrium as compared to $\frac{\partial Y}{\partial G}$.

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