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I was reading Keynesian Economics and came across this relation: $$\frac CY > \frac {dC}{dY}$$ provided that $$Y = C+S$$ where Y is total income of an individual, C is Consumption of the individual and S is Saving of that individual.

Basically the relation is between Marginal Propensity to Consume (MPC) $(\frac {dC}{dY})$ and Average Propensity to Consume (APC) $(\frac CY)$ and it is given that $APC>MPC$.

I was wondering how to prove this mathematically that $\frac CY > \frac {dC}{dY}$

Edit 1: As pointed out in comments by @TonyK, this question is probably more of a real world assumption: "if your income doubles, then your spending will less than double". This argument looks right to me.

Edit 2: (Justification of above relation but not a mathematical proof) As pointed out in comments by @Jam, "Wikipedia also says, in the article on MPC that "In a standard Keynesian model, the MPC is less than the average propensity to consume (APC) because in the short-run some (autonomous) consumption does not change with income. Falls (increases) in income do not lead to reductions (increases) in consumption because people reduce (add to) savings to stabilize consumption. Over the long-run, as wealth and income rise, consumption also rises; the marginal propensity to consume out of long-run income is closer to the average propensity to consume.". Which explains where the assumption comes from."

ATK
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  • (You've mixed up marginal and average. Remember averages are quotients and marginal means derivative of.) Do $C$ and $Y$ satisfy some kind of equation or any other kind of relationship from which to derive? – anon Nov 23 '19 at 17:42
  • Is this mathematically provable? It looks like a real-world assumption: if your income doubles, then your spending will less than double. More psychology than maths. – TonyK Nov 23 '19 at 18:02
  • @runway44 I have corrected the same and added the base relation between $Y$ and $C$. – ATK Nov 23 '19 at 18:03
  • @TonyK I think you are right that it looks more psychological assumption than mathematics – ATK Nov 23 '19 at 18:05
  • Yes, you are right I think. I edited the question and removed those conditions. But now it becomes less of a mathematical question to ask here it seems. – ATK Nov 23 '19 at 18:10
  • My economics isn't strong enough to verify it but Wikipedia says (without a source) that APC can be more or less than MPC, depending on the shape of the MPC curve. (Link). – Jam Nov 23 '19 at 18:22
  • You are right. My class notes mentioned the same. I should probably delete this question now. – ATK Nov 23 '19 at 18:25
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    Wikipedia also says, in the article on MPC that "In a standard Keynesian model, the MPC is less than the average propensity to consume (APC) because in the short-run some (autonomous) consumption does not change with income. Falls (increases) in income do not lead to reductions (increases) in consumption because people reduce (add to) savings to stabilize consumption. Over the long-run, as wealth and income rise, consumption also rises; the marginal propensity to consume out of long-run income is closer to the average propensity to consume.". Which explains where the assumption comes from. – Jam Nov 23 '19 at 18:27
  • Yes, I just checked the same. I should have checked that before but I thought about how to prove it mathematically. – ATK Nov 23 '19 at 18:38
  • Can you put that in solution, I can accept that. – ATK Nov 23 '19 at 18:50
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    I think since not really a math answer, I shouldn't. But keeping the question up for future people with the same question as you should suffice imo. If you're still interested in the topic, this paper (Link) has a good overview and a further reading section. – Jam Nov 23 '19 at 18:57

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