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# chapter4B - Total Revenue Marginal revenue and Price...

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Total Revenue, Marginal revenue and Price Elasticity TR=PQ MR=dTR/dQ TR is max when MR=0, and η= -1

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There is an important relationship between the price elasticity and total expenditure on a good. If demand is price-elastic, MR is positive and decreases in price result in higher TR If demand is price-inelastic, MR is negative and increases in price result in higher TR TR is max when demand is unit-elastic, and MR = 0, that is, dTR/dQ = 0 (FOC). Prove that MR=0 when demand is unit-elastic for a linear D curve: see pp. 100
The Relationship Generalized We can extend our analysis to a general demand curve It follows that total revenue is Also marginal revenue is 0 / ), ( ) 4 ( < = dQ dP Q f P Q Q f Q P TR ) ( ) 5 ( = = + = + = + = + = η 1 1 1 1 ) ( ) / ( 1 ) ( ) ( / (6) P P Q f Q dQ dP Q f Q f Q dQ dP MR Q P dP dQ

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Demand Elasticity and MR Notice that: If η=-1 then MR=0, and FOC for the existence of a max TR are met. If η<-1, then MR=positive, and if η>-1, then MR=negative The MR formula is useful for understanding monopoly and firms with market power. -1 as P MR B -1 as P MR (7A) < + = + = η η η η 0 1 1 ) 7 ( 0 1 1
bQ a P - = ) 1 ( 2 ) 2 ( bQ aQ Q P TR - = = Another relationship between the price elasticity and MR marginal revenue always falls exactly twice as fast as demand for the case of linear demand.

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