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Econ 299 Chapter2c - 2.2.2 Elasticities We have already...

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1 2.2.2 Elasticities We have already seen how the derivative, or the slope, can change as x and y change -even if a slope is constant, changes can have different impacts at different points -For example, given a linear demand for Xbox 360’s, a $100 price increases affects profits differently at different starting prices:
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2 2.2 Xbox 360 Example Price increase from $0 to $100 Xbox Demand 0 1000000 2000000 3000000 4000000 5000000 6000000 0 100 200 300 400 500 600 700 800 900 Xbox Price Xbox Demand New Income
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3 2.2 Xbox 360- Example Price increase from $500 to $600 Xbox Demand 0 1000000 2000000 3000000 4000000 5000000 6000000 0 100 200 300 400 500 600 700 800 900 Xbox Price Xbox Demand Old Income New Income
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4 2.2.2 Elasticities $0 to $100 Old Revenue: $0 New Revenue: 4.5 million sold X $100 each $450 million (INCREASE) $500 to $600 Old Revenue: 2.5 million sold X $500 each $1.25 Billion New Revenue: 1.5 million sold X 600 each $0.9 Billion (DECREASE)
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5 2.2.2 Elasticities -to avoid this problem, economists often utilize ELASTICITIES -elasticities deal with PERCENTAGES and are therefore more useful across a variety of points on a curve ELASTICITY = a PROPORTIONAL change in y from a PROPORTIONAL change in x Example: elasticity of demand: E = Δy/y / Δx/x = (Δy/Δx) (x/y) = (dy/dx) (x/y)
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6 2.2.2 Elasticity Example Let y=12x+7 Find elasticities at x=0, 5, and 10 1) dy/dx = 12 2) f(0) = 12(0) +7 = 7 3) f(5) = 12(5) + 7 = 67 4) f(10) = 12(10) +7 = 127 Next we apply the formula:
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7 2.2.2 Elasticity Example Let y=12x+7 Find elasticities at (x,y)=(0,7) (5,67) and (10,127) E = dy/dx * x/y 1) E (0)= 12* 0/7 = 0 2) E (5)= 12 * 5/67 = 0.90 3) E (10)= 12 * 10/127 = 0.94
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8 2.2.2 Elasticity Interpretation What does an elasticity of 3 mean? => for every 1% increase in x (or the independent variable), there will be a 3% increase in y (or the dependent variable) In our example, a 1% increase in x caused a: 1)0% increase in y 2)0.90% increase in y 3)0.94% increase in y Question: Is an increase in y good or bad?
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9 2.2.2 Inelastic vrs. Elastic: The Boxer’s or Briefs debate An elasticity of less than 1 (in absolute terms) is inelastic. That is, y responds less than x in percentage terms. An elasticity of greater than 1 (in absolute terms) is elastic. That is, y responds more than x in percentage terms. -This has policy implications…
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10 2.2.2 Elastic Xboxes Let q=5,000,000-5,000p Or q=5,000-5p Where q is Xbox’s demanded in 1000’s p is price of an Xbox Find elasticities at p=0, 100, and 500 1) dq/dp = -5 2) Q(0) =5,000 3) Q(100)=4,500 4) Q(500)=2,500
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11 2.2.2 Elastic Xboxes Find elasticities at p,q =(0,5000), (100,4500) and (500,2500) E = dq/dp * p/q 1) E = -5* 0/5000 = 0 2) E = -5 * 100/4500 = -0.11 (inelastic) 3) E = -5 * 500/2500 = -1 (unit elastic)
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12 2.2.2 Elastic Xboxes From these values, we know that demand for Xboxes 360’s is INELASTIC below $500 and ELASTIC above $500 How does this impact profits? Total Revenue = p*q(p) dTR/dp = q(p)+p*dq/dp = q( 1+p/q*dq/dp) = q (1+ E)
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13 2.2.2 Making Microsoft Money If E = -1, dTR/dp = 0; a change in price won’t affect profits If |E| < 1 (inelastic), dTR/dp>0, increases in prices increase profits If |E| > 1 (elastic), dTR/dp<0, increases in prices decrease profits Therefore, price increases are revenue enhancing up to a price of $500.
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14 2.2.2 Elasticity Production Rule If demand is inelastic Raise Price If demand is elastic Decrease Price If demand is unit elastic Price is perfect (usually)
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15 2.2.2 More Elasticity Examples Let q = 100-2p 1) Find Elasticities at p=5, 20 and 40 2)
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