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elasticity deffinitions aec 303_Page_3

# elasticity deffinitions aec 303_Page_3 - If the Demand...

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Unformatted text preview: If the Demand curve is set up to have Price on the left hand side and Quantity on the right hand side, then instead of QdZAPb the demand equation would be Pb = (1/A)Qd Therefore P = (1/A)“de”b The elasticity is still the same, but the coefﬁcient that Qd is raised to is the inverser or 1 over the Demand elasticity not the demand elasticity. So pay close attention to whether price P is on the left or right hand‘ side of equals sign. If price is on the left hand side the power to which Qd is raised is the Elasticity of Demand for all values of Qd. If the equation is set up so that Qd is on the left hand side and P is raised to a power,- then the correct Elasticity of Demand is 1 over the power to which Qd is raised. For example, if Qd =AP'2 than we know that the Elasticity of Demand is —2 everywhere on this demand curve since Qd is on the left hand side of the = Sign. But if P=CQd'°'5 We know that the Elasticity of Demand also must be -l/0.5 0r —2. These are actually the same demand curves but the ﬁrst has Qd on the left and the second has P' on the left. Also A in the ﬁrst Demand curve is equal to =(1/C)'0‘5 Tricks with logs: There are even simpler ways to prove that for any (demand) function of the general form Qd=APb that the elasticity of demand will be equal to the coefﬁcient b. Recall from basic calculus that for the function y= log x that the derivative of log of x = dy/dx = l/x. Now suppose that we have a function of the form (4) Qd=APb Let’s ﬁrst take logs of both sides (5) log(Qd)=logA+ blogP Don’t be alarmed here as log(Qd) is just a variable. We could call it y or 2 if you want to. LogA is a constant, since A is a constant like c. The derivative of any constant is always zero. We want to ﬁnd dlode/dlogP Think—First lets just call this equation y = c + bx You wouldn’t have any difﬁculty ﬁnding dy/dx for this equation, since the derivative of c is zero and the derivative of bx is simply b. Same deal here exactly to ﬁnd the derivative of equation 4 dlog(Qd)/dlog(P) = 0 +b = b = Ed for Q . Note that in equation (4), b is the power to which P is raised! ...
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