Elasticity from mathematical demand curves. Before we saw linear and nonlinear demand curves. We return to them to get elasticity values from them. . Linear demand. A linear demand curve might be of the form Q x = a 0 + a x P x + a y P y + a M M.
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Before we saw linear and nonlinear demand curves. We return to them to get elasticity values from them.
A linear demand curve might be of the form
Qx = a0 + axPx + ayPy + aMM.
To evaluate the price elasticity of demand from a certain point on the demand curve you would need to have a Px and Qx point to start from. The elasticity is then axPx / Qx
As an example say we have
Qx = 1000 - 3Px + 4Py - .01M and we start at
Px = 1 and Q = 300. Then around the price = 1 the Ed = -3(1)/300 = -.01
Cross price and income elasticities are found in a similar way.
Note: take the coefficient of the relevant term, multiply by the original value of the relevant price or income and divide by the starting quantity.
Say when Py = 2 Qx = 400. From the previous screen, the cross price elasticity would be 4(2)/400 = .02 so we have an example of substitutes.
Demand may not be a linear function. A popular nonlinear form takes the form
Qx = cPxBxPyByMBMHBH. An example would be
Qx = 10Px-1.2Py3M.5H.3 . An interesting thing about this form is if you take the natural log (sometimes written Ln)of each side you get
log Qx = 10 – 1.2 log Px + 3 log Py+ .5 log M + .3 log H.
This nonlinear demand is said to be linear in logs.
When the nonlinear demand is written in natural log form the coefficients in the equation are themselves elasticities. From the previous screen the price elasticity of demand is –1.2, the cross price elasticity is 3, and the income elasticity is .5.
So from the previous screen good x is elastic in the range investigated, is a normal good and in regard to good y is a substitute.