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Economics Homework Ten - Model (view source)
Revision as of 22:44, November 26, 2009
, 22:44, November 26, 2009proof complete
:::P<sub>ed</sub>=((Q<sub>2</sub>-Q<sub>1</sub>)/Q<sub>1</sub>)/((P<sub>2</sub>-P<sub>1</sub>)/P<sub>1</sub>)
:::P<sub>ed</sub>=((Q<sub>2</sub>-Q<sub>1</sub>)/Q<sub>1</sub>)/((P<sub>2</sub>-P<sub>1</sub>)/P<sub>1</sub>)
::But the above formula is simply an approximation, and the higher precision provided by calculus is needed to prove this rigorously. So we rewrite the above equation using calculus:
::But the above formula is simply an approximation, and the higher precision provided by calculus is needed to prove this rigorously. So we rewrite the above equation using calculus, such that P<sub>ed</sub> equals:
:::<math>P<sub>ed</sub>=\frac{dQ}{dP}\frac{1/Q<sub>1</sub>}{1/P<sub>1</sub>}</math>
:::<math>\frac{dQ}{dP}X\frac{1/Q}{1/P}</math>
::where you can think of "dQ" as replacing a small change in Q (Q<sub>2</sub>-Q<sub>1</sub>) and "dP" replacing a small change in P (P<sub>2</sub>-P<sub>1</sub>) in the above equations.
::where you can think of "dQ" as replacing a tiny change in Q (i.e., Q<sub>2</sub>-Q<sub>1</sub>) and "dP" replacing a tiny change in P (i.e., P<sub>2</sub>-P<sub>1</sub>) in the above equations. "dQ/dP" is the "differential" of Q with respect to P (i.e., the tiny change in Q based on a tiny change in P).
::We can now solve the differential equation for P<sub>ed</sub> by substituting in Q=30/P:
:::<math>\frac{d}{dP}\frac{30}{P}X\frac{1/Q}{1/P}</math>
::differentiating in the first half and substituting P=30/Q in the second half yields:
:::<math>\frac{-30}{P^2}X\frac{30}{Q^2}</math>
::substituting in Q=30/P and P=30/Q yields:
:::<math>\frac{-Q}{P}X\frac{P}{Q}</math>
and thus P<sub>ed</sub>= -1. By convention, because price elasticity is always negative, the negative sign is dropped and we have our answer: elasticity is 1.
9. Provide, in your own words, the best definition of "public good" that you can.
9. Provide, in your own words, the best definition of "public good" that you can.