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Expert: - 10/7/2009


Question
Hi,

Received the following during an Economics lecture. Its to explain elasticity. Could you please explain what it means?

Thanks!

Sincerely
Kelly


Let:
P = Price
Q = Quantity
PED = Price elasticity of demand
k = constant
I = Income


When PED = -1:

Q = K/P

lnQ = ln k - ln P

The coefficient is -1, which corresponds to PED.


Likewise, we can deduce that

Income elasticity is dlnx/ dlnI

and Cross price elasticity is dlnQx/ dlnPy.


I have no idea how he got any of this, let alone what it means! Any help would be appreciated. Thank you so much!

Answer
When k is used, it is in both K and k, but I will assume these are the same.

When the elasticity of demand is a constant, the PED value is a constant,
so as was said, PED = -1.

Since Q = K/P, that is quantity = contant/price (and this from the linear assumption),
take the ln() of both sides and get ln(Q) = ln(K/P).

Note that the ln(A/B) = ln(A) - ln(B), so ln(K/P) = ln(K) - ln(P).

For "income elasticity", I found http://economics.about.com/cs/micfrohelp/a/income_elast.htm

For "cross price elasticity", I found http://economics.about.com/cs/micfrohelp/a/cross_price_d.htm

Is there more help you need?  It would be great to send a more detailed description of which lines you don't understand.
For example, "I don't understand
lnQ = ln k - ln P
Income elasticity is dlnx/ dlnI
and Cross price elasticity is dlnQx/ dlnPy" for example, and what it is you don't understand.

To me, it looks like dlnx / dlnI is the derivative of ln(x) with respect to ln(I)
and dlnQx / dlnPy is the derivative of the ln(Q sub x) with respect to ln(P sub y).

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