What is the difference between Coefficient of Regression and Elasticity

Question

I am studying elasticity of demand and how to get the optimal price from elasticity using regression. I have referred Rbloggers and medium blogs to understand the concepts. But still I have a doubt. Say I have a linear equation as below

Sales of Eggs = 137.37 – (16.12)Price.Eggs + 4.15 (Ad.Type) – (8.71)Price.Cookies 

Mean of Price.Eggs= 4.43,
Mean of Price.Cookies= 4.37,
Mean of Sales of Eggs= 30

We can deduce the equation as : increase in sales of eggs increases the price of cookies by 8.71 and price of eggs by 16.12.

But in the case of elasticity, we calculate the formula and the elasticity of price of eggs is -2.38 and elasticity of price of cookies is -1.27 which also tells the unit increase in value with respect to dependant variable. What is the difference between these two ? I know the values are different but both meant the same right ? Please advice and correct if I am wrong

Answer 1

Well it depends. I'm going to simplify the model a bit to one product (eggs for example):

Assuming elasticity is not constant and the demand curve is linear:  
E = Elasticity  
Q = Quantity Demanded
P = Price  
t = time  
b0 = constant  
b1 = coefficient (slope)  

See the breakdown for elasticity here

Picture a graph of the Demand Curve with Q on the vertical axis and P on the horizontal axis - because we're assuming Quantity Demanded will change in response to changes in Price.

I can't emphasize this enough - in the case where demand is linear and elasticity is not constant along the entire demand curve:

The coefficient (slope) is the change (difference) in the dependent variable (Q) divided by the change in the independent variable (P) measured in units - it is the derivative of your linear equation. The coefficient is the change in Q units with respect to a change in P units. Using your eggs example, increasing the price of eggs by 1 unit will decrease the quantity demanded of eggs by 16.12 units - regardless of whether the price of eggs increases from 1 to 2 or 7 to 8, the quantity demanded will decrease by 16.12 units.

From the link above, you can see that Elasticity adds a bit more information. That is because elasticity is the percent change in Quantity Demanded divided by the percent change in Price - ie the relative difference in Quantity Demanded with respect to the relative difference in Price. Let's use your eggs model but exclude Ad.Type and Price.Cookies

Sales of Eggs = 137.37 - 16.12 * Price.Eggs

"P"     "Qd"    "E"  
1.00    121.25  -0.13  
2.00    105.13  -0.31  
3.00    89.01   -0.54  
4.00    72.89   -0.88  
5.00    56.77   -1.42  
6.00    40.65   -2.38  
7.00    24.53   -4.60  
8.00    8.41    -15.33  

See graph of Demand Curve vs Elasticity

In the table you can see that as P increases by 1.00, Qd decreases by 16.12 regardless if it's from 1.00 to 2.00 or 7.00 to 8.00.

Elasticity, however, does change rather significantly relative to changes in price, so even if the change in units for each variable remains the same, the percent change for each variable will change.

A price increase from 1 to 2 is a 100% increase and would result in a change in quantity demanded from 121.25 to 105.13 which is a 13% decrease.

A price change from 7 to 8 is a 14% increase and would result in a quantity demanded change from 24.53 to 8.41 which is a 66% decrease.

If you're interested in learning more about different ways to measure elasticity I highly recommend These lecture slides especially slide 6.26.