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 dependent 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

It is questionable. I'll simplify the model by limiting it to just one product (eggs, for example):

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

Because we're expecting that Quantity Demand would fluctuate in reaction to changes in Price, imagine a graph of the Demand Curve with Q on the vertical axis and P on the horizontal axis.

I can't stress this enough: in the scenario of linear demand with elasticity that varies over the demand curve:

The derivative of a linear equation is the change (difference) in the dependent variable (Q) divided by the change (difference) in the independent variable (P) measured in units. The coefficient is the ratio of the change in Q units to the change in P units. Using your eggs as an example, increasing the price of eggs by 1 unit reduces the number demanded by 16.12 units - regardless of whether the price rises from 1 to 2 or from 7 to 8, the quantity demanded falls by 16.12 units.
Elasticity, as you can see from the link above, offers a little more information. Because elasticity is defined as the percent change in Quantity Demanded divided by the percent change in Price, or the relative difference in Quantity Demanded vs the relative difference in Price. Let's take your eggs model and remove Ad.Type and Price from it. 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  

As you can see in the table, as P increases by 1.00, Qd reduces by 16.12, whether it's from 1.00 to 2.00 or from 7.00 to 8.00.
Elasticity, on the other hand, increases dramatically in response to price fluctuations, thus even if the change in units for each variable remains constant, the percent change for each variable will change.
A price rise from 1 to 2 is a 100% increase, resulting in a fall in demand from 121.25 to 105.13, a 13 percent decrease.
A price rise from 7 to 8 is a 14 percent increase, resulting in a 66 percent decrease in demand from 24.53 to 8.41.