Price sensitivities of demand differ depending on the product and the market. We can model our demand function to reflect how we think quantity responds to price. The following post will use calculus. If you don’t like calculus, the important thing to remember is that changes in the natural log approximately equal percentage changes. And so, by logging either quantity or price, you are saying that you think the relevant change for your purpose is a percentage change rather than a unit change. I’ll add another post to describe how to maximize profits when demand is linear or semilog.

Linear Model: Q = aP + b

To see the implications of the linear model, let’s take the derivative with respect to P.

dQ/dP = a dQ = a*dP

units change in Q = a*(units change in P)

In the linear model, a $1 change in price leads quantity to change by ‘a’ units, where ‘a’ is the slope of the demand curve. In other words, a $1 change in price gives the same absolute change in quantity regardless of the price level it is being measured.

Price Sensitivity = (dQ/Q/dP/P) = (P/Q)(dQ/dP) = (Pa)/Q = a(P/Q) which is not constant and varies over the demand curve depending upon the P, Q coordinates. Also Price sensitivity = (aP)/(aP +b) if we substitute aP+b for Q

Log-log model: ln(Q) = aln(P) + b

First let’s remember that: (d ln(Q))/dQ = 1/Q , and ln(Q) = dQ/Q = (change in Q)/Q = % change in Q if we take %

Now, if we differentiate the log-log model, we have: (d ln(Q))/(d ln(P)) = a d ln(Q) = a*d ln(P) dQ/Q = a * dP/( P) % change in Q = a* (% change in P)

In the log-log model, a 1% change in price gives an ‘a%’ change in quantity. At any beginning price level, the same percent increase in price will lead to the same percent decrease in quantity regardless of the point on the demand curve – i.e., the price elasticity is constant.

Price Elasticity = (dQ/Q)/dP/P) = ‘a’ which is a constant equal to the slope of the log log demand curve.

Semilog model (or log-linear model): ln(Q) = aP + b d ln(Q)/dP = a d ln(Q) = a*dP dQ/Q = a*dP

% change in Q = a*(unit change in P)

In the semi-log model, a $1 change in price leads to the same % change in quantity regardless of the point on the demand curve. Price Elasticity is not constant as it depends on the Price at the point where it is measured.

Price Elasticity = (dQ/Q)/dP/P) = (a*dP)/(dP/P) = aP