In: Economics
A supermarket is a local monopoly that sells many grocery
products, using markup pricing.
That is, its price for a product is P = c(1 + m), where c is how
much it pays for each unit of the
product from producers, and m is the markup it adds to the product
when selling to consumers.
Products may differ in the price elasticity of demand.
Q= Under what conditions will markup pricing maximize the supermarket’s profit?
The formula for the product price of the monopolist is given as,
Let's derive the profit function for the monopolist.
Profit = P×Q - C
Here P = price, Q = quantity and C = cost of production
Now maximizing profit with respect to Q by differentiating profit function with respect to Q and putting it equal to 0
0 = (dP/dQ × Q + P×1) - dC/dQ
0 = P(dP/dq × Q/P + 1) - dC/dQ
Here, dP/dQ ×Q/P = 1/ed where ed = price elasticity of demand
And dC/dQ = marginal cost of production c
Substituting them back in equation we get,
0 = P(1/ed + 1) - c
P(1/ed + 1) = c
P/ed + P = c
P - c = -P/ed
(P - c)/P = -1/ed
Here, (P - c)/P = markup over price for the monopolist, m.
m = -1/ed
So as we can see that a monopolist will always choose a markup m > 0. And for that to happen we need a restriction on demand Q.
For m > 0, ed < -1
So the monopolist will keep on producing as long as the demand is inelastic or in other words ed < -1.
The reason behind this is simple, if the demand is inelastic the monopolist can exploit this fact by charging higher markup above marginal cost of production. Because demand will not change by much due to increase in price since demand is inelastic. So the monopolist will keep on producing as long as demand is inelastic.
So the monopolist will never operate at a point where demand Q is inelastic because if demand is inelastic monopolist can always increase its profit by increasing the price. Which monopolist was not maximizing its profit at previous point.
I hope I was able to help you, thank you.