Question

Bob’s Board Game café faced the following demand curve:

Q^d = 280 - 60ln(P)

a) Calculate general elasticity (fully simplify and leave no q in your final answer)

b) Calculate elasticity at a price of $40 and $80. How would a firm increase revenue at these prices? Why?

c) Use elasticity to calculate the price that maximizes revenue.

Answer 1

Point elasticity is the price elasticity of demand at a specific point on the demand curve instead of over a range of it. It uses the same formula as the general price elasticity of demand measure, but we can take information from the demand equation to solve for the “change in” values instead of actually calculating a change given two points. Here is the process to find the point elasticity of demand formula:

Point Price Elasticity of Demand = (% change in Quantity)/(% change in Price)

Point Price Elasticity of Demand = (∆Q/Q)/(∆P/P)
Point Price Elasticity of Demand = (P/Q)(∆Q/∆P)

Where (∆Q/∆P) is the derivative of the demand function with respect to P. You don’t really need to take the derivative of the demand function, just find the coefficient (the number) next to Price (P) in the demand function and that will give you the value for ∆Q/∆P because it is showing you how much Q is going to change given a 1 unit change in P.

(b)Here is an example demand curve: Q = 280 - 60P
Given this demand curve we have to figure out what the point price elasticity of demand is at P = $40 and P = $80.

First we need to obtain the derivative of the demand function when it's expressed with Q as a function of P. Since quantity goes down by 50 each time price goes up by 1,

This gives us (∆Q/∆P)= -60

Next we need to find the quantity demanded at each associated price and pair it together with the price: (40, -2120), (80, -4620)

e = -60(40/-2120) = 1.13
e = -60(80/-4620) = 1.03. it means (ep>1)

Relatively elastic demand refers to the demand when the proportionate change produced in demand is greater than the proportionate change in price of a product. The numerical value of relatively elastic demand ranges between one to infinity.

Mathematically, relatively elastic demand is known as more than unit elastic demand (ep>1). For example, if the price of a product increases by 20% and the demand of the product decreases by 25%, then the demand would be relatively elastic.

The demand curve of relatively elastic demand is gradually sloping, as shown in Figure-4:

It can be interpreted from Figure-4 that the proportionate change in demand from OQ1 to OQ2 is relatively larger than the proportionate change in price from OP1 to OP2. Relatively elastic demand has a practical application as demand for many of products respond in the same manner with respect to change in their prices.

For example, the price of a particular brand of cold drink increases from Rs. 15 to Rs. 20. In such a case, consumers may switch to another brand of cold drink. However, some of the consumers still consume the same brand. Therefore, a small change in price produces a larger change in demand of the product.

Total revenue is calculated as the quantity of a good sold multiplied by its price. It is a measure of how much money a company makes from selling its product, before any costs are considered. Obviously, the goal of a company is to maximize profits, and one way to do this is by increasing total revenue. The company can increase its total revenue by selling more items or by raising the price.

Why Elasticity Matters

Price elasticity of demand and total revenue are closely interrelated because they deal with the same two variables, P and Q. If your product has elastic demand, you can increase your revenue by decreasing the price of that good. P will decrease, but Q will increase at a greater rate, thus increasing total revenue. If the product is inelastic, then you can actually raise prices, sell slightly less of that item but make higher revenue. As a result, it is important for management to know whether its product has inelastic or elastic demand.

(c)At the -1 price elasticity of demand point, the gain from increasing price will be exactly canceled out by the loss from decreasing quantity of demand.

In other word, the gain from price effect is exactly canceled out with the loss from quantity effect.

If the price elasticity of demand is -7(elastic), that means, if P increases by 1%, Qd will decrease by 7 %.

Similarly, if P decreases by 1%, Qd will increase by 7%. And If the firm decrease price, then the loss from price effect is smaller than the gain from quantity effect, so this change increase firm's profit.

If the price elasticity of demand is -0.7(inelastic), that means, if P increases by 1%, Qd will decrease by 0.7%.

Similarly, if P decreases by 1%, Qd will increase by 0.3%. So If the firm increase price, then the gain from price effect is larger than the loss from quantity effect, so this change increase firm's profit.

The rule for maximizing profit for monopolistic firm.

If the firm faces elastic point of demand, then decrease price.

If the firm faces inelastic point of demand, then increase price.