Consider the following three demand curves (where P is price in dollars and Q is quantity in units): (A) Q = 200 – P (B) Q = 100 - 0.5P (C) Q = 200 - 0.5P At a price of $20, what is the price elasticity of demand for each of the three demand curves?  

Expert Answers

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The Price Elasticity of Demand or "PED" equation measures the percentage change in quantity demanded over the percentage change in price:

PED = (dQ/Q)/(dP/P)

These symbols equate to the following:

PED = Price Elasticity of Demand

dQ = change in quantity

Q = quantity

dP = change in price

P = price

We can simplify the equation to:

PED = (dQ/dP)*(P/Q)

(dQ/dP) is equal to the slope of your equation or the (m) in Y=mx+b.

Problem (A) we have a price of $20 and the following equation:

Q = 200 - P

We can rearrange into y=mx-b format to get:

Q = -1P+200

Notice that our slope is -1 which is equal to (dQ/dP) or m.

Plug in $20 for P and solve for Q:

Q = -1(20)+200

Q = 180

(dQ/dP) or m = -1

P = 20

Now we just plug these values into the PED equation:

PED = (dQ/dP)*(P/Q)

PED = -1 * (20/180)

PED = -0.111

Problem (B) we have a price of $20 and the following equation:

Q = 100 -0.5P

We can rearrange into Y=mx+b format to get:

Q = -0.5P +100

Again, -0.5 is our slope which is equal to (dQ/dP) or m.

Plug in $20 for P and solve for Q:

Q = -0.5(20) +100

Q = 90

(dQ/dP) = -0.5

P = 20

Now that we have our values we plug them into the PED equation and solve:

PED = (dQ/dP) * (P/Q)

PED = -0.5 * (20/90)

PED = -0.5 * 0.222

PED = -0.111

Problem C we have a price of 20 and the following equation:

Q = 200 - 0.5P

We can rearrange into Y=mx+b format to get:

Q = -0.5P + 200

Again, notice that our slope is -0.5 which is equal to (dQ/dP) or m.

We plug in our price (P) of 20 and solve for Q:

Q = -0.5(20) + 200

Q = 190

(dQ/dP) = -0.5

P = 20

We take our values and plug them into the PED equation and solve:

PED = (dQ/dP) * (P/Q)

PED = -0.5 * (20/190)

PED = -0.5 * (.105)

PED = -0.053

Conclusion

Problem (A) the PED = -0.111

Problem (B) the PED = -0.111

Problem (C) the PED = -0.053

Now, since the PED is always negative, we can ignore the negative signs and then put our answers to the following test to determine if they are elastic, unitary, or inelastic.

PED > 1 = elastic

PED = 1 = unitary

PED < 1 = inelastic

All of the answers are less than 1 and are therefore inelastic.

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The price elasticity of demand is the percentage change in the quantity demanded (q) over the percentage change in price (p).

Price Elasticity of Demand = (dQ/Q)/(dP/P) = (dQ/dP)*(P/Q)

Where Q is the quantity and P is the income/price.

Notice that (dQ/dP) is the change in Q over P, which can be thought of as the slope of the demand curve. All of your equations are in y = mx + b (you can rearrange them if you do not see this, i.e. Q = 200 – P can be rearranged as Q = –1P + 200), where m is the slope, or (dQ/dP).

P = $20 in each of the following equations.

  1. *See photo 1 for the problem solved + work involved*

  2. *See photo 2 for the problem solved + work involved*

  3. *See photo 3 for the problem solved + work involved*

*You can drop the negative signs because price elasticity of demand is always negative, the negative sign is usually ignored as it is expected.

**If an elasticity equation yields a number

> 1, the curve is elastic

= 1, the curve is unitary

< 1, the curve is inelastic

All of these are inelastic.

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The price elasticity of demand is the percentage change in demand for a percentage change in price.

If the demand is Q and price is P, the price elasticity of demand at any price p is Ep = P/Q*(dQ/dP)

The first demand curve is Q = 200 - P.

dQ/dP = -1

At P = 20, Q = 200 - 20 = 180

The price elasticity of demand in this case is (20/180)*-1 = -1/9

The second demand curve is Q = 100 - 0.5P

dQ/dP = -0.5

At P = 20, Q = 90

The price elasticity of demand is (20/90)*-0.5 = -1/9

The third demand curve is Q = 200 - 0.5P

dQ/dP = -0.5

At P = 20, Q = 190

The price elasticity of demand is (20/190)*-0.5 = -1/19

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Using the first question as an example;

(A) Q=200-P

We differentiate the above equation with respect to P.

dQ/dP= -1

So we substitute dQ/dP = -1 and Q = 200-P into the price elasticity of demand equation which is (dQ / dP)*(P/Q)


Price elasticity of demand: = (-1)*(P/(200-P)
Price elasticity of demand: = -1P/(200-P)

We are interested in getting the price elasticity of demand at the price(P) of $20. So we substitute P=20 into our price elasticity demand equation = -1P/(200-P).

= -20/(200-20)

= -20/180

=-0.1 which essentially means demand is inelastic

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