We read your photo as:

Given demand and supply functions Qd = 20 − 2p and Qs = 10 + 3p, determine the equilibrium price and quantity and calculate the point price elasticity of demand at equilibrium.

Economics

ScanSolve AI Answer

Answers

3 parts
i

GH¢2\text{GH¢2}

16 kilos\text{16 kilos}

ii
-0.25
iii
0.375

Step-by-step solution

7 steps

Here are the solutions to the demand and supply functions problem:

Given the demand and supply functions:
Demand function: Qd=20−2pQ_d = 20 - 2p
Supply function: Qs=10+3pQ_s = 10 + 3p
where pp is price in cedis and QQ is quantity in kilos.

Part i· 3 steps

Determine the equilibrium price and quantity

  1. Set demand equal to supply to find the equilibrium price.

    At equilibrium, Qd=QsQ_d = Q_s.

    20−2p=10+3p20 - 2p = 10 + 3p
  2. Solve for pp.

    20−10=3p+2p20 - 10 = 3p + 2p 10=5p10 = 5p p=105p = \frac{10}{5} p=2p = 2

    The equilibrium price is GH¢2.

  3. Substitute the equilibrium price into either the demand or supply function to find the equilibrium quantity.

    Using the demand function:

    Qd=20−2(2)Q_d = 20 - 2(2) Qd=20−4Q_d = 20 - 4 Qd=16Q_d = 16

    Using the supply function (to verify):

    Qs=10+3(2)Q_s = 10 + 3(2) Qs=10+6Q_s = 10 + 6 Qs=16Q_s = 16

    The equilibrium quantity is 16 kilos.

    The equilibrium price is GH¢2\boxed{\text{GH¢2}} and the equilibrium quantity is 16 kilos\boxed{\text{16 kilos}}.

Part ii-0.25

Calculate point price elasticity of demand at equilibrium

The formula for point price elasticity of demand (EdE_d) is:

Ed=dQddp×pQdE_d = \frac{dQ_d}{dp} \times \frac{p}{Q_d}
  1. Find the derivative of the demand function with respect to price (pp).

    Given Qd=20−2pQ_d = 20 - 2p, the derivative dQddp\frac{dQ_d}{dp} is the coefficient of pp.

    dQddp=−2\frac{dQ_d}{dp} = -2
  2. Substitute the values of dQddp\frac{dQ_d}{dp}, equilibrium price (pp), and equilibrium quantity (QdQ_d) into the elasticity formula.

    Equilibrium p=2p = 2
    Equilibrium Qd=16Q_d = 16

    Ed=−2×216E_d = -2 \times \frac{2}{16} Ed=−2×18E_d = -2 \times \frac{1}{8} Ed=−28E_d = -\frac{2}{8} Ed=−14E_d = -\frac{1}{4} Ed=−0.25E_d = -0.25

    The point price elasticity of demand at equilibrium is −0.25\boxed{-0.25}.

Part iii0.375

Calculate the price elasticity of supply at equilibrium.

The formula for point price elasticity of supply (EsE_s) is:

Es=dQsdp×pQsE_s = \frac{dQ_s}{dp} \times \frac{p}{Q_s}
  1. Find the derivative of the supply function with respect to price (pp).

    Given Qs=10+3pQ_s = 10 + 3p, the derivative dQsdp\frac{dQ_s}{dp} is the coefficient of pp.

    dQsdp=3\frac{dQ_s}{dp} = 3
  2. Substitute the values of dQsdp\frac{dQ_s}{dp}, equilibrium price (pp), and equilibrium quantity (QsQ_s) into the elasticity formula.

    Equilibrium p=2p = 2
    Equilibrium Qs=16Q_s = 16

    Es=3×216E_s = 3 \times \frac{2}{16} Es=3×18E_s = 3 \times \frac{1}{8} Es=38E_s = \frac{3}{8} Es=0.375E_s = 0.375

    The price elasticity of supply at equilibrium is 0.375\boxed{0.375}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question