Question

Market demand (D) and supply (S) are:D: P = 60-2QandS: P = 5 +Q. Let Qe = Quantity at equilibrium and Pe = Priceat equilibrium.(a) Compute Qe and Pe and graph the D and S functions in the same graph with P on the vertical axis. (b) Show that at Q1 = 9, Net Market Benefits (NB) are less than NB at Qe. What is the Dead-Weight Loss (DWE) at Q1 = 9? (c)Show that at Q2 = 20, Net Market Benefits (NB) are less than NB at Qe. What is the Dead-Weight Loss (DWE) at Q2 = 20? (d) Compute price elasticity of demand at Qe. (e) Compute price elasticity of supply at Qe.

Answer 1

Solution:

Market demand, D: P = 60 - 2Q

Market supply, S: P = 5 + Q

(a) Equilibrium is attained where quantity demanded equals the quantity supplied, that is, where the demand and supply curve intersects. Then,

60 - 2Qe = 5 + Qe

3Qe = 55

Qe = 55/3 = 18.33 units

Pe = 5 + Qe = 5 + 18.33 = $23.33 per unit

Net market benefits is the total surplus generated. With the equilibrium, consumer surplus = area of triangle above equilibrium price, below demand curve = (1/2)*(Qe)*(60 - Pe) = (1/2)*18.33*(60 - 23.33) = $336.08

Producer surplus = area of triangle below equilibrium price, above supply curve = (1/2)*(Qe)*(Pe - 5) = (1/2)*18.33*(23.33 - 5) = $168

So, net benefits (that is total surplus) = consumer surplus + producer surplus = 336.08 + 168 = $504.08

(b) At Q = 9 units:

Price paid by buyers = 60 - 2*9 = $42

Price received by sellers = 5 + 9 = $14

So, Consumer surplus = area of trapezium above equilibrium price, below demand curve = (1/2)*((60 - Pe) + (42 - Pe))*Q

Consumer surplus = (1/2)*((60-23.33)+(42-23.33))*9 = (1/2)*(36.67+18.67)*9 = $249.03

And producer surplus = area of trapezium below equilibrium price, above supply curve = (1/2)*((Pe - 5) + (Pe - 14))*Q

Producer surplus = (1/2)*((23.33 - 5)+(23.33 - 14))*9 = (1/2)*(18.33+9.33)*9 = $124.47

So, net benefits = 249.03 + 124.47 = $373.5 which is clearly less than $504.08

Dead weight loss in this case equals the loss of total net benefits = 504.08 - 373.5 = $130.58

(c) At Q = 20 units:

Price paid by buyers = 60 - 2*20 = $20

Price received by sellers = 5 + 20 = $25

Notice that the maximum price the consumers are willing to pay at this quantity is lower than the minimum price sellers are willing to receive. So, there will be surplus in economy, that is there will be extra quantity as a wastage to the society. This wastage is the measure of dead weight loss in this case.

So, Consumer surplus = area of triangle above equilibrium price, below demand curve (same as before) net of triangle below equilibrium price and above demand curve = 336.08 - (1/2)*(Q - Qe)*(Pe - P) = 336.08 - (1/2)*(20-18.33)*(23.33-20) = $333.3

And producer surplus = area of triangle below equilibrium price, above supply curve (same as before) net of triangle above equilibrium price and below supply curve = 168 - (1/2)*(Q - Qe)*(P - Pe) = 168 - (1/2)*(20-18.33)*(25-23.33) = $166.61

So, net benefits = 333.3 + 166.61 = $499.91 which is again clearly less than $504.08

Dead weight loss in this case equals the loss of total net benefits = 504.08 - 499.91 = $4.17

(d) Price elasticity of demand at equilibrium = =

= -2 (from the demand function)

So, required elasticity = (1/(-2))*(23.33/18.33) = -0.636

(e) Price elasticity of supply at equilibrium = =

= 1 (from the supply function)

So, required elasticity = (1/1)*(23.33/18.33) = 1.27