Question

Take Home 3 Math 1520 1. Consumer and Producer Surplus. Round answers to the nearest whole number. a) Find the consumer and producer surpluses at the equilibrium (X,P). P.(x)=6278 - Demand price Ps(x)=3V102 +Supply price b) Find the new equilibrium and surpluses for the outward shift in demand P. (x)=6783- Demand price Ps(x)=3 102+x Supply price

Answer 1

In each case,

• In equilibrium, pD = pS
• Consumer surplus (CS) = Area between demand curve and price
• Producer surplus (PS) = Area between supply curve and price

(a)

6$\dpi{80} \sqrt{}$(78 - x) = 3$\dpi{80} \sqrt{}$(102 + x)

Squaring,

36.(78 - x) = 9.(102 + x)

2808 - 36x = 918 + 9x

45x = 1890

x = 42

p = 6$\dpi{80} \sqrt{}$(78 - 42) = 6$\dpi{80} \sqrt{}$36 = 6 x 6 = 36

When x = 0, pD = 6$\dpi{80} \sqrt{}$78 = 6 x 8.83 = 52.98 (vertical intercept of demand curve)

CS = (1/2) x (52.98 - 36) x 42 = 21 x 16.98 = 356.58

When x = 0, pS = 3$\dpi{80} \sqrt{}$102 = 3 x 10.1 = 30.3 (vertical intercept of supply curve)

PS = (1/2) x (36 - 30.3) x 42 = 21 x 5.7 = 119.7

(b)

6$\dpi{80} \sqrt{}$(83 - x) = 3$\dpi{80} \sqrt{}$(102 + x)

Squaring,

36.(83 - x) = 9.(102 + x)

2988 - 36x = 918 + 9x

45x = 2070

x = 46

p = 6$\dpi{80} \sqrt{}$(83 - 46) = 6$\dpi{80} \sqrt{}$37 = 6 x 6.08 = 36.48

When x = 0, pD = 6$\dpi{80} \sqrt{}$83 = 6 x 9.11 = 54.66 (vertical intercept of new demand curve)

CS = (1/2) x (54.66 - 36.48) x 46 = 23 x 18.18 = 418.14

When x = 0, pS = 3$\dpi{80} \sqrt{}$102 = 3 x 10.1 = 30.3 (vertical intercept of supply curve)

PS = (1/2) x (36.48 - 30.3) x 46 = 23 x 6.18 = 142.14