hw 9 question 7

Question

Consider the following supply and demand functions, where price is expressed as a function of quantity.

S(x) = 3x + 8

D(x) = -4x + 16

1.  Find the equilibrium quantity, x_E.
2. Find the equilibrium price, p_E
3. Find the consumer's surplus at the equilibrium point.
4. Find the producer's surplus at the equilibrium point.

Answer 1

set Demand = Supply

D= S

3x+8 = -4x +16

7x = 8

x = 8/7= equilibrium quantity

equilibrium price is found by substituting equilibrium quantity into either the D or S equation

3(8/7) + 8 = 24/7 + 56/7 = 80/7 = 11 3/7 = equilibrium price

-4(8/7) +16 = -32/7 + 112/7 = 80/7 = 11 3/7

consumer surplus = the area above the price line P= 80/7 and below the Demand curve, bounded by the y or P axis.

P-4x +16 has a y intercept = 16 16-80/7 = 112/7. - 80/7 = 32/7

Consumer Surplus = CS = the area of a right triangle with base = 8/7 and height 32/7. Area =bh/2 = 8/7(32/7)/2 = 4/7(32/7) = 128/49

CS = 128/49 = about 2.61

PS = area below the price line but above the supply curve, bounded by the y or P axis

S= 3x + 8 has y intercept 8 = 56/7 80/7-56/7 = 24/7 = height of the right triangle

base of the right triangle = 8/7. PS = bh/2 = (8/7)(24/7)/2 = 4/7(24/7) = 96/49

PS = 96/49 = about 1.96

consumers gain more collectively than the producers in this market trade at equilibrium price