Help solving this three-part word problem?

Question

At $0.49 per bushel, the daily supply for wheat is 305 bushels, and the daily demand it 515 bushels.  When the price is raised to $0.73 per bushel, the daily supply increases to 545 bushels, and the daily demand decreases to 435 bushels.  Assume that the price-supply and price-demand equations are linear.

 

A.) Find the price-supply equation

B.) Find the price-demand equation

C.) Find the equilibrium price and equilibrium quantity

 

I'm particularly having trouble with part C. but if you could work this entire problem out for me I would appreciate it!

Answer 1

A.)  Since the price supply equation is linear, it will have the general form

 

       y = ax + b 

 

Let y = the number of bushels and x the price per bushel:

 

       y = a($0.73) + b ⇒ 545 bushels = ($0.73)a + b

       y = a($0.49) + b ⇒ 305 bushels = ($0.49)a + b

       

 

       Subtract the two equations

 

        (545 bushels - 305 bushels) = a($0.73 - $0.49)

 

       240 bushels = a($0.24)

 

       ∴ a = (240 bushels)/($.24) = 1000 bushels per $

 

        To find the value of b, plug the value of a into either of the original equations:

 

        y = 545 = (1000)($0.73) + b

 

        b = 545 - 730 = -185

 

       So the price-supply equation may be written as: y = (1000)x - 185

 

B.)   For the price-demand equation use the same procedure

 

       y = a($0.73) + b ⇒ 435 bushels = ($0.73)a + b
       y = a($0.49) + b ⇒ 515 bushels = ($0.49)a + b

 

       (435 bushels - 515 bushels) = a($0.73 - $0.49)

 

       -80 bushels = ($0.24)a

 

       a = (-80 bushels)/($0.24) = -333.33

 

       435 bushels = ($0.73)(-333.33) + b

 

       b = 435 bushels - ($0.73)(-333.33) = 678.33

 

       The price-demand equation is: y = -333.33x + 683.33

 

C.)   The equilibrium price and quantity are at the point where these two lines intersect.

 

       y = (1000)x - 185

       y = -333.33x + 683.33

 

       1000x - 185 = -333.333x + 683.333

 

       x(1000 + 333.333) = 683.333 + 185

 

       1333.333*x = 868.333

 

       x = (683.333)/(1333.333) = $0.65 (equilibrium)

 

       y = (1000)($0.65) - 185 = 466.25 bushels (equilibrium)

 

       

 

      

Answer 2

A.) Points for price-supply, (p,s): ($0.49/b, 305 b), ($0.73/b, 545 b)

 

s(p) - 305 = ((545-305)/(0.73-0.49))(p - 0.49)

s(p) - 305 = (240/(0.24))(p - 0.49)

s(p)  = 1000(p - 0.49) + 305

s(p) = 1000p - 490 + 305

s(p) = 1000p - 185 b

 

B.) Points for price-demand, (p,d): ($0.49/b, 515 b), ($0.73/b, 435 b)

 

d(p) - 515 = ((515-435)/(0.49-0.73))(p - 0.49)

d(p) - 515 = (80/(-0.24))(p - 0.49)

d(p) = -(1000/3)(p - 0.49) + 515

d(p) = -1000p/3 + 490/3 + 1545/3

d(p) = -1000p/3 + 2035/3 b

 

C.) Use Substitution:

1000p - 185 = -1000p/3 + 2035/3

Multiply by 3:

3000p - 555 = -1000p + 2035

4000p = 555 + 2035 = 2590

p = 259/400 = 2.59/4 = $0.6475/b

 

s($0.6475/b) = d($0.6475/b) = 1000(0.6475) - 185 = 647.5 -185 = 462.5 b