The quantity demanded x each month or Russo Espresso Makers is 250 when the unit price p is $138. The quantity demanded each month is 1000 when the unit price is $108. The suppliers will market 700 espresso makers when the unit price is $78. At a unit price of $108, they are willing to market 2200 units. Both the supply and demand equations are known to be linear.a) Find the demand equation. p = ?b) Find the supply equation. p = ?c) Find the equilibrium quantity and the equilibrium price.? units$ ?

Question

The quantity demanded x each month or Russo Espresso Makers is 250 when the unit price p is $138. The quantity demanded each month is 1000 when the unit price is $108. The suppliers will market 700 espresso makers when the unit price is $78. At a unit price of $108, they are willing to market 2200 units. Both the supply and demand equations are known to be linear.a) Find the demand equation.  p  =  ?b) Find the supply equation.  p  =  ?c) Find the equilibrium quantity and the equilibrium price.? units$ ?

Samsonitew7b · Accepted Answer

Step 1(a). Let the demand equation be   p  =  m  d  +  b, where p is the unit price, d is the demand, m is the slope of the demand line and b is its y-intercept. Then   m  =  (  138  −  100  )      /    (  250  −  1000  )  =  −  38      /    750  =  −  19      /    375 so that the demand equation becomes   p  =  −  (  19      /    375  )  d  +  b. Further, on substituting   p  =  108 and   d  =  1000 in this equation, we get   108  =  −  (  19      /    375  )  ⋅  1000  +  b  =  −  152      /    3  +  b so that   b  =  108  +  152      /    3  =  432      /    3  =  144. Hence, the demand equation is   p  =  −  (  19      /    375  )  d  +  144.Step 2(b). Let the supply equation be   p  =  n  s  +  c, where p is the unit price, d is the supply , n is the slope of the supply line and c is its y-intercept. Then   n  =  (  108  −  78  )      /    (  2200  −  700  )  =  30      /    1500  =  1      /    50 so that the supply equation becomes   p  =  (  1      /    50  )  s  +  c. Further, on substituting   p  =  108 and   s  =  2200 in this equation, we get   108  =  (  1      /    50  )  ⋅  2200  +  c  =  44  +  c so that   c  =  108  −  44  =  64. Thus, the supply equation is   p  =  (  1      /    50  )  s  +  64.Step 3(c ). The equilibrium occurs when the supply is equal to the demand. Further, from the demand equation, we get   d  =  (  375      /    19  )  (  144  −  p  ) and similarly, from the supply equation, we get   s  =  50  (  p  −  64  ). Thus, for the equilibrium, we have   d  =  s or,   (  375      /    19  )  (  144  −  p  )  =  50  (  p  −  64  )     or  ,  50  p  +  (  375      /    19  )  p  =  (  375      /    19  )  ⋅  144  +  50  ⋅  64     or  ,  (  1325      /    19  )  p  =  (  54000  +  60800  )      /    19  =  114800      /    19 so that   p  =  114800      /    1325  =      $    86.64.( on rounding off to the nearest cent). Thus, the equilibrium price is $86.64.on substituting   p  =  114800      /    1325 in the supply equation, we get   (  1      /    50  )  s  +  64  =  114800      /    1325     or  ,  s  =  50  (  114800      /    1325  −  64  )  =  50  ⋅  (  114800  −  84800  )      /    1325  =  50  ⋅  30000      /    1325  =  1132 ( on rounding off to the nearest whole number). Thus, the equilibrium quantity is 1132.

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