1.) Ralph owns a department store where he mainly sells Knives. After doing rese

Question

1.) Ralph owns a department store where he mainly sells Knives. After doing research he finds out he can sell 100 knives when he charges $12 per knife. Also, if he only charges $10 per knife he is able to sell 200 knives. Let P represent the price per knife (in dollars) and Q represent the # of knives. Assume the demand equation is linear.

a.) Find an expression for the demand equation. (HINT: Since the relationship is linear we know p = mq + b)

b.) What's the revenue when q knives are sold? (HINT: What's the price per knife when we sell q? It depends on q.)

c.) Suppose the supply equation is given by p = 4/50q. Find the equilibrium price and quantity.

Explanation / Answer

The demand equation is linear (and in terms of price)
Thus Price 'P' = mq + b;
We know that when P = 12, Q = 100
12 = 100m+ b
We also know that when P=10, Q= 200
10 = 200m + b

Subtracting second equation from the first we get:
12-10 = 100 m + b - (200m +b)
2 = -100m
m= -2/100 = -1/50;
Substituting m = -1/50 in 10 = 200 m + b we get:
10 = 200 (-1/50) + b
10 = -4 + b
b=14;

Thus, the linear relation between demand Q and price P is
P = - Q/50 + 14; or Q = (14-P)*50

b) When q knives are sold, revenue = p*q = price per unit * number of units sold
Revenue = (-Q/50 + 14 ) (Q) = -Q2/50 + 14Q

c) Supply equation is p=4/50q or 50pq = 4;
Equlirbium condition happens when the supply price is equal to the demand side price;
Demand side quantity = supply side quantity

-Q/50 + 14 = 4/50 Q
5Q/50 = 14
Q= 14*50/5 = 14*10 = 140
When Q=140, P = 4/50 * 140 = 56/5 = 11.2

Thus, equilibrium price = 11.2$ per knife and equilibrium quantity = 140;

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