Assume that the equation for demand for bread at a small bakery is Qd = 60 - 10Pb + 3Y, where Qd is the quantity of bread demanded in loaves, Pb is the price of bread in dollars per loaf, and Y is the average income in the town in thousands of dollars. Assume also that the equation for supply of bread is Qs = 30 + 20Pb - 30 Pf, where Qs is the quantity supplied and Pf is the price of flour in dollars per pound. Assume finally that markets clear, so that Qd = Qs.
a. If Y is 10 and Pf is $1, solve mathematically for equilibrium Q and Pb. (Hint: substitute all exogenous variables into the equations and set Qd = Qs to find equilibrium Pb first and then subsitute back equilibrium Pb to the equations fo find equilibrium Q)
b. If the average income in the town increases to 15, solve for the new equilibrium Q and Pb.
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a. To find the equilibrium quantity (Q) and price (Pb), we need to set the quantity demanded (Qd) equal to the quantity supplied (Qs) and solve for Pb.
Qd = Qs
Substituting the given equations:
60 - 10Pb + 3Y = 30 + 20Pb - 30Pf
Now, substitute the given values Y=10 and Pf=1:
60 - 10Pb + 3(10) = 30 + 20Pb - 30(1)
60 - 10Pb + 30 = 30 + 20Pb - 30
Combine like terms:
20Pb + 10Pb = 60 + 30 - 30 - 30
30Pb = 30
Divide both sides by 30:
Pb = 1
Now, substitute this value back into the equation for Qd or Qs:
Qd = 60 - 10(1) + 3(10)
Qd = 60 - 10 + 30
Qd = 80
Therefore, the equilibrium quantity is Q = 80 loaves of bread and the equilibrium price is Pb = $1 per loaf.
b. To find the new equilibrium Q and Pb when the average income (Y) increases to 15, we will follow the same steps as above.
Qd = Qs
Substituting the given equations:
60 - 10Pb + 3Y = 30 + 20Pb - 30Pf
Now, substitute the new value Y=15 and Pf=1:
60 - 10Pb + 3(15) = 30 + 20Pb - 30(1)
60 - 10Pb + 45 = 30 + 20Pb - 30
Combine like terms:
20Pb + 10Pb = 60 + 45 - 30 - 30
30Pb = 45
Divide both sides by 30:
Pb = 1.5
Now, substitute this value back into the equation for Qd or Qs:
Qd = 60 - 10(1.5) + 3(15)
Qd = 60 - 15 + 45
Qd = 90
Therefore, the new equilibrium quantity is Q = 90 loaves of bread and the new equilibrium price is Pb = $1.5 per loaf.
a. To solve for the equilibrium quantity and price, we need to set the quantity demanded equal to the quantity supplied. We are given the demand equation Qd = 60 - 10Pb + 3Y and the supply equation Qs = 30 + 20Pb - 30Pf.
Step 1: Substituting the given values in the demand equation, we have Qd = 60 - 10Pb + 3(10) = 60 - 10Pb + 30 = 90 - 10Pb.
Step 2: Substituting the given values in the supply equation, we have Qs = 30 + 20Pb - 30(1) = 30 + 20Pb - 30 = 20Pb.
Step 3: Setting Qd = Qs, we get 90 - 10Pb = 20Pb.
Step 4: Adding 10Pb to both sides, we get 90 = 30Pb.
Step 5: Dividing both sides by 30, we get Pb = 3.
Therefore, in equilibrium, the price of bread (Pb) is $3.
Step 6: Substituting Pb = 3 back into the demand equation (Qd = 90 - 10Pb), we have Qd = 90 - 10(3) = 90 - 30 = 60.
Therefore, in equilibrium, the quantity of bread demanded (Qd) is 60 loaves.
b. To find the new equilibrium quantity and price when the average income increases to 15, we follow a similar process as in part a.
Step 1: Substituting the new income value in the demand equation, we have Qd = 60 - 10Pb + 3(15) = 60 - 10Pb + 45 = 105 - 10Pb.
Step 2: Setting Qd = Qs, we get 105 - 10Pb = 20Pb.
Step 3: Adding 10Pb to both sides, we get 105 = 30Pb.
Step 4: Dividing both sides by 30, we get Pb = 3.5.
Therefore, in the new equilibrium, the price of bread (Pb) is $3.50.
Step 5: Substituting Pb = 3.5 back into the demand equation (Qd = 105 - 10Pb), we have Qd = 105 - 10(3.5) = 105 - 35 = 70.
Therefore, in the new equilibrium, the quantity of bread demanded (Qd) is 70 loaves.