Sample Problems from Class:
2. Consider the general demand function:
Qd = 710 - 25P + 0.25M -14PR
a. Derive the equation for the demand function when M = $50,000 and PR = $15.
b. Interpret the intercept and slope parameters of the demand function derived in part a.
c. Sketch a graph of the demand function in part a. Where does the demand function
intersect the quantity-demanded axis? Where does it intersect the price axis?
d. Using the demand function from part a, calculate the quantity demanded when the
price of the good is $250 and when the price is $300.
e. Derive the inverse of the demand function in part a. Using the inverse demand
function, calculate the demand price for 7,000 units of the good. Give an interpretation
of this demand price.
7. Consider the general supply function:
Qs = 1,000 + 20P - 9PI + 25F
where Qs = quantity supplied, P = price of the commodity, PI = price of a key input in the
production process, and F = number of firms producing the commodity.
a. Interpret the slope parameters on P, PI, and F.
b. Derive the equation for the supply function when PI = $480 and F = 60.
c. Sketch a graph of the supply function in part b. At what price does the supply curve
intersect the price axis? Give an interpretation of the price intercept of this supply
curve.
d. Using the supply function from part b, calculate the quantity supplied when the price of
the commodity is $1,000 and $1,500.
e. Derive the inverse of the supply function in part b. Using the inverse supply function,
calculate the supply price for 40,000 units of the commodity. Give an interpretation of
this supply price.
13. Suppose that the demand and supply functions for good X are
Qd = 298 - 8P
Qs = -32 + 4P
a.
b.
c.
d.
Find the equilibrium price and quantity?
What is the market outcome if price is $35? What do you expect to happen? Why?
What is the market outcome if price is $29? What do you expect to happen? Why?
What happens to equilibrium price and quantity if the demand function becomes
Qd = 373- 6P while the supply remains the same?
e. What happens to equilibrium price and quantity if the supply function becomes
Qs = -40 + 5P , for the original demand function Qd = 298 - 8P ?
Answers:
2. a. Qd = 710 - 25P + 0.25(50,000) -14(15) = 710 - 25P +12,500 - 210 = 13,000 - 25P
b. Intercept parameter: If price were 0, consumers would take 13,000 units of the good for
free. Slope parameter: A $1 increase in price, all else constant, will cause consumers to
buy 25 fewer units per period.
c. The following sketch plots price on the vertical axis and quantity demanded on the
horizontal axis. Demand is a straight line that intersects the price axis at $520 and the
quantity demanded axis at 13,000 units.
d. When P = $250, Qd = 13,000 – 25(250) = 6,750 units. When P = $300, Qd = 13,000 –
25(300) = 5,500.
e.
P = 520 – (1/25)Qd or P = 520 – 0.04Qd . To calculate demand price: P = 520 – 0.04(7,000)
= $240. The maximum price consumers will pay to buy 7,000 units per period is $240.
7. a. For P: A $1 increase in the price of the good or service, all else constant, will increase
quantity supplied by 20 units per period.
For PI : A $1 increase in the price of a key input, all other factors affecting producers
held constant, will decrease quantity supplied by 9 units per period.
For F: If one more firm begins producing the commodity, all other things held constant,
the quantity supplied of the commodity will increase by 25 units each period.
b. Qs = 1,000 + 20P - 9(480) + 25(60)
= 1,000 + 20P - 4,320 +1,500
= -1,820 + 20P
c. The sketch show below plots price on the vertical axis and quantity supplied on the
horizontal axis. Supply is a straight line that intersects the price axis at $91 and the
quantity supplied axis at –1,820 units. Only the segment of the supply line at and above
P = $91 is economically meaningful. The price-intercept of the supply curve, $91, is
found by setting Qs = 0 in the supply function and solving for P. Since the price intercept
is $91, it follows that the price below which firms will quit producing the commodity is
$91.
d. Qs = -1,820 + 20(1,000) = 18,180
Qs = -1,820 + 20(1,500) = 28,180
e. To derive the inverse supply function, take the supply equation in part b and solve
algebraically for P as a function of Qs. The inverse supply equation is P = 91+ 0.05Qs .
Supply price for 40,000 units: P = 91+ 0.05(40,000) = $2,091. The minimum price
producers will accept to produce 40,000 units per period is $2,091.
13. a. At equilibrium, quantity demanded equals quantity supplied. So 298 - 8P = -32 + 4P .
Solving for price, we find P = $27.50. Substituting $27.50 into either the demand or
supply equation, we find Q = 78.
b. When price is out of equilibrium at the price $35:
Qd = 298 - 8(35) = 18
Qs = -32 + 4(35) = 108
There is a excess quantity supplied of 90 units. Due to the excess supply, producers will
lower price in order to avoid accumulating unwanted inventories. The price will fall,
reducing the excess supply, until equilibrium is reached at a price of $27.50.
c. When price is out of equilibrium at the price $29:
Qd = 298 - 8(29) = 66
Qs = -32 + 4(29) = 84
There is a excess quantity supplied of 18 units. Due to the excess supply, producers will
lower price in order to avoid accumulating unwanted inventories. The price will fall,
reducing the excess supply, until equilibrium is reached at a price of $27.50.
d. In equilibrium, 373 – 6P = –32+ 4P. The new equilibrium is P = $40.50 and Q = 130.
e. In equilibrium, 298 – 8P = –40 + 5P. The new equilibrium is P = $26 and Q = 90.