91Ó°ÊÓ

The inverse demand for carbon-steel chef's knives is given by \(P=120-\frac{1}{2} Q^{D}\) where \(P\) is the price of a chef's knife and \(Q^{D}\) is the quantity of chef's knives desired per week, in thousands. The inverse supply of chef's knives is given by \(P=20+2 Q^{S}\) where \(Q^{S}\) is the quantity of chef's knives offered for sale each week, in thousands. a. Accurately graph the inverse supply and inverse demand curves, with \(P\) on the vertical axis and \(Q\) on the horizontal axis. b. What are the buyers' and sellers' choke prices in your graph? How can you find those same choke prices by looking at the inverse demand and inverse supply equations? c. Equate inverse demand and inverse supply to find the market equilibrium quantity of chef's knives sold. d. Plug the quantity you found in (c) into the inverse demand curve to solve for the equilibrium price. Then double- check your work by plugging that same quantity into the inverse supply curve.

Step by step solution

01

Identify Choke Prices

To find the buyers' choke price, set the demand quantity \(Q^D = 0\) in the inverse demand equation \(P = 120 - \frac{1}{2}Q^D\). This yields \(P = 120 - \frac{1}{2} \times 0 = 120\). Thus, the buyers' choke price is 120. For sellers' choke price, set the supply quantity \(Q^S = 0\) in the inverse supply equation \(P = 20 + 2Q^S\), giving \(P = 20 + 2 \times 0 = 20\). Thus, the sellers' choke price is 20.

02

Graph Inverse Demand and Supply Curves

Plot the inverse demand curve from Step 1 (\(P = 120 - \frac{1}{2}Q^D\)) and the inverse supply curve (\(P = 20 + 2Q^S\)) on the same graph. The price \(P\) is on the vertical axis and the quantity \(Q\) is on the horizontal axis. The demand curve is downward sloping starting at 120 (buyers' choke) when \(Q^D = 0\) and the supply curve is upward sloping starting at 20 (sellers' choke) when \(Q^S = 0\).

03

Find Market Equilibrium Quantity

Set the inverse demand and supply equations equal to find equilibrium: \[120 - \frac{1}{2}Q^D = 20 + 2Q^S\]Assuming market equilibrium \(Q^D = Q^S = Q\), solve for \(Q\):\[120 - \frac{1}{2}Q = 20 + 2Q\] \[100 = 2.5Q\] \[Q = 40\] (in thousands). Thus, the equilibrium quantity is 40,000 knives.

04

Find Equilibrium Price Using Demand Equation

Substitute \(Q = 40\) into the inverse demand equation: \[P = 120 - \frac{1}{2} \times 40 = 120 - 20 = 100\]. Thus, the equilibrium price is 100.

05

Confirm Equilibrium Price Using Supply Equation

Substitute \(Q = 40\) into the inverse supply equation: \[P = 20 + 2 \times 40 = 20 + 80 = 100\]. The price 100 matches the equilibrium price found using the demand equation, confirming correctness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Demand

Inverse demand is a concept that helps us understand how the demand for a product changes as its price varies. It is expressed in the form of an equation where price is a function of the quantity demanded. For the carbon-steel chef's knives, this relationship is given by the equation: \(P = 120 - \frac{1}{2} Q^{D}\).Here, \(P\) is the price of the knives, while \(Q^{D}\) represents the quantity demanded per week, measured in thousands. As we can see, the price (\(P\)) decreases when the quantity demanded (\(Q^{D}\)) increases, which is typical of most demand curves. It's a downward-sloping line on a graph, and it shows that buyers are generally willing to pay less as more of an item becomes available.

Inverse Supply

Inverse supply is the counterpart to inverse demand. It expresses how the supply of a product changes with its price. The inverse supply equation for the chef's knives is: \(P = 20 + 2 Q^{S} \).In this context, \(P\) is the price, and \(Q^{S}\) is the quantity supplied per week, also in thousands. Unlike the demand curve, the supply curve is upward sloping. This means as the price increases, suppliers are more willing to produce and sell more knives. The supply curve begins at a price of 20 when no knives are produced, highlighting the minimum price at which suppliers are willing to start selling.

Choke Price

The choke price is a crucial concept in understanding market dynamics. For the demand side, the choke price is the price where the quantity demanded begins at zero. In the exercise, the inverse demand equation showed us a choke price of 120. This indicates that buyers will not purchase any knives if the price is at or above 120. Similarly, for the supply side, the choke price is where suppliers are willing to sell the first unit on the market. From the inverse supply equation, this price was found to be 20. At this price, suppliers are indifferent to supplying knives, often setting a floor for market entry.

Equilibrium Quantity

Equilibrium quantity is where the quantity demanded equals the quantity supplied. It's a point in the market where there is no tendency for change, as buyers and sellers have reached an agreement on the amount of product that satisfies both parties.To find this, we solve the equations for inverse demand and inverse supply simultaneously. In this exercise, equating \(120 - \frac{1}{2}Q\) and \(20 + 2Q\) yielded \(Q = 40\) (in thousands). Thus, the equilibrium quantity is 40,000 knives, indicating a balance between supply and demand.

Equilibrium Price

The equilibrium price is the price at which the quantity of goods supplied equals the quantity demanded. This price ensures that there is no surplus or shortage in the market. In the exercise, after finding the equilibrium quantity to be 40, we substitute this value into either the inverse demand or supply equation. Using the inverse demand equation, we found the equilibrium price to be 100, which was confirmed by plugging the quantity back into the inverse supply equation. This price of 100 ensures that the market for chef's knives clears, with no excess or shortage.