Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Suppose consumers will buy \(1,000,000\) T-shirts if the selling price is $$\$ 15$$, but for each $$\$ 1$$ increase in price, they will buy 100,000 fewer T-shirts. Moreover, suppose vendors will order \(2,000,000\) T-shirts if the selling price is $$\$ 20$$, and for every $$\$ 1$$ increase in price, they will order an additional 150,000 . (a) Express the number \(Q\) of T-shirts consumers will buy if the selling price is \(p\) dollars. (b) Express the number \(K\) of T-shirts vendors will order if the selling price is \(p\) dollars. (c) Determine the market price-that is, the price when \(Q=K\).

Step-by-Step Solution

Answer

The market price is approximately $11.67.

Step 1: Determine Consumer Demand Function

Consumers buy 1,000,000 T-shirts at \(15, and for each \)1 increase in price, they buy 100,000 fewer. We have:At \(\\)15: Q = 1{,}000{,}000\(\For each \)\\(1 increase, \)Q decreases by 100,000:$$Q = 1{,}000{,}000 - 100{,}000(p - 15)$$Simplifying gives:$$Q = 2{,}500{,}000 - 100{,}000p$$.

Step 2: Determine Vendor Supply Function

Vendors order 2,000,000 T-shirts at \(20, and for each \)1 increase in price, they order 150,000 more T-shirts. We have:At \(\\)20: K = 2{,}000{,}000\(For each \)\\(1 increase, \)K increases by 150,000:$$K = 2{,}000{,}000 + 150{,}000(p - 20)$$Simplifying gives:$$K = 200{,}000p - 1{,}000{,}000$$.

Step 3: Set Demand Equal to Supply

To find the market price, set \(Q = K\):\[2{,}500{,}000 - 100{,}000p = 200{,}000p - 1{,}000{,}000\]Solve for \(p\):\[2{,}500{,}000 + 1{,}000{,}000 = 200{,}000p + 100{,}000p\]\[3{,}500{,}000 = 300{,}000p\]\[p = \frac{3{,}500{,}000}{300{,}000} = 11.67\]Therefore, the market price is approximately $11.67.

Key Concepts

Consumer Demand Function

The consumer demand function is a critical concept in economics, helping us understand how the quantity of a product demanded by consumers changes with price. In this scenario, consumers initially wish to buy 1,000,000 T-shirts when the price is \(\\) 15\( each. However, as the price increases by each \)\\( 1\), the demand decreases by 100,000 T-shirts. This relationship can be expressed using a linear function:

• At the starting price of \(\\) 15\(, the quantity demanded is 1,000,000 T-shirts.
• The slope of this demand function represents the change in quantity demanded per \)\\( 1\) change in price, which is -100,000.

We summarize this relationship with: \[ Q = 2{,}500{,}000 - 100{,}000p \] Where \( Q \) is the quantity of T-shirts and \( p \) is the price in dollars. Here, 2,500,000 is the y-intercept, indicating the projected demand if the price were hypothetically zero, reflecting the underlying consumer interest in T-shirts. This function effectively captures the sensitivity of consumer demand to price changes.

Vendor Supply Function

The vendor supply function in economics reflects how suppliers adjust the quantity of goods they are willing to offer based on changes in price. For this exercise, vendors initially supply 2,000,000 T-shirts when the selling price is \(\\) 20\(. For each additional \)\\( 1\) increase in price, vendors are willing to supply an extra 150,000 T-shirts due to expected higher profits. The vendor supply function is also represented as a linear relationship:

• At a base price of \(\\) 20\(, the supply is 2,000,000 T-shirts.
• The function's slope is positive at 150,000, indicating the incremental increase in T-shirt supply per \)\\( 1\) increase in price.

Accordingly, the vendor supply function can be expressed as:\[ K = 200{,}000p - 1{,}000{,}000 \] Here, \( K \) represents the number of T-shirts vendors will supply and \( p \) signifies the price per T-shirt. The y-intercept is negative, indicating if the price were zero, they would theoretically not supply any T-shirts. This reflects the dependency on pricing for supply levels.

Market Price Determination

Market price determination occurs when the quantity of goods consumers want to buy equals the quantity that vendors wish to sell, achieving an equilibrium in the market. This balance is vital for ensuring that neither excess supply (leading to unsold inventory) nor excess demand (causing shortages) persists. To find the equilibrium price in our example, set the consumer demand function equal to the vendor supply function:\[ 2{,}500{,}000 - 100{,}000p = 200{,}000p - 1{,}000{,}000 \] Solving this equation provides the equilibrium price \( p \), where:\[ 2{,}500{,}000 + 1{,}000{,}000 = 300{,}000p \] \[ 3{,}500{,}000 = 300{,}000p \] \[ p = \frac{3{,}500{,}000}{300{,}000} = 11.67 \] Thus, the market price is approximately $11.67. At this price, the quantity of T-shirts consumers buy equals the quantity suppliers provide, ensuring a balanced market without oversupply or shortage.