Question

In Example \(13.3,\) we computed the general short-run total cost curve for Hamburger Heaven as \\[ 400 \\] a. Assuming this establishment takes the price of hamburgers as given \((P),\) calculate its profit function (see the extensions to Chapter 13 ), \(I T^{*}(P, V, W)\) b. Show that the supply function calculated in Example 13.3 can be calculated as \(d T T^{*} / d P=\) \(q(\text { for } w=v-4)\) c. Show that the firm's demand for workers, \(L\), is given by \(-d i T^{*} / d w\) d. Show that the producer surplus calculated in Example 13.5 can be computed as e. Show how the approach used in part (d) can be used to evaluate the increase in pro ducer surplus (and in short-run profits) if Prises from \(\$ 1\) to \(\$ 1.50\)

Short answer

In conclusion, the exercise involved calculating the profit function for Hamburger Heaven, deriving the supply function from the profit function, showing the relationship between the firm's demand for workers and the profit function, and evaluating the change in producer surplus and short-run profits when the price of hamburgers changes. By following the step-by-step solutions provided, we demonstrated these relationships and showed how the company's decisions and profitability are affected by changes in prices and demand for workers.

Step-by-step solution

Identify the total revenue formula

For a price-taking firm, the total revenue (TR) is the product of the price (P) and the number of units of goods sold (q): TR = P * q

Identify the total cost formula

The given total cost (TC) formula for Hamburger Heaven is: TC = 400

Calculate the profit function

A firm's profit can be represented by the difference between its total revenue and total cost. Hence, the profit function IT*(P, V, W) can be expressed as: IT* = TR - TC Substituting in the formulas for TR and TC from Steps 1 and 2, we get: IT*(P, V, W) = P * q - 400 #b. Deriving the supply function from the profit function#

Calculate dIT*/dP

To find the relationship between the profit function and the supply function, we need to compute the derivative of the profit function with respect to P, which is: dIT*/dP = d(P * q - 400) / dP

Apply the given condition

The exercise specifies that w = v - 4. We can substitute this condition into the derivative: dIT*/dP = q (for w = v - 4) From this derivation, we can see that the exercise is asking us to demonstrate that the supply function can be represented by q under the specific condition of w = v - 4. #c. Showing the firm's demand for workers from the profit function#

Calculate dIT*/dw

To find the relationship between the profit function and the firm's demand for workers, we need to compute the derivative of the profit function with respect to w, which is: dIT*/dw = d(P * q - 400) / dw

Show that the firm's demand for workers is -dIT*/dw

Given that the firm's demand for workers, L, is given by -dIT*/dw, we can rewrite the formula obtained in Step 1 to: L = -dIT*/dw This shows the relationship between the firm's demand for workers and the profit function. #d. Calculating the producer surplus# The producer surplus calculated in Example 13.5 is the amount by which the income from selling q units exceeds the cost of producing that many units. To calculate this surplus, we can employ the approach used in parts (a) and (b) for the profit function.

Compute the producer surplus

To calculate the producer surplus, we can use the formula: PS = TR - TC where TR is the total revenue and TC is the total cost. Substitute the formulas for TR and TC from part (a) (TR = P * q; TC = 400): PS = P * q - 400

Evaluate the change in producer surplus

To evaluate the increase in producer surplus (and in short-run profits) as a result of an increase in the price of hamburgers from \(1 to \)1.50, we need to substitute these new values into the producer surplus formula obtained in Step 1 and compute the difference: PS1 = 1 * q - 400 PS2 = 1.50 * q - 400 Change in producer surplus = PS2 - PS1 Change in producer surplus when the price of hamburgers increases from \(1 to \)1.50 can now be calculated using the given information.

Key concepts

Supply Function

The supply function is an essential concept in economics that describes how the quantity of a product that a firm is willing to supply changes in relation to price changes. In this case, if we examine a firm like Hamburger Heaven, its supply function shows how many hamburgers it will provide when the price of each is given.

Using the information from the example, we recognize that:

• The firm is a price-taker, meaning it accepts the market price for its hamburgers without setting it.
• The total cost given is constant, at 400.
• The supply function is derived from the profit function by taking the derivative with respect to price.

For Hamburger Heaven, the supply function can be simplified under the condition given in the exercise: when the wage cost, denoted by 'w', equals 'v - 4'. By deriving the profit function with respect to the price, the relationship becomes clear:

• The derivative \(\frac{dIT^{*}}{dP}\) equals the quantity supplied (\(q\)).
• This indicates that Hamburger Heaven supplies hamburgers in direct response to changes in price, assuming other factors (like production capacity and costs) remain constant.

Demand for Workers

The demand for workers, in this context, refers to how many employees a firm like Hamburger Heaven needs based on its economic conditions and production decisions. This demand is influenced heavily by labor costs, reflected in wages, and how these costs affect the firm's profits.

Here's how to understand it:

• The exercise specifies that the firm's demand for labor is linked to the derivative of the profit function.
• Essentially, the firm's labor demand is given by \(-\frac{dIT^{*}}{dw}\), meaning it's influenced by changes in wages.
• This relationship depicts how an increase in wages might reduce the firm's demand for workers, as rising costs would negatively impact profitability.

By obtaining the derivative of the profit function respect to wage, the form of demand highlights how crucial labor costs are to staffing decisions. This helps us comprehend why, under certain economic conditions, firms adjust their workforce to balance between optimal production and cost efficiency.

Producer Surplus

Producer surplus is a measure of producer welfare. It's the difference between what producers are willing to accept for a good versus what they actually receive. For Hamburger Heaven, it's crucial to ensure the production cost allows for a healthy surplus.

Calculating the producer surplus involves:

• Subtracting total costs from total revenue to find the surplus.
• Employing the profit function's earlier deductions for revenue \(P \times q\) and cost (400).

Producer surplus can also demonstrate the impact of market price changes: energy:

• Consider a price increase from \(1 to \)1.50. The surplus increment is obtained by calculating the difference in revenue provided by new and old prices.
• This analysis showcases that a higher market price translates directly into increased producer surplus, highlighting increased profitability.

Deliberating these changes in producer surplus enables better strategic planning and identification of favorable price points for stable and increasing profit margins.