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 \(\$ 1.50\)

Step by step solution

01

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

02

Identify the total cost formula

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

03

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#

04

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

05

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#

06

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

07

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.

08

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

09

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.

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