Firms and customers II

Materials for class on Wednesday, February 14, 2018

Contents

Slides

Download the slides from today’s lecture.

Raspberry Cordials

• Raspberry cordials, empty file
• Raspberry cordials, filled out

Finding the cheapest cost of production

Cost function

Total costs

If we decompose total costs into fixed and variable costs, we see that the rise in costs is driven almost entirely by increasing variable costs.

Average costs

The optimal point on the ATC curve occurs when Q = 3.93. This is also not coincidentally where the MC curve intersects the ATC curve. The optimal price at this quantity is $11.52 per gallon of cordial, but the firm won’t necessarily be able to set the price at that point on its own (unless it’s a monopoly; and even then, they’ll set it higher).

IMPORTANT NOTE: Because we’re dealing with curves and not lines, calculating marginal values with Excel by subtracting the previous value from the current value will not be 100% accurate. The only way to get perfectly accurate marginal values is to use calculus to find the instantaneous derivative at exactly that point.

Finding the optimal price and quantity

The firm’s quantity decision depends on the market demand for raspberry cordial. The demand curve for this market looks like this:

With this demand curve, we can find the price and quantity that would produce the maximum revenue, assuming there were no costs to production.

The firm can maximize its revenue by producing 5.5 gallons of cordial, which would create $151.25 in revenue.

However, this doesn’t take into account the firm’s costs. The firm’s profit maximizing point is defined as \(MC = MR\), so we need compare marginal costs and marginal revenues and calculate total profit (π) across all quantities of output.

Again, note that using chunky marginal values by subtracting previous values won’t be as accurate as calculus-based instant marginal values.

The point where \(MC = MR\) can’t be seen in the table, since it happens between 4 and 5 gallons. If the firm produces 4.26 gallons of cordial at a price of $12.38 per gallon, it will achieve its maximum profit of $94.43.

Elasticity of demand

Finally, we calculated the elasticity of demand of raspberry cordials. Recall the formula for elasticity:

\[ \begin{align} \varepsilon &= -\frac{\% \text{ change in demand}}{\% \text{ change in price}} \\ &= - \frac{\% \Delta Q}{\% \Delta P} \end{align} \]

Remember that \(\% \Delta Q = \frac{Q_{\text{new}} - Q}{Q}\) and that \(\% \Delta P = \frac{P_{\text{new}} - P}{P}\) (or just \(\frac{\text{new} - \text{old}}{\text{old}}\)). We can also write \(Q_{\text{new}} - Q\) as \(\Delta Q\), or just the change in \(Q\) (and also \(\Delta P\)) This means we can rewrite the equation like so:

\[ \begin{align} \varepsilon &= - \frac{\% \Delta Q}{\% \Delta P} \\ &= - \frac{\frac{Q_{\text{new}} - Q}{Q}}{\frac{P_{\text{new}} - P}{P}} \\ &= - \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}} \end{align} \]

We can then simplify this huge hairy fraction by multiplying both the numerator and denominator by the inverse of the denominator, \(\frac{P}{\Delta P}\):

\[ \begin{align} \varepsilon &= - \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}} \times \frac{\frac{P}{\Delta P}}{\frac{P}{\Delta P}} \\ &= - \frac{\Delta Q}{Q} \times \frac{P}{\Delta P} \\ &= - \frac{\Delta Q}{\Delta P} \times \frac{P}{Q} \end{align} \]

That’s the final version of the price elasticity of demand formula: \(\varepsilon = - \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}\). Conveniently, \(\frac{\Delta Q}{\Delta P}\) is also the slope of the demand curve.

Demand is elastic as long as the slope of the revenue function is positive, and demand is inelastic when the slope of revenue is negative, as seen here:

Feedback for today

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