Look again at Figure 11.17 (p. 432 ). Suppose that the marginal costs c1 and c2 were zero. Show

Look again at Figure 11.17 (p. 432 ). Suppose that the...

Key Concepts

Marginal Cost

Marginal cost represents the cost of producing one additional unit of a product. When marginal costs are zero, the firm does not incur additional production expenses when selling extra units or bundles. This condition allows the firm to focus directly on revenue maximization through optimal pricing strategies without the constraint of additional costs.

Mixed Bundling

Mixed bundling gives customers the option to purchase either the complete set or individual components separately. While this strategy can cater to different customer preferences, it may be less profitable when the marginal costs are zero because selling items individually might force lower pricing to capture buyers who value only part of the bundle, thereby reducing overall revenue.

Pricing Strategy Under Zero Marginal Cost

In a zero-marginal-cost scenario, the pricing strategy is aimed at setting a bundle price that maximizes consumer participation while capturing the highest aggregate willingness-to-pay. The optimal bundle price is often set at the level where the bundle attracts all consumers who have a value above that threshold, thereby maximizing profits since there are no additional costs incurred for each extra bundle sold.

Bundling Strategies

Bundling is a method by which multiple products or services are packaged together and sold as a single combined unit. This approach can capture additional consumer surplus and reduce transaction costs, potentially improving overall profitability compared to selling items separately.

Pure Bundling

Pure bundling involves offering only the package of products as a single unified product, with no option for purchasing items individually. This method can be advantageous when marginal costs are zero, as it allows the firm to extract the full combined willingness-to-pay from consumers without diluting the value of the bundle through lower individual pricing.

Transcript

00:02 For this problem, we have the derivative of p over the derivative of c equals negative 200 over c plus 3 to the three halves.

00:11 And then we also have p of c if p equals 10 and c equals 61.

00:18 The first part of the problem is we need to find the profit function.

00:21 To do this, we have to rearrange this formula and separate the variable.

00:27 So we have d of p equals negative 200 over c plus 3.

00:37 To the three halves times d of c and then from here we can integrate this so we would have the integral of dp equals and we take the 200 out the negative 200 out so we have negative 200 the integral of 1 over c plus 3 to the 3 halves dc and then from there we can go on to do p equals negative 200 c plus three to the three halves plus one over negative three halves plus one plus c1 and then we keep putting this down and making it simple we're simplifying this negative 200 times c plus 3 to the negative to the one half over negative one half because these cancel out plus c1 and p equals negative 200 to the minus 2 times c plus 3 because we can put this and flip it over and put it on top to the 1 half plus c1 and these are negative that this and this are negative and then we can reduce this even further and put it into the formula of p equals 400 over the square root of c plus 3 plus c1 so you multiply this the negatives cancel and since this is a negative 1 half, it can go back down to the bottom.

03:33 And then you just take the square root of it down there, since it's squared.

03:38 And then you just add that.

03:43 Then from here, we can plug in the p equals 10 and the c equals 61.

03:51 So we'd have 10 equals 400 over 61 plus 3 plus c1.

04:09 So we're looking for c1 now.

04:12 From here, what we get is 10 equals 400 over the square root of 64, which is 8.

04:29 So we have 10 equals 400 over 8 plus c1.

04:42 Then that can reduce down, so we have 10 equals 50 plus c1.

04:51 And then we can subtract the 50 onto this side, so we'll have c1 equals negative 40.

05:03 And then we can plug that back into the formula that we put up there of pc equals 400 over c plus 3 and square to c plus 3 minus 40.

05:27 And that is the solution to part a.

05:33 For part b, for part b, what we need to do is we need to put in p of 0 into this and solve for c...

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