# Economic Analysis of Availability of Follow-on Protein Products. Appendix A: Memo on Entry Modeling A-1

William B. Vogt

The purpose of this memo is to provide a theoretical framework and empirical suggestions for the "Economic analysis of the availability of follow-on protein products." This document is based on the Lewin/i3 Innovus technical specifications for that analysis, on the entry framework of Bresnahan & Reiss (JPE, 1992), and on the empirical work of Grabowski, Ridley, & Schulman.

1. The Entry Model

The key equations for step 1, "world with FoPPs," are the entry equations:

N=S/N(PN - AVCN)d(PN) - FN           (1)

The number of ÕNis greater than zero. If we ignore the fact that N must be an integer, we can write:

N S(PN - AVCN)d(PN)/ FN           (2)

The number of entrants is going to be the N given by approximation 2, rounded down to the nearest integer. The notation for this equation is defined in Table 1.

Table 1: Notation for entry model

Symbol Meaning
N Profits of Nth entrant
S Market size (in people)
N Number of entrants under consideration
PN Price ofr the drug of the Nth entrant
AVCN Average variable cost of the Nth entrant
dN Per-capita demand at price PN
FN Fixed cost of entry of Nth entrant

Since this is a static, one-period model, when we pass to thinking about the real world, everything must be recast in net present value (NPV) terms. Fixed costs are paid only once, but profits are earned for several periods, so we should think of the whole numerator in approximation 2 as multiplied by the necessary factor to put it in NPV terms: for an infinitely lived product with constant profits, we would multiply the numerator by 1/r, where ris the typical entrant's cost of capital.

Now, if this factor is constant across drugs, it is of no particular concern. However, if there is some reason to believe that biologic drugs will be on the market longer/shorter or that they will experience less/more entry over time by competing branded drugs, then something should be done to boost/shrink market size for the biologics.

1.1  Simple manipulation of the entry equation

Equation 2 can yield some interesting results. If we ignore price effects of entry, we can see that the "first-order" effect of a change in the fixed cost of entry is to change the number of entrants by the same percentage, δlnN/δlnFN=1 or the fixed-cost elasticity of entry is roughly one. Similarly, the average margin elasticity of entry is one and the market size elasticity of entry is one.

It can be useful to rewrite this equation in terms of revenue and price-cost margin:

N  = (S/N) R(PN - AVCN)d(PN) - FN

N  = (1/N) [Sd(PN)PN] [PN -AVCN/PN]- FN

N  = (1/N) R(PN)PCM(PN) - FN

(3)

Equating profits to zero and doing a little algebra yields:

ln(N) ≈ ln Rev(PN) + ln PCM(PN) - ln FN     (4)

Now, suppose then that we start with some small-molecule generic market about which we know a lot, and we consider how some biologic "generic" market is going to differ. As long as the differences in the various quantities are small (and maintaining the assumption of no price effects of entry), a reasonable approximation of the percent difference in the number of entrants (in the long run) is going to be:

∆%N ≈ ∆%Rev + ∆%PCM - ∆%FN     (5)

Unfortunately, we cannot expect the differences to be small (especially regarding fixed costs), so we must write the formula properly:

Nbiologic / Nsmallmol ≈ Revbiologic / Revsmallmol PCMbiologic / PCMsmallmol Fsmallmol / Fbiologic       (6)

that there is a small-molecule market with 10 generic entrants. Suppose further that we are interested in predicting the number of entrants in a biologic market (somehow similar in the view of the analyst, e.g., same disease treated). Say the biologic and small molecule markets have the data in Table 2, with revenue and price-cost margin measured pre-patent-expiry.

Table 2: Calculating relative entry levels
Variable Small Molecule Biologic Ratio

Revenue (SdP)

\$1BB \$600MM 0.6

Markup (PCM)

50% 75% 1.5

Fixed Cost (F)

\$100MM \$300MM 0.3
Total (multiplying)     0.3

The rows in that table correspond to the terms in Equation 6. Multiplying down the last column of the table is multiplying across that equation, and the total at the bottom is the ratio on the left-hand-side of the equation. This simple analysis would lead us to estimate that there will be 30% as many, or three biologic entrants. This framework assumes no reactions of price to entry and, therefore, no differential reactions of price to entry and, therefore, no reaction of demand to entry, etc.

1.2  Introducing price effects

To do the entry model properly requires recognizing that prices in the generic market decline with entry. We expect the prices of generic drugs to drop with entry of more generics, as demonstrated in the relevant economic literature. As price falls with entry, price-cost margins will also fall. Revenues may fall or rise, depending on the price elasticity of demand; if demand for prescription drugs is price inelastic, then revenue will fall with entry.

We can approximate the response of revenues and price-cost margins to entry with constant elasticity functional forms as follows. For revenues,
Rev(PN) = AN, and, for price-cost margin, PCM(PN) = BN. Since we will often know the revenue and price-cost margin for the innovator drug before generic entry, it is helpful to recast these equations relative to the innovator's revenues and price-cost margins pre-expiry, thus
Rev(PN) / Rev(P1) = N and PCM(PN) / PCM(P1) = N. Notice the negative signs on d and g, as these parameters measure the percent fall in revenues and price-cost margins with entry.

A little algebra on equation 4 leads us to:

ln (N) ln Rev(P1) +δ ln N + ln PCM1+ γ lnN - ln FN                            (7)

This equation can be solved for N:

ln (N) ln Rev(P1) + ln PCM1- ln FN / 1 + γ + δ                                    (8)

From this, we can calculate the elasticity of entry:

η = 1 / 1 + γ + δ                                                                                       (9)

We now need estimates of d and g, the entry elasticity of price-cost margin and the entry elasticity of revenue, respectively. The price elasticity of revenue is equal to one minus the price elasticity of demand, 1 - e. We know that the price elasticity of price-cost margin is equal to (1 -PCM)/PCM. Thus, if we know the price-cost margin, the price elasticity of demand, and the entry elasticity of price, we can calculate the elasticity of entry, h.

A reasonable estimate of the price elasticity of demand from the literature is e = 0.4.

Grabowski et al. (hereafter GRS) find that each entrant reduces prices by 9%. Since they see eight entrants on average, this corresponds to an entry elasticity of price of 0.72. However, since GRS made their estimate on small-molecule generics and since biologic generics are likely to be more differentiated than small-molecule generics, the entry elasticity of price is likely to be smaller than this.

1.3  Modifying price effects & entry elasticity for biologics

We can get an idea of how much smaller using standard models of product differentiation. The CES model of Dixit and Stiglitz has a pricing equation of:

PN = c (1 + ημ / n - 1)                                                                                    (10)

The logit product differentiation model has a very similar pricing equation:

PN = c + ημ / n - 1                                                                                          (11)

In each case, m is a parameter controlling how differentiated the products are. It is a little easier to work with the logit model. The cross-price elasticities of demand are proportional to m in this model, as is the elasticity of price with respect to N.

So, in the logit model at least, if we think that generic biologics are only half as substitutable with one another as are generic small-molecule drugs, then we should reduce the entry elasticity of price by one-half for biologicals, to 0.36.

1.4  Putting it together

First, we calculate gand d. The entry elasticity of revenue, d, is just the entry elasticity of price, 0.36, times the price elasticity of revenue, 0.6, or 0.22. The entry elasticity of price-cost margin is just the entry elasticity of price, -0.36, times the price elasticity of price-cost margin, which is 0.43 for a price-cost margin of 70%, or 0.15. This makes the entry elasticity, h, equal to 1/(1 + 0.22 + 0.15), or 0.73.

Returning to the example of Table 2, we found that entry would be only 30% of the small-molecule-drug level for the similar biologic. That now has to be modified appropriately to deal with h < 1. The improved analysis leads us to change Equation 6 as follows:

Nbiologic / Nsmallmol = (Revbiologic / Revsmallmol    PCMbiologic / PCMsmallmol    Fsmallmol / Fbiologic )n             (12)

To calculate the actual difference between the small-molecule and biologic examples, we need to take 30% to the 0.72 power. Since 0.30.72 = 0.42, we conclude that there will be 42% as many entrants, or about 4 instead of about 10.

2 Applying the method

The method we describe above allows us to compare the expected number of generic entrants between a small-molecule drug and a similar biologic drug, based on differences in revenue, price-cost margin, and fixed costs of entry. In practice, we apply this model to each biologic market by, first, predicting the number of drugs that would enter that market were it a small-molecule market given the market's revenue. Then, we modify that number of entrants to take account of the differences between small-molecule and biologic drugs using the formula:

Nbiologic = Nsmallmol (PCMbiologic / PCMsmallmol    Fsmallmol / Fbiologic )n                                             (13)

The revenue term is omitted since revenue was used to predict the number of small-molecule entrants.

3 Differences with Grabowski et al.

The method we describe above is different from that described by GRS in several ways.

First, we derive our estimate of hrather than estimating it. GRS estimate h using data from small-molecule drugs and then assume that this h also applies to biologic drugs. Because we believe that: (1) biologic generic drugs are likely to be more differentiated than small-molecule generics, (2) price-cost margins for biologics are likely to be different than those for small-molecule drugs, and (3) these differences lead to differences in the entry elasticity as a theoretical matter (as discussed above), it is important to adjust the entry elasticity, accordingly.

Second, we apply our estimate of the entry elasticity to the differences in fixed costs and to the differences in price-cost margins between small-molecule drugs and biologic generics. Again, as discussed above, this difference is mandated by the standard theory of entry. Just as the higher fixed costs of biologic generics discourage entry, the higher price-cost margins of biologics encourage entry, and we need to account for tradeoffs between these two forces.

Third, we apply our adjustment formula in a multiplicative form, as in Equation 13, rather than in a partially linear form, as GRS do in their equations 2 and 6. For comparison, their Equations 2 and 6 would imply that, for a small molecule and biologic differing only in fixed costs:

(Nbiologic - Nsmallmol ) / Nsmallmol = η(Fsmallmol / Fbiologic ) / Fsmallmol                                                    (14)

whereas, our formula would yield (again for drugs differing only in fixed costs):

Nbiologic = Nsmallmol (Fsmallmol / Fbiologic )n                                                                                                (15)

It is apparent that GRS's equations are a linear approximation to ours by noting that:

η = Nbiologic / Fbiologic    Fbiologic / Nbiologic

≈ (Nbiologic - Nsmallmol) /Nsmallmol / (Fsmallmol- Fbiologic) / Fbiologic                                                           (16)

and observing that the above expression is equivalent to their formulation. There are two related problems with the GRS formulation. First, the approximation used above is derivative- based and is therefore only valid for small differences in fixed costs. Since we expect large differences in fixed costs between biologic and small-molecule drugs, this renders the approximation suspect. Second, for any proportional difference in fixed costs greater than 1/h, the GRS model predicts negative numbers of entrants. Our formulation never predicts negative numbers of entrants, although it can predict zero entrants if Nbiologic is less than one, since we always round fractional numbers of entrants down. A prediction of zero entrants can sometimes be reasonable, such as for sufficiently high difference in fixed costs; whereas a prediction of a negative number of entrants is not.

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