I’ve had a think about this and I think a better way to vividly illustrate what I am getting at would actually be with pursuing an analogy considering production.
The reasons are two-fold. First (as Rothbard himself correctly emphasises in MES), money can be thought of as the most convertible good in existence for the conduct of action within a market economy. Where we might place it on a value scale against all the ends we might utilise it to purchase other items with concerns the valuen placed on keeping a sum of it for cash-holding as an end. Hence, a nice way to illustrate the effects of rising “costs” would be to consider the decisions of how to allocate a convertible capital/consumer good. Secondly, since money itself is quite a distinguishing and complicating factor of the analysis subject to its own variations in purchasing power based on expectations and actions based on them, I think a better way to isolate what I am getting at with regard to a “pure income effect” would be to utilise an example not explicitly considering money. This would mean the phenomena is more universal than just for indirect exchange economies, and this would make sense given that necolassical value and price theory is itself pretty much “money-less” in an essential sense.
So let us explicitly consider a simple example. lets say we have a convertible good, e.g. cups of water to use a scenario very similar to the type Menger originally used to illustrate marginal utillity in his Principles. We first of all recognise the elemntary but important point that it is not means that are ultimately valued but ends. This of course is the same for indirect means; capital goods whose value is iindirectly imputed from lower order goods from their ultimate ends.
Hence say we have 3 cups of water. One is needed for each of the following purposes ranked in descending order of importance: drinking to keep oneself alive for one day, keeping one’s household pet alive, watering your plant for the day. Now, going via this traditional example we may say a cup of water has been lost, and of course one must decide where to apply the remaining cups. The essential aspect of the decision however is a choice among ends however; which one to forsake and which one to keep. The marginal end forsaken out of the 3 if forced to keep just 2 by the acting individual, and the value of this purpose imputes the value of any of the cups of water.
Now, let us modify the example only slightly. Say that you fall ill, such that actually at least 2 cups would be required to sustain and keep yourself alive for the day. You still have 3 cups of water, and the requirements for the other purposes have not changed. This example of course does not employ a situation in which the same number of units of water for each purpose, but this in no way detracts from our abillity to analytically realise the resultant consequences, once we focus our attention on the choice scenario regarding the ultimate determinats of valuation; the ends (if you want a good example of how the Austrian framework is flexible enough to deal with such scenarios, and avoid the traps caused by the diminishing marginal utillity forumulations used by mathematical economists, see Mises HA, p.128). We resolve the consequences in this situation by focusing on the possible tradeoffs among ends. After ensuring the possibility of the most highly ranked end it is no longer possible to realiseb both the second most valuable and third most valuable end, hence the latter is the the one forsaken. I hope this is abundantly obvious. Furthermore, the end for whom the requirements have increased is not the one that suffers a diminishing in its realisation as a result of a “price” change, half of the effect considered above.
We need only slightly alter the scenario to forsee the possibillity of not only a restriction in the decided realisation of an alternative end as a result of a rise in the requirements to realise one end, but an actual increase in the “demand” for the realisation of the more “costly” end. Hence we might consider a scenario where let’s say we have the cups of water (with the actor still posessing only 3 of them) represented by w able to produce goods A and B that help realise a set of ends E1-E3. Lets say the rankings of the ends and the requirements for the achievement of each one may be as follows:
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1w → A → E1.
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2w → B → E2
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1w → A → E3.
Now if the requirements to realise the production of good A somehow change for the individual (one conceivable reason for such a change might be a change undergone by a complementary capital good required for A’s realisation), from 1w to 1.5w our actor is faced with a tradeoff between ends. First and foremost, it becomes abundantly clear that ends E1 and E2 cannot both be realised together. Faced with the choice between them E2 is forsaken. Yet due to the discontinuous nature of the tradeoffs and means-end relations this leaves the opportunity of applying the remainder 1.5w to another purpose. If as above, the next most valued end that can be realised with them also involves the production of A, then considering as Mises abundantly demonstrated, action involves the exchange of conceivable states of affairs a scenario in which E1 and E3 can both be realised, would be willingly exchanged for one where only E1 could be realised.
Now this example helps illustrate some important points. First, we see that the income effect, and the resultant consequence through which a “Giffen good effect” can be observed is by no means at all characteristic only of the conditions of a market economy, but actually a general part of value theory. Secondly, having derived and accounted for this possibility in valuational terms, as praxeologists we can also succeed over neoclassicals in thus explaining why the income effect may generally occur as a result of chainging prices in a market economy (so that rising/falling prices of goods can cause demand reductions/increases in other goods), we can also effectively account for why the observation of the second part of this effect (causing the label of Giffen Good) is so rarely observed.
The reasons are due to the fact that money itself is the most divisible and convertible good on the market, thus due to the very large number of other consumption possibilities made available by the extra funds remaining after abstaining from end like E2, it is much more likely ceteris paribus that they would be used to purchase more of alternative items as opposed to more of A. Furthermore, money allows for the division and joint ownership and contracts of use of other goods, loans etc. and all sorts of in betweens as you pointed out that might make the observation of the second part of this effect unlikely.
But this is no reason to deny the possible existence of such tradeoffs valuationally, as well as their more general significance.