Applying concepts of producer’s theory to real-life problems is seldom easy. Let us try to do this with what I am doing now: posting on a blog.
Blog posts are assumed to be a function of time (t) and mental effort (e) in a Cobb-Douglas function. One will stipulate the cost of one unit of time spent in writing a post as the value of leisure forgone (traditionally, the wage rate). As for the cost of effort, it is the opportunity cost of spending part of my daily endowment of mental capacity with the post, which I could otherwise employ in studying, with marginal returns in the form of higher future wages (because of higher grades). Assuming decreasing marginal utility of future salary with respect to grades (and, by the chain rule, to effort), there is an increasing cost of effort (as we employ more effort on the blog, we put less effort on studying and the marginal opportunity cost increases). For simplicity, the cost of effort is assumed to come in a quadratic form.
Independency of the costs of mental effort and time are assumed from their definition: when using more time or more mental effort, the blogger is forgoing two independent things: on one hand, leisure; on the other, mental capacity for other mentally demanding activities. Besides, in blogging, as in other intellectual activities (like studying), there are tasks which require mostly time (like purely descriptive posts or studying accounting) and tasks which require one to be essentially productive (like drawing conclusions from what one sees or solving problem sets). A tired or time-constrained blogger can still post something according to his (fairly independent) endowments of time and mental capacity.
One should note that posts in a blog like this one are public goods: there is both non-rivalry in consumption (given that the hosting server is far from capacity) and impossibility of exclusion (as it is the case with the regular blog, the available technology does not allow the author to block a given IP; however, if the blog features a paid website, exclusion is made possible). Such description suits the de facto impossibility of the blogger to charge each viewer his/her marginal valuation of reading one post (or, as most often happens, any price whatsoever).
Anyhow, the blogger would not need to stipulate a price on posts: he derives utility from simply having people reading his work (say, in a linear fashion, U(b) = A.b + U0(C), where A is some constant, b is one post entry and C is consumption; note that, as consumption depends on goods’ prices and income, both not affected by blogging, it is mere aesthetical to have here the second parcel). To convert this into monetary units, one uses λ, the marginal utility of income when solving the consumer’s optimization problem by the Lagrangian method (and which is used to convert changes in the indirect utility function to changes in the consumer surplus).
Utility arises, however, from post views, rather than from number of posts (and, here, non-rivalry works in favor of the blogger’s “revenue”). Assuming that the number of views comes from the quality of the posts (which by turn depends, again, on time and effort), b can be interpreted as a quality-adjusted measure of blog posts. Hence, a higher b can mean a higher quality post (with many views) or many average-quality posts (with fewer views each).
All this leads to the following optimization problem and optimal quantities:
In the end, the model predicts that a blogger will produce more as the value he attributes to blog views (A) increases and less as the money value of utility (1/ λ) decreases and as the opportunity cost of time increases. The choice problem is drawn below and illustrates the fact that, although the production function presents constant returns to scale, costs increase non-linearly in e.
A few comments on the market structure: there is free entry and a large number of small bloggers and consumers, but each blogger writes as he wishes (both in quantity and in quality) and often without regarding other’s posts. Moreover, each blogger has his/her own “price” (A). Therefore, the market most resembles one of monopolistic competition. Although there’s non-rivalry in consumption, viewers’ time is scarce and bloggers must compete for it. A further development of the model would require some assumptions on the functional form of demand, which should include features such as taste for variety and the abovementioned time constraint. One suggestion could be a kind of linear Hotelling model with quantity rather than price competition. In the meanwhile, let us keep writing.
João Garcia (#618)