Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Structural Models and Economic Computation/IV-06
Demand, Supply, Bundles, and Welfare · Chapter IV-06

Viewing-Time Optimization and Set-Valued Demand

A bundle's value can arise from an internal allocation problem rather than a sum of item values. With concave viewing benefits and a time constraint, optimal attention depends jointly on all available channels. Reference-bundle increments approximate this set value only under restrictions that make marginal contributions portable across sets.

Conceptual map

  1. IV-06.01Time allocation
  2. IV-06.02KKT systems
  3. IV-06.03Proportional allocation rules
  4. IV-06.04Reference bundles
  5. IV-06.05Purchase and choice sets

1. Bundle value can be an internal allocation optimum

When items compete for a time budget, the value of a set depends jointly on every available item. A marginal increment at one reference set need not travel to another.

For a finite channel set \(S\), choose viewing times \(t_{j}\ge0\) to maximize \(\sum _{j\in S}a_{j}\sqrt{t_{j}}\) subject to \(\sum t_{j}\le T\), where \(T\ge0\) and \(a_{j}\ge0\).

Proposition 1 · Proportional time allocation

If \(T>0\) and at least one \(a_{j}>0\),

\[t_{j}^{*}=T a_{j}^{2}/\sum _{k\in S}a_{k}^{2}, \quad W(S)=\sqrt{T\sum _{k\in S}a_{k}^{2}}.\](1)

Proof. For positive-weight channels, the KKT condition is \(a_{j}/(2\sqrt{t_{j}})=\lambda\). Solving and imposing the binding time constraint gives the allocation. Substitution gives the value. Zero-weight channels receive zero time. ∎

2. Marginal values depend on the reference bundle

Let \(T=1\) and channel weights be \(a_{1}=3\) and \(a_{2}=4\). Then \(W\)({1})=3, \(W\)({2})=4, and \(W\)({1,2})=5. Channel two adds four to the empty set and only two to {1}. Channel one adds three to the empty set and one to {2}.

Optimal joint viewing times are 9/25 and 16/25. Treating standalone values as additive would assign bundle value seven and ignore competition for time.

3. Concavity generates diminishing increments

Equation (1) composes the modular weight \(\sum a_{j}^{2}\) with a concave square root. The resulting set function is monotone and submodular: adding a fixed channel produces a weakly smaller increment when the reference set has a larger accumulated squared weight.

A purchase model can add price and random utility around \(W(S)\), but its probabilities must use exact set values or an approximation whose reference domain is stated. Reference increments are local summaries of a set function, not primitive item utilities.

4. Empty sets, zero weights, and additional constraints

Failure case · Reference increment reused as a portable item value

The increment of channel two is four at the empty set and two at {1}. Substituting the first number into every bundle overstates value whenever other channels already use the time budget.

Define \(W(\varnothing)=0\). At \(T=0\) every feasible allocation is zero. If all weights vanish, every feasible allocation has value zero and the proportional formula has a zero denominator. Minimum viewing blocks, channel-specific caps, or nonconcave benefits change the KKT regimes and can make the optimizer set valued.

5. Implementation, exercises, and sources

Enumerate finite bundles, solve the internal allocation, and verify primal feasibility, multiplier signs, complementary slackness, and value. Compare exact set values with any reference-increment approximation on every evaluated bundle.

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Exercises

  1. Reproduce all values and times in Section 2.
  2. Prove submodularity directly from the square-root increment.
  3. Add an upper viewing cap of one half to channel two and resolve the two-channel problem.
Partial solutions

1. Squared weights sum to 25, so times are 9/25 and 16/25 and value is five. 2. The increment \(\sqrt{x+w}-\sqrt{x}\) decreases in \(x\). 3. Channel two binds at one half; allocate the remaining half to channel one and compare the KKT multipliers.

  1. Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Chapter 5.KKT conditions for concave resource allocation.
  2. Satoru Fujishige, Submodular Functions and Optimization.Concave-over-modular set functions.
  3. Hal R. Varian, Microeconomic Analysis, chapters on constrained consumer choice.Resource allocation and indirect value.

6. Audit checkpoint

State the bundle universe, time budget, benefit functions, zero weights, empty-set convention, allocation constraints, KKT regime, exact set value, reference bundle, approximation error, purchase shock law, and solver residuals.

7. Scope boundary

The chapter covers a finite concave viewing-allocation model and its set value. Strategic content supply and large-scale bundle estimation require separate models.

Prerequisites