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Register a new userOptimal Pricing under Mixed Logit Choice
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In this paper we consider the problem of pricing multiple differentiated products. This is challenging as a price change in one product, not only changes the demand of that particular product, but also the demand for the other products. To address this problem, customer choice models have recently been introduced as these are capable of describing customer choice behavior across differentiated products. In the present paper the objective is to obtain the revenue-maximizing prices when the customers decision making process is modelled according to a particular customer choice model, namely the mixed logit model. The main advantage of using the mixed logit model, also known as the random coefficients logit model, for this purpose is its flexibility. In the single-product case we establish log-concavity of the optimization problem under certain regularity conditions. In addition, in the multi-product case, we present the results of our extensive numerical experiments. These suggest that the mixed logit model, by taking unobserved customer heterogeneity and flexible substitution patterns into account, can significantly improve the attainable revenue.
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We consider assortment optimization over a continuous spectrum of products represented by the unit interval, where the sellers problem consists of determining the optimal subset of products to offer to potential customers. To describe the relation between assortment and customer choice, we propose a probabilistic choice model that forms the continuous counterpart of the widely studied discrete multinomial logit model. We consider the sellers problem under incomplete information, propose a stochastic-approximation type of policy, and show that its regret -- its performance loss compared to the optimal policy -- is only logarithmic in the time horizon. We complement this result by showing a matching lower bound on the regret of any policy, implying that our policy is asymptotically optimal. We then show that adding a capacity constraint significantly changes the structure of the problem: we construct a policy and show that its regret after $T$ time periods is bounded above by a constant times $T^{2/3}$ (up to a logarithmic term); in addition, we show that the regret of any policy is bounded from below by a positive constant times $T^{2/3}$, so that also in the capacitated case we obtain asymptotic optimality. Numerical illustrations show that our policies outperform or are on par with alternatives.
The study of network formation is pervasive in economics, sociology, and many other fields. In this paper, we model network formation as a choice that is made by nodes in a network to connect to other nodes. We study these choices using discrete-choice models, in which an agent chooses between two or more discrete alternatives. One framework for studying network formation is the multinomial logit (MNL) model. We highlight limitations of the MNL model on networks that are constructed from empirical data. We employ the repeated choice (RC) model to study network formation cite{TrainRevelt97mixedlogit}. We argue that the RC model overcomes important limitations of the MNL model and is well-suited to study network formation. We also illustrate how to use the RC model to accurately study network formation using both synthetic and real-world networks. Using synthetic networks, we also compare the performance of the MNL model and the RC model; we find that the RC model estimates the data-generation process of our synthetic networks more accurately than the MNL model. We provide examples of qualitatively interesting questions -- the presence of homophily in a teen friendship network and the fact that new patents are more likely to cite older, more cited, and similar patents -- for which the RC model allows us to achieve insights.
This paper extends the project initiated in arXiv:2002.00201 and studies a lifecycle portfolio choice problem with borrowing constraints and finite retirement time in which an agent receives labor income that adjusts to financial market shocks in a path dependent way. The novelty here, with respect to arXiv:2002.00201, is the fact that we have a finite retirement time, which makes the model more realistic, but harder to solve. The presence of both path-dependency, as in arXiv:2002.00201, and finite retirement, leads to a two-stages infinite dimensional stochastic optimal control problem, a family of problems which, to our knowledge, has not yet been treated in the literature. We solve the problem completely, and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if state constraints are present. Note that, differently from arXiv:2002.00201 , here the HJB equation is of parabolic type, hence the work to identify the solutions and optimal feedbacks is more delicate, as it involves, in particular, time-dependent state constraints, which, as far as we know, have not yet been treated in the infinite dimensional literature. The explicit solution allows us to study the properties of optimal strategies and discuss their financial implications.
Adopting a probabilistic approach we determine the optimal dividend payout policy of a firm whose surplus process follows a controlled arithmetic Brownian motion and whose cash-flows are discounted at a stochastic dynamic rate. Dividends can be paid to shareholders at unrestricted rates so that the problem is cast as one of singular stochastic control. The stochastic interest rate is modelled by a Cox-Ingersoll-Ross (CIR) process and the firms objective is to maximize the total expected flow of discounted dividends until a possible insolvency time. We find an optimal dividend payout policy which is such that the surplus process is kept below an endogenously determined stochastic threshold expressed as a decreasing continuous function $r mapsto b(r)$ of the current interest rate value. We also prove that the value function of the singular control problem solves a variational inequality associated to a second-order, non-degenerate elliptic operator, with a gradient constraint.
We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a consumption constraint. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion $0<alphale1$ of her consumption habit, which is the exponentially-weighted average of past consumption rates. $alpha=1$ prohibits the habit process to decrease and corresponds to the completely addictive model. $alpha=0$ makes the habit-formation constraint moot and corresponds to the non-addictive model. $0<alpha<1$ leads to partially addictive models, with the level of addictiveness increasing with $alpha$. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility. Assuming that the individual invests in a risk-free market, we formulate and solve an infinite-horizon, deterministic control problem to maximize the discounted CRRA utility of the consumption-to-habit process subject to the habit-formation constraint. Optimal consumption policies are derived explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either: (1) larger than a high threshold; or (2) below a low threshold and the agent is less risk averse. Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.
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