On the Price of Commitment Assets in a General Equilibrium Model with Credit Constraints and Tempted Consumers

1 Introduction

Since the seminal work of Strotz (1956), there is now an extensive literature stressing the importance of temptation and self-control problems in explaining individual behavior in economic models. When studying dynamic models with such dynamically inconsistent preferences, theoretical economist have developed various solutions methods to explain the behavioral observations that have been found in the empirical literature. ^[1] Two predominant frameworks analyzed in the dynamic choice literature include models of the quasi-hyperbolic discounter and the GP model of self-control (see Gul, Pesendorfer (2001)). Economists attempted to analyze and compare various implications of both frameworks in specific applications including: characterizing consumption dynamics of consumers facings temptations (see Laibson (1997) or Balbus, Reffett, and Woźny (2015) for the β–δ model and Gul and Pesendorfer (2004a) for the GP representation), temptation implications on the optimal taxation (see Krusell, Kuruşçu, and Smith (2002) for the β–δ model or Krusell, Kuruşçu, and Smith (2010) for the GP framework), or the optimal incentive schemes for tempted agents (Yilmaz (2013) for the β–δ model and Woźny (2015) for the GP representation) among many others.

Surprisingly, only few papers analyzed temptation/self control implications on equilibrium prices, demand/supply of commitment assets, left alone the existence and properties of a general equilibrium at the theoretical level. Few notable exceptions include Herings and Rohde (2006), Luttmer and Mariotti (2006, 2007), Gabrieli and Ghosal (2013), Dziewulski (2015) or examples presented in Kocherlakota (2001) and Gul and Pesendorfer (2004b). These authors discuss problems with equilibrium existence, characterize competitive equilibrium allocations or present some examples of equilibria, when households exhibit self control problems or have time inconsistent preferences. Importantly to say, all of the above mentioned papers concentrate on the exchange economies or power utilities (see Luttmer and Mariotti 2003).

Specifically and importantly to our research, Gabrieli and Ghosal (2013) showed that competitive equilibrium may not exists in a three period exchange economy (without commitment assets) with a single β–δ (time-inconsistent) consumer. This resulted from (generic) non-monotonicity and non-convexity of the sophisticated consumer’s date 1 (implied) preferences (i. e. ones obtained by incorporating date 2 and 3 optimal decisions). Demand calculated in their example do not vary with prices sufficiently to cover the part of domain with fixed supply. Next Kocherlakota (2001) showed ^[2] that in the same economy, but with commitment asset and credit constraints, the demand for one of the assets: (liquid) savings in the final date and (illiquid) date-1 commitment asset must be zero. Intuitively, if the commitment asset is relatively cheep, the first period self must leave the second period self which such level of capital (to finance date 2 consumption) that no further savings in date 2 is necessary. If on the other hand, the price of the commitment asset is too high, then the investment in the commitment asset is zero and all consumption in date 3 is covered from savings in date 2. These result from the fact that, if the second period self is unconstrained, than any replacement between assets, such that the consumption in date 2 and 3 is unchanged, leaves the cost of self control (lost efficiency from the date 1 perspective) unchanged, but may reduce costs. The corner demand result of Kocherlakota (2001) causes, however, problems with respect to existence of equilibrium of the exchange economy. Similarly to the result of Gabrieli and Ghosal (2013) the date 1 implied preferences, although concave, are not strictly monotone. In fact, unless the prices are such that consumers are indifferent between investing in the short term or commitment asset, and the number of consumers is large, the demand will not equate supply and equilibrium will not exists. Following the technical argument of Rogerson (1988), he argued that in order for the competitive equilibrium to exists the (large) group of agents must divide into those investing in the commitment asset only and those investing in the liquid assets only. In such case asymmetric equilibrium may exist but as he argued, such equilibrium allocation is not supported by empirical observations. ^[3] Importantly, note that the nonexistence example of Gabrieli and Ghosal (2013) cannot be remedied using the large economy framework as the market clearing prices do not belong to the convex hull of demand.

Finally, Gul and Pesendorfer (2004b) (Section 5.1) provided an example showing that, if we model consumer’s choice using GP framework, in the competitive equilibrium of the exchange economy, we may observe agents possessing strictly positive demands of both assets, i. e. commitment and short term bonds. The intuition used to understand the result in the β–δ model is not longer useful here. Even if the commitment asset is expensive, it may be used to cover some, but not all, date 3 consumption. On the contrary to the β–δ model, in the GP framework any change in assets such that date 2 and date 3 consumption is unchanged, changes the cost of self-control, as it depends on the level of date 2 liquid but not illiquid asset. Finally to mention, GP model does also allow for the corner solution, if prices are appropriate. As a result, Gul and Pesendorfer (2004b) claimed that GP representation/model possesses an advantage over the β–δ one, as it can model non-corner choices of all three assets and, as a result, a (symmetric) equilibrium in the exchange economy with finite number of consumers may exist.

The aim of this paper is to extend the analysis of Kocherlakota (2001) (or Gabrieli and Ghosal (2013)) and Gul and Pesendorfer (2004b) (Section 5.1) to a three period production economy with commitment assets and credit constraints, and verify, if the results of both, the above mentioned models, remain discrepant. Specifically, we prove the existence of equilibrium in both models and study how the equilibrium allocations and prices differ between both models.

From this perspective the contribution of this paper is threefold. First, we argue that, when the production side of the economy is incorporated into both of the above mentioned models (with commitment (illiquid) asset and credit constraints), the equilibrium allocation is characterized by the positive consumption of the commitment asset and corner consumption of one of the short term assets. Contrary to the examples for the exchange economy, the results of both three-period models with production are hence similar. Still, there are differences between both models, especially seen if we consider more than three periods. Second, in the paper we provide examples, when solutions of both models are indeed discrepant. Three and finally, we prove the equilibrium existence in the three-period production economy of a representative agent. The reason we restore equilibrium existence in our model is the joint presence of commitment assets together with credit constraints and active firms. In fact, the corner allocation results we obtain is necessary to avoid problems mentioned by Gabrieli and Ghosal (2013) and presence of the firm helps to avoid problems identified by Kocherlakota (2001), i. e. allow demand/supply to vary sufficiently with prices to clear the market.

In the rest of the paper, we first study the firm’s (Section 2) and then the consumer’s maximization (for GP and β−δ) (Section 3) and summarize our results with equilibrium analysis (Section 4). Section 5 discusses few extensions of our basic model by allowing for more (liquid) assets, and more than three periods.

2 Firm’s Behavior

Consider a firm possessing technology given by a production function F, defined over capital and labor inputs. We assume F: R+×R+→R+ has constant returns to scale, is weekly concave, is strictly monotone and strictly concave with each argument separately for a positive input of the other argument, twice continuously differentiable and satisfies Inada conditions. Moreover, both inputs are required for production: Fk,0=0 and F0,l=0. Firm has no endowment of capital. Each date firm hires labor lt at wage wt∈R++ as well as rents capital kt≥0 and returns 1+rt at the end of the period. In date 1 firm can also rent a (long term) capital ^[4]a. Once deciding to do so, a is used in date 2 and 3 and is returned with interest ra at the end of date 3. ^[5] There is no capital depreciation and rt,ra∈R++ Firm discounts its profits using qt=11+rt+1. Price p of (date 1) consumption good is normalized to 1. A problem of the firm is then to solve:

As asset returns are finite the Inada conditions assumed on F imply that l1,l2,l3 and k₁ are interior. However, as a and k₂ (and k₃) are substitutes it is not clear is all of these are interior. Knowing that we obtain the first order conditions:

where μii=1,2 are Lagrange multipliers associated with non-negativity constraints: −ki≤0 and μa with −a≤0. We start with the following lemmas:

Both lemmas characterize the demand of liquid and illiquid assets for given factor prices. For the case of 1+ra=1+r21+r3 the solution exists but it not unique.

We now shortly comment that the use of commitment asset shifts profits between the periods. So, denote by πt date t optimal profits, i. e. π1=Fk1,l1−w1l1−r1k1, π2=Fa+k2,l2−w2l2−r2k2 and π3=Fa+k3,l3−w3l3−r3k3−ra. Clearly π1=0, but for a positive demand of illiquid assets, profit is shifted from π₃ to π₂. The next lemma formalizes this result.

3 Consumer’s Behavior

We now turn to characterize the consumer’s side of the market, starting with the GP preferences and then moving to the β−δ ones. Consumer is endowed with initial capital k1>0 and one unit of free time each period. As we assume no disutility from labour, its supply is fixed and lt=1 in both models.

4 Competitive Equilibrium

We now turn to compare the equilibrium implications for both representations of temptation preferences: GP and β – δ one. We define:

Hence, by a competitive equilibrium we mean a list of quantities (solving both the consumer and the producer problems) and prices such that markets for all assets clear.

We start by providing few results characterising the equilibrium allocations.

For β−δ consumer, if 1+ra=1+r21+r3, then the thesis is clear from lemma 6. Also for 1+ra<1+r21+r3 the thesis is clear by lemma 2 and 5, as otherwise we would obtain a contradiction. ■

This implies that in equilibrium we cannot observe positive allocation of all three assets, i. e. ak2k3=0.

If in equilibrium k2=0 and a=0, then π2+w2=0=c2. For r2>0 the consumer can increase k₂ by ε, get positive income and afford positive c₂. He is willing to do so by Inada conditions. Otherwise consumer’s maximization is violated. The other case of r₂=0 cannot be an equilibrium price, markets cannot clear as firm requires infinite demand of a.

If in equilibrium k₃=0 and a=0, then π3+w3=0=c3. Similarly as above consider r3>0, but in such a case the consumer can increase k₃ by ε, get positive income and afford positive c₃. He is willing to do so by Inada conditions and thus violates consumer’s maximization.

For β−δ consumer the thesis is clear for 1+ra<1+r21+r3 from lemma 2. For 1+ra=1+r21+r3 suppose the contrary, i. e. a=0. But this implies that π3+w3=0. As above for ra>0, by Inada condition, consumer wants to transfer some amount to date 3, contradicting a=0. ■

Let us stress, that the existence of competitive equilibrium is not obvious here. Firstly, both types of consumers ^[7] have utility functions that are not strictly monotone nor concave. This properties can imply equilibrium nonexistence (see Gabrieli and Ghosal (2013) or Kocherlakota (2001)). For these reasons, we now provide a proof, when the competitive equilibrium indeed exists.

Observe that ζ1 is continuous and strictly decreasing with a and k₂ and strictly increasing with μ₃. As a result the equation ζ1k2,a∗k2,μ3,μ3=0 determines a continuous function a*, that is strictly decreasing with k₂ and strictly increasing with μ₃. Moreover, substituting z:=a+k2 we have that the function ζ˜1 is strictly decreasing with z and strictly increasing with k₂, where ζ˜1k2,z,μ3=−u′f(k1)+k1−z+βδ21+f′z−k2+1+μ3+f′z−k2f′(z)u′fz−k2+z−k2. Hence, although a^* is strictly decreasing with k₂, k2→k2+a∗k2,μ3 is strictly increasing with k₂. Observe that a∗k2,μ3∈0,fk1+k1−k2. Next, consider k2→ζ2k2,a∗k2,μ3,μ2 and observe that this is continuous and strictly decreasing. Finally, limk2→fk1+k1ζ2k2,a∗k2,μ3,μ2=−∞. Hence, if ζ20,a∗0,μ3,μ2≥0, then we obtain the unique k2∗μ2,μ3>0 but if ζ20,a∗0,μ3,μ2<0, we set k2∗μ2,μ3=0. In both cases we obtain the unique a∗k2∗μ2,μ3,μ3.

We next consider the function Φ, that fixed points are competitive equilibrium prices r₂, r₃, r_a (normalized to consumption good price). For this reason, we let the price of the consumption good be denoted by p and consider a simplex Δ⊂R4, where for a vector r2,r3,r4,p∈Δ its elements sum up to one. So take any r2,r3,r4,p∈Δ and for r2p, r3p, rap consider the firm’s maximization problem (1) with fixed lt=1, k₁, r1=f′k1. Then, maximization problem (1) is strictly concave with linearly independent constraints. Hence, there exists the unique Lagrange multiplier μ2′=μ2r2p,r3p,rap and μ3′=μ3r2p,r3p,rap that (from Berge maximum theorem) is continuous with parameters r2p,r3p,rap. Next for such Lagrange multipliers μ2′, μ3′ solve ζ1k2,a,μ′3=0 and ζ2k2,a,μ′2≤0 as above to determine the unique k2∗μ′2,μ′3 and unique a∗k2∗μ′2,μ′3,μ′3, both continuous with μ2′, μ3′ and hence r₂,r₃,r_a. In the final step update prices r2′=μ2′+f′a∗k2∗μ′2,μ′3,μ′3+k2∗μ′2,μ′3,1 and r3′=μ3′+f′a∗k2∗μ′2,μ′3,μ′3 and ra′=f′a∗k2∗μ′2,μ′3,μ′3+1+μ′3+f′a∗k2∗μ′2,μ′3,μ′3f′a∗k2∗μ′2,μ′3,μ′3+k2∗μ′2,μ′3. Observe that 1+r′a≤1+r′21+r′3.