Textbook Producer Theory: a Behavioral Version

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1 Textbook Producer Theory: a Behavioral Version Xavier Gabaix NYU Stern, CEPR and NBER November 22, 2013 Preliminary and incomplete Abstract This note develops a behavioral version of textbook producer theory, building on earlier work. The producer chooses her inputs to minimize cost, subject to a production constraint. However, she is not attentive to all prices. We derive the input demand function. The Slutsky matrix is no longer symmetric non-salient prices are associated with anomalously small cross-elasticities. The result holds also when there are adjustment costs, as the traditional Slutsky symmetry still holds when there are adjustment costs. The paper also endogenizes the producers inattention. Hence, we obtain a testable deviation from full rationality in the textbook producer s problem. 1 Introduction This note develops a behavioral version of the basic producer s theory. It is the counterpart of the behavioral version of consumers theory developed in Gabaix 2013a). In this setup firms do not pay full attention to prices. This is for two reasons. First, there are many indeed, thousands) of prices to consider, and the informational requirements to consider all of them all of the time seems huge. Second, the many imperfections in price setting under the rubric of sticky prices) do indirectly suggest some imperfections in input price processing. It may be useful to record those results, first, because they offer the first treatment of textbook microeconomic theory of producer behavior Varian 1992, Mas-Colell, Whinston and Green 1995). Second, because those predictions might lead to the search for their counterpart in empirical data. Measuring inattention isn t easy, but it may be doable. Third, it may be that experimentally the producer s problem is easier to implement than the consumer s problem. Hence, a behavioral version of the producer s problem may be useful to study broader issues of bounded rationality and inattention. xgabaix@stern.nyu.edu. I am grateful to the CGEB, INET, the NSF grant SES ) for financial support. 1

2 This paper directly draws from Gabaix 2013a), which defines a sparse max, where the agent maximizes while paying limited attention to variables, and endogenizes that attention. Much like in the traditional theory, the results for the producer s theory are in some sense a simpler version of the results for the consumers. As a results, it may be a simpler, purer laboratory to study bounded rationality. Reis 2006) analyzes a dynamic producer s problem, and in particular how often producers revise their plans. His analysis is silent about the basic propositions studied below. 2 Producer Theory 2.1 Basic theory The basic concept is the cost function C p, y): C p, y) = min z p z subject to F z) y 1) The firm chooses the input mix z, to minimize the cost p z of producing y units of the good. Production function F is the strictly concave. We call z p, y) the resulting factor demand. The price of good i is The firm perceives the price p i = p d i + x i p s i = p d i + m i x i for an vector m i [0, 1] which is an attention factor.. We state the demand by a sparse firm. We assume that F is twice continuously differentiable around z. Proposition 1 The demand of an inattentive firm is: z s p, y) = z r p s, y) where p is the true price and p s the perceived price. Indeed, the firm simply perceives the price p s, rather than p. Proof. This is a simple application of the sparse max in a somewhat degenerate form, where the misperception does not affect the constraint). We form L z, m, x, λ) = p s z + λ F z) y). The optimum z is clearly the solution of min z p s z s.t. F z) y, i.e. it is z r p s, y). However, in the traditional model, we have D p z r p, y) = D r ppc p, y), a symmetrical matrix, so that in plain terms: z i p, y) = z j p, y) p i for all i, j 2) 2

3 Again, this is quite a surprising relation. The impact of the rise in price of factor j on the demand for factor i, is equal to the impact of the rise in price of factor j on the demand for factor i. We can surmise that many business people would be surprised by this. Perhaps this is because they adopt a less rational approach. inattentive firms. Accordingly, let us see the analogue of equation 2) for Proposition 2 Slutsky asymmetry due to inattention) Evaluated at the default price, the input demand satisfies when m i 0) z s i Hence, the input demand matrix zs i is generally not symmetric, unlike in the traditional model. Columns corresponding to less salient prices are smaller. The intuition is quite simple: prices that are salient high m j ) have a big impact on the demand of factor i, zi s / ), while prices that are not attended to low m j ) have a small impact on factors. Proof. zi s p, y) p d k + m k x k )k=1...n, y) p=p x d j x=0 zi r p k ) = m k=1...n, y) j by the chain rule p=p d = m j z r i p, y) i.e. zs i p d,y) zi = m r p d,y) j The above proposition is implies that an inattentive firms is quite different from a firm with adjustment cost. Even with adjustment costs, a firm would satisfy Slutsky symmetry. 1 The Slutsky prediction is robust to adjustment costs m j We now show that the Slutsky prediction is robust to adjustment costs Slutsky symmetry also holds when there are adjustment costs this result must have been already noted many time, but for completeness we provide its derivation here). Consider the generalized cost function with adjustment costs: C a p, y) = min z p z + a z, z d) subject to F z) y 3) 1 If there is an adjustment cost g z), then C p, y) = min z p z+g z) subject to F z) y. By the envelopes theorem, C p p, y) = z, so z p = C pp, so z p is symmetrical: 3 z i = zj p i.

4 where a z, z d) is the adjustment cost when going from inputs z d to z, minimum at z = z d for instance, it might be some quadratic function z z d 2 ). Let us call z p, y) the resulting factor demand when there are adjustment costs. Proposition 3 The results hold with adjustment costs) The results of Propositions 2 hold even with adjustment costs. At the default price, we have z i = Ca p,y;a), and p i p=p d z s i while the demand of the rational producer, even with adjustment costs, satisfies Slutsky symmetry: m j z r i = zr j p i 4) Proof. We can write C a p, y) = p z + a z, z d) λ p, y) F z) y) for a Lagrange multiplier λ p, y). By the envelope s theorem, we have even when p p d ) hence, C a p i p, y) = z i p, y; a) z r i = C a p i = C a p j p i = C a p i p j by Young s theorem which says that we can invert the order of the derivatives, f xy = f yx ). Because likewise C a p i p j, we obtain zr i. For the inattentive demand, the reasoning is exactly like in the proof of Proposition Producer s allocation of attention Let s now endogenize attention. We use the same attention function A v) as in Gabaix 2013a): it is between 0 and 1, and weakly increasing in v. Proposition 4 In the basic consumption problem, assuming that price shocks are uncorrelated, attention to price i is: σ 2 ) m pi i = A ψ p 2 i p i z i /κ i where ψ i is the own-elasticity of demand for input i. Attention is greater for goods whose price is more volatile, that have a higher price-elasticity of demand, and that have a higher cost share. 4 5)

5 Proof The Lagrangian is: and the action is the input vector, z. So, L = p + x) z + λ F z) y) 6) L zz = λf zz and the loss term is Λ ij = z xi L zz z xi σ ij = λz pi F zz z pj σ ij Now the f.o.c. of min z L is. p = λf z, so, differentiating w.r.t. p, I n = λf zz z p + F z λ p, i.e. I n = λf zz z p + p λ p λ Also, by homogeneity of degree 0 of z p), z p p = 0. So, pre-multiplying the last displayed equation by z p z p = λz p F zzz p + z p p λ p λ = λz pf zz z p + 0 Hence, Λ ij = λz pi F zz z pj σ ij gives: Λ ij = z i p j σ ij 7) Recall that this is still symmetric i.e., Λ ij = Λ ji ), because zp i j and σ ij are symmetric. When price shocks are uncorrelated: under the traditional model) Λ ii = z i p i σ 2 i = σ2 i p 2 i p i zp i i p i z i z i = σ2 p i ψ p 2 i p i z i i Hence, attention is σ 2 ) m pi i = A ψ p 2 i p i z i /κ i 2.3 Revisiting some other classic results when do not pay full attention to prices We now revisit Shepard s lemma classic results. Proposition 5 Shephard s lemma holds at the default price: C s p = z s, but away from this price, the lemma needs to be modified: C s p i p, y) = zi s p, y) + p p s ) C r p i p s, y) m i. 5

6 Its proof is immediate. We give the link between the sparse and traditional cost function. Proposition 6 The cost function C s p, y) satisfies, at the default price: C s = C r, C s p = C r p but C s p i p j = C r p i p j m i + m j m i m j ) 8) Proof of Proposition 6 Cost-minimization problem 1) has the Lagrangian transforming it into a maximization of p z and using the notation z for the consumption of factors): L z, p, λ) = p z + λ F z) y) and the cost is C= v, for v the value function. Proposition 23 of Gabaix 2013a, online appendix) gives: v pp = L pp z p L zz z p + 2b s λ s = 0 z p L zz z p + 0 = z p L zz z p The strategy given in Proposition 1 gives z s p = z p M, where M = diag m 1,..., m n ). Proposition 21 of Gabaix 2013a, online appendix) gives: vpp s = v pp + ) z s ) p z p Lzz z s p z p = v pp + I M) z pl zz z p I M) vpp s = v pp I M) v pp I M) As the cost is C= v, we have C s pp = C pp I M) C pp I M) Coordinate-wise, this becomes: C s p i p j = C pi p j 1 m i ) C pi p j 1 m j ) = C pi p j m i + m j m i m j ) 3 References Gabaix, Xavier. 2013a. A Sparsity-Based Model of Bounded Rationality Applied to Basic Consumer and Equilibrium Theory Working Paper, NYU. Mas-Colell, Andreu, Michael Whinston, and Jerry Green Microeconomic Theory Oxford University Press. Reis, Ricardo Inattentive producers. Review of Economic Studies, 73, Varian, Hal Microeconomic Analysis, 3rd edition. Norton. 6