Technical Appendix to "Sequential Exporting"

Transcription

1 Not for publication Technical ppendix to "Sequential Exporting" acundo lbornoz University of irmingham Héctor. Calvo Pardo University of Southampton Gregory Corcos NHH Emanuel Ornelas London School of Economics bstract Here we show that our main results and empirical predictions are robust to: i adopting a demand function of the form pq = max {d q, 0} Non-negative prices, and to ii an arbitrary positive correlation across time or destinations Non-negative correlation. - Non-negative Prices Here we show as a result of forcing prices to be non-negative pq = max {d q, 0}, optimal export quantities in t = increase, while volumes in t = remain unaffected. Since expected export profits also increase, there is also more entry. Intuitively, such a demand function "convexifies" the revenue function, providing implicit insurance to the risk neutral producer against the event of negative prices, inducing the producer to take more risk, producing larger volumes conditional on entry, and becoming more propense to enter. ecause the surviving threshold in t = remains unchanged >, there is also more exit. Therefore our empirical predictions and 3 are if anything, strengthened. Since optimal export quantities in t = increase, while volumes in t = remain unaffected, predicted average second year growth is lower, but still positive as long as minimum marginal costs lie above expected willigness to pay. Hence, also our empirical prediction survives. More entry and larger volumes in t = translate into higher expected first period operational profits, inducing more experimentation. nd because expected first period operational profits are Technically, it just introduces a first order stochastically dominant SD shift in first period profitability, irrespective of destinations. I

2 Not for publication larger, some firms that would have entered sequentially, now enter simultaneously, as well as some non-entrants now will rather enter sequentially than not. Therefore our propositions and obtain, and so do their implications for trade policy proposition 3. Thus, avoiding negative prices has no effect on the expected value of information either across periods or destinations. This is why in the main text we impose the minor technical restriction d > E, instead of exposing the reader to the cumbersome technicalities displayed here. Proposition irst period export volumes are larger under a non-negative price restriction Proof. We want to show that: q j qj where: { } q j arg max E max d q, 0 q c+ j q q 0 q j arg max E d q q c+ j q q 0 The corresponding necessary and suffi cient OCs are, under the assumption of independence between demand d and supply c shocks: { } { } E j {d>q } qj +E max d q j, 0 }{{} MR p0 q j E + d q j }{{} MR = E c+ j }{{} MC = E c+ j }{{} MC Noting that E { {d>q} q } = qe { {d>q} } = q Kq q, q d, d, and { } { E max d q, 0 max E d q }, 0 = {Ed>q } E d q E d q, it follows that the marginal revenue is larger under the non-negative price restriction, MR p 0 q MRq, q d, d II

3 Not for publication because the marginal revenue is a non-increasing function of the quantity. Since the marginal cost remains the same MC, we have that q j qj. To be able to say if there is more or less sequential entry, we would need to know how do expected profits compare under the non-negative price restriction relative to its absence. irst, notice that: Proposition Conditional on entry, expected first period operational profits are larger when imposing a non-negative price restriction. Proof. Expected first period operational profits under a non-negative price restriction are: Ψq j ; j V j = { max E q 0 { max q 0 { max q 0 { } } max d q, 0 q c+ j q { max E d q } }, 0 q E c+ j q E d q q E c+ j q } = Ψ q j ; j V j Where the second inequality follows from the convexity of the max operator and Jensen s inequality, { and the third from noting that max E d q }, 0 = {Ed>q } E d q E d q, q. 3 Second, it is also true that: Corollary 3 Operational profits under a non-negative price restriction are larger 4 Proof. Notice that the definitions of V j and of W ; in the main text remain unchanged by the imposition of a non-negative price-restriction. The reason being that they constitute the ex-ante evaluation of ex-post optimal entry decisions, which rule out negative prices, i.e. = p 0 rom Leibniz s rule, we have that MR p0q q = Kq = MRq q, q 3 fter some tedious algebra, it can be shown that expected first period operational profits are equal to Ψq j ; j = Pd > q j q j + V j. 4 In the case of imperfect correlation across destinations, second period optimal output of sequential entrants is based on the conditional expectation of prices. s a result, prices can also be negative and the non-negative price restriction also constraints second period optimal outputs to be larger than they would absent the restriction. ut because profits are larger, the new entry cutoff would also allow for more entry, and a similar reasoning applies. III

4 Not for publication : V j = j j j dg = E j j { j > j } = Pr j > j E j j j > j ; W ; = + dg = Pr > + E > +. Therefore, the previous corollary implies that: Ψq j ; j Ψ q j ; j, j s a result: Corollary 4 oth sequential and simultaneous entry strategies display higher profits under a nonnegative price restriction. restriction, Therefore, the fixed cost entry thresholds under a non-negative price and Sm, are less binding. Proof. Defining Ψq j ; j Ψ j, Π Ψ +W ;, Π Sm Ψ +Ψ, the previous corollary implies: Π Π and Π Sm Π Sm Since the profit function is decreasing in the sunk entry cost, we immediately have: The definition of Sm and the previous corollary imply that: Sm +W ; Sm = Ψ Ψ = Sm +W ; Sm Since d +W ; d = G + 0, we immediately have that Sm Sm. irms that in the absence of a non-negative price restriction did not enter, now adopt a sequential entry strategy, and some of the previous sequential entrants, now would rather enter simultaneously. Therefore: IV

5 Not for publication Corollary 5 Proof. > Sm, i.e. Proposition survives a non-negative price restriction = Ψ + W ; > Ψ Ψ > Ψ W ; Sm = Sm where the weak inequality follows from the assumption that, and the strict inequalities obtain because under perfect positive correlation, the option value of entering sequentially is strictly positive, W ; > 0,. Consequently, our empirical predictions entry and 3 exit prevail, and are even reinforced by the adoption of a non-negative price restriction. The next proposition shows that under an economically reasonable condition, also prediction holds despite of being weakened: Proposition 6 Empirical prediction holds if c Ed. Proof. rom the OC we obtain the following expression for q j : q j = E j +λ {E> j +λ} P d > q j where Prd > q j Kq j, and λ Prd q j E need to show that: c Ed = Eq j qj 0 d d q j 0, q j d, d. We Noting that Eq j = E q j = E > j j, omitting the non-negativity restriction on quantities in the profit maximization problem, the above implication is equivalent to: The proof proceeds in 3 steps. c Ed = E > j j Step : Simplifying the RHS of the above implication. E j +λ Prd > q j fter cancelling common terms and rearranging, we can express the RHS as : Prd > q j E > j E Prd q j E d d q j + j V

6 Not for publication by definition of λ. Since E = Prd > q j E d > q j + Prd q j E this expression into the above inequality and rearranging yields: Prd> q j {E > j } E d > q j Prd q j {E d q j, plugging } d q j E d d q j j Substituting in the definition of = d c, and taking advantage of the assumption of independence between demand and supply shocks, we get: Prd> q j {E d d > c + j E d d > q j +Ec E c c < d j} Prd q j { Ec j} Noting that the proof of empirical prediction in the online appendix implies that: Prd > q j { Ec E c c < d j} 0, we can then move this term to the RHS of the inequality to obtain, after some simplifications: Prd>q j {E d d > c + j } E d d > q j Therefore the RHS of the inequality is negative. Step : The LHS of the inequality is positive if c + j > q j, c. { Ec E c c < d j} Prd q j { E c c < d j + j} It follows from an extension of the proof of empirical prediction in the online appendix: 5 = E > E >,,, Step 3: c > Ed = c + j > q j, c. Notice that c + j c + j + j j Ec Prd > q j c Prd > q j = c Ec j Prd > q j 5 The proof proceeds similarly to the proof of empirical prediction in the online appendix: integrate by parts both expressions and subtract them to obtain E > E > = because G. is a non-decreasing function. G > d + G G G d 0 G G VI

7 Not for publication and also that Ed Ec j Prd > q j = E j j +λ Prd > q j E Prd > q j = qj Since the inequality must be true for all realizations of c, if c > Ed it must be true that Ed Ec j Prd>q j and therefore that c, c + j > q j, completing the proof. c Ec j Prd>q j > - Non-negatively correlated export profitabilities Here we show that our results generalize to the case of positive but imperfect statistical dependence between random variables and. To keep the model symmetric, we assume distributions G and G are identical, although this is not essential. Upper-bar variables denote the counterparts to the variables in the main text under perfect correlation. or brevity, we denote E = u by E, where u denotes a particular realization of the random variable. -. Output choice: Output decisions in at all times and in at t = are taken in the same way as in the main text. Output choice in at t = takes into account the realization of. rom the convexity of the max function and Jensen s inequality, max q 0 q q dg dg max q 0 q q dg, where dg = dg dg. Expected profits are larger when an optimal production decision in is made taking into account the experience acquired in. Hence, optimal output is q E = {E > }. -. Value of the sequential exporting strategy: The conditional expectation of random variable can be expressed as E = E +u E d du G w = u dw, - u=u 0 }{{} ϖ VII

8 Not for publication where ϖ captures the statistical dependence between and. 6 t t = a firm enters market if E = u E / +. - Define u ; as the that solves - with equality. The firm enters market at t = if u ;. Plugging - in - yields u ; E +ϖu E =, which is strictly decreasing in. Comparing u ; with its analog under perfect correlation, we have that E = E implies lim u ; =. ϖ Expressed in t = 0 expected terms, entering market at t = yields profits, { W ; E max max E q q }, 0 { q 0 } E = E { > ϖ} {E > } = E ϖ dg, where ϖ / + ϖ ϖ ϖ E is the cutoff realization of export profitability in above which a sequential exporter enters in at t =. or expositional clarity, notice that if and follow a bivariate normal distribution with parameters E, E, σ, σ, ρ, the cutoff varies with ϖ = ρ as follows: Thus, when E d ρ = E / + dρ ρ. > / + the cutoff rises as ρ increases, implying a lower value from experimentation. This simply reflects the fact that, if E > / +, it is optimal to enter market already at t =. 6 The proof of - can be found at the end of this appendix. Conversely, when E < / + the cutoff falls as ρ rises, VIII

9 Not for publication implying a higher value from experimentation. This indicates that experimentation becomes more worthwhile as the statistical dependence between and increases. Experimentation is most valuable in the case of perfect correlation assumed in the main text, when it is worth W ;. Experimentation is least valuable when and are independent, when it has no value. 7 Derivation of -: Here we show how the conditional expectation can be expressed as a function of the unconditional expectation, as in -. Integrating by parts both expectations and taking the difference we obtain: E = u E = G w G w = u dw = G w G w = u dw Since G w G w, = G w G = G w, w,, because G =. y definition, G w = G w = u dg u, which inserted above yields: E = u E = G w = u dg u G w = u dw = G w = u dg u G w = u dg u dw }{{} = G w = u G w = u dg u dw. Now assuming that G w. C,, by the mean-value theorem, we obtain: u 0, : G w = u G w = u = u u E = u E = u u d du G w = u = d du G w = u u=u 0 u=u 0 dg u dw 7 Under independence between and, entry in conveys no information about profitability in. Thus, if it is not worthwhile to enter market at t =, it is not worthwhile entering at t = either. Conversely, if it pays to enter market at t =, it must pay to enter also at t =, to avoid forgoing profits in the first period. Thus, under independence waiting to enter at t = is never optimal. or a formal proof of this statement, see.n. 4 below. IX

10 Not for publication Since the term d du G w = u u=u0 is a constant, it follows that: E = u E = E u d = u E du G w = u u=u0 dw d du G w = u u=u0 dw We use Lehmann s 966, p.43-4 definition of regression dependence, which is in our context: Definition 7 non-increasing non-decreasing in u. is positively negatively regression dependent on if G w = u is Our assumption of statistical dependence between and implies regression dependence. Thus we can sign the integrand in the last equality above. inally by rearranging the last equality, we obtain -: if and are positively associated, d du G w = u u=u0 0 and d du G w = u u=u0 0, w so that d du G w = u u=u0 dw 0. Now if export profitability in was better than expected u E, expected export profitability to increases E = u E. Example: normal distribution. Consider a joint normal distribution of and. It is enough to compute 8 : where G w = u w = + σ exp π ρ d du G w = u ρ u=u 0 dw s E +ρ σ σ u E ds is the conditional distribution of, such that = u NE +ρ σ σ u E, σ ρ. We note that 9 : i dg s = u is a continuous function of s, u R, ii d du dg s = u exists and is continuous, and iii w dg s = u ds is continuous. Therefore we can differentiate 8 lthough expression - is defined for random variables on bounded supports, we conjecture that it can be extended to random variables over unbounded supports as long as their c.d.f., say G, possess an absolute moment of order ψ > 0, i.e if and only if 966, p.49. ψ G + G is integrable over, +, see Lemma in eller 9 acts i - iii are stated without proof, but since exp x is continuous, positive and bounded above by an integrable function exp x + : exp x + dx = e, on R, the proofs are left to the interested reader. R σ X

11 Not for publication inside the integral: d du G w = u = w = w d du exp { dg s = u ds σ π ρ ρ ρ σ σ σ ρ s E +ρ σ σ u E s E +ρ σ σ u E σ σ } ds which substituted above yields: + d du G w = u and hence the well-known relationship: = ρ σ σ G w = u, u=u 0 dw = + ρ σ σ G w = u 0 dw = ρ σ σ E = E +ρ σ σ E -3 which is a particular case of - where ϖ ρ σ σ. -.3 Choice of export strategy extension of Proposition, main text: s in the main text, is the fixed cost that makes a firm indifferent between exporting sequentially and not exporting, whereas Sm makes a firm indifferent between simultaneous and sequential exporting strategies: : Ψ +W ; =, Sm : Ψ W ; Sm = Sm Since Ψ j is monotonically decreasing in j and, and since W ; is non-negative, there is a non-degenerate interval of fixed costs where firms choose the sequential export strategy. -.4 Comparing imperfect with perfectly correlated export profitabilities Here we show that when profitabilities are non-negatively regression dependent, the option value of learning one s export profitability in market by entering in market first, W ;, is bounded by the option values in the two polar cases of i.i.d. correlation above. distributions below and perfect positive XI

12 Not for publication We start with the lower bound. With i.i.d. marginal distributions of and we have E = E = E and therefore ϖ = 0. ccordingly, the entry condition - becomes E / + so that lim ϖ 0 W ; = {E> / + } {E> } E. ut then entering market sequentially is dominated by a simultaneous entry strategy at t = : lim W ; < Ψ. The reason is that by entering at t = the firm only sacrifices ϖ 0 positive expected profits, V, because under independence, export experience in is useless in..hence lim ϖ 0 W ; = 0, and the firm will never adopt a sequential entry strategy. igure below illustrates this case. 0 Consider now the upper bound. Under perfect positive correlation between and, the 0 nalytically, we only need to examine whether there are values of such that Π Sm Π when ϖ = 0 : Ψ + Ψ Ψ + lim ϖ 0 W ; Cancelling terms and substituting the expression for lim ϖ 0 W ;, E Ψ {E> / + } {E> } ccording to the first indicator function, we must distinguish two cases: i if E > / +, the inequality reduces to V 0, which is false. Hence, there is no value of that satisfies it. ii If E / +, the inequality reduces to Ψ 0, meaning that the only values of that satisfy the inequality are those for which early entry in is not worth e = 0. Since late entry in is worth only when Ψ V, V > 0 and the above inequality imply that: Ψ > Ψ V, a contradiction. Therefore, there is no value of either that satisfies the inequality. Consequently, the sequential entry strategy is never adopted. XII

13 Not for publication Π Ψ + Ψ Π Sm Ψ + Ψ V Π simultaneous entry only enter and will never enter no entry E 0 = Sm = Ψ = Ψ igure : With independent export profitabilities ϖ = 0, a firm will never enter sequentially. term that captures the degree of statistical dependence ϖ in expression - becomes : d du G w = u u=u 0 dw =. Plugging this condition into expression -, and since E = E = E, E =, we obtain that as ϖ : lim W ; = W ; ϖ Under perfect positive correlation between and, G w if w u = u = 0 if w < u, which is a Heavyside step function or unit step function T w u = u δw sds, where δw s denotes a Dirac + if w = s delta function δw s = such that δw sdw =, s,. Since d T w u = δw u du 0 otherwise we have: d du G w = u u=u0 dw = δw u 0dw =. XIII

14 Not for publication which is the expression in the main text. inally, notice that: { } W ; = E max max q q, 0 q 0 { } = E E max max q q, 0 q 0 { E max maxe q 0 q q }, 0 { = E max max E q q }, 0 q 0 = W ;, ϖ 0 where the inequality obtains from applying twice Jensen s inequality and the convexity of the max {.} operator, while the third equality above follows from the law of iterated expectations, i.e. E f = E E f. Therefore: 0 W ; W ; s in the main text, those bounds on the option values correspond to sunk entry cost thresholds above which the exporter prefers to enter sequentially Sm, as illustrated in igure. Hence, the region defined by Proposition where it is optimal to adopt a sequential entry strategy shrinks as the statistical dependence of export profitabilities across the two destinations is reduced from Notice that in figure in accordance with notation in the main text Π Π ϖ= whereas, Π Π ϖ=0. lso notice from the figure that Π > Π, ϖ=. The only non-trivial point is to prove that Π 0 = V Π 0 = Ψ V convexity of the max {.} operator: V = E max qq = E q0 max E qq = q0 {E> } which follows from the application of Jensen s inequality and the {> } E Ψ V XIV

15 Not for publication Π Ψ + Ψ Ψ +V Ψ + Ψ V Π Π Π Sm Sm Sm = Ψ W ; Sm ϖ = ϖ = = Ψ + W ; no sequential entry entry simultaneous entry only enter no entry 0 Sm ϖ = 0 = Ψ ϖ = 0 = Ψ igure : ounds on sunk entry thresholds, Sm and, as a function of the statistical dependence ϖ between export profitabilites. perfect to no correlation: Sm Ψ + W ; Ψ W ; Sm = Ψ Ψ + W ; + W ; Sm Ψ Ψ +W ; +W ; Sm Sm >ϖ>0 Ψ Ψ Sm ϖ=0-3 References eller, W. 966, n Introduction to Probability Theory and Its pplications, Vol. II, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons. Lehmann, E. L. 966, "Some Concepts of Dependence", The nnals of Mathematical Statistics 375, XV