There is broad consensus among international and development economists that restrictive trade policy regimes result in substantial economic costs. Examples of the costs most frequently cited are higher domestic prices or inferior quality of products sold, inefficient resource allocation, administrative costs of policy implementation, and, as has been more recently argued, the costs of lobbying for such policies. Consequently, economists have come to agree that import liberalization, in general, would foster productive efficiency and increase consumer welfare. Relaxing or removing trade barriers is, therefore, now a key element in the advice given to developing countries by academics and international institutions alike.
While the focus has mainly been on studying the costs of protecting final output, an important cost that has not received attention in the theoretical literature is one that emerges in trade regimes with quotas on imported inputs. Quotas on imported inputs in manufacturing industries of developing countries are often based on installed capacity. As a result, firms in these industries create excess capacity, which has been observed to persist for prolonged periods of time. This creation of idle capacity exacerbates the problem of capital shortages in developing countries. This paper presents a simple theoretical model to explain the relationship between excess capacity and quotas on imported inputs.
Explanations of capacity underutilization that are typically given are the variability in demand conditions coupled with the observation that capital investments are irreversible decisions. Such reasons are valid in explaining variability in capacity utilization, but are inadequate in explaining the persistence of excess capacity over time. It will be shown that excess capacity is a natural outcome under certain trade regimes, irrespective of demand conditions or the reversibility of capital decisions. This study, therefore, provides a direct link between input quotas and excess capacity in developing economies.
Many governments in developing countries issue licenses for imported inputs on the basis of installed capacity and not actual production undertaken.1 Two major empirical studies in the 1970s—the first by the Organization for Economic Cooperation and Development (Little, Scitovsky, and Scott (1970)), and the second by the National Bureau for Economic Research (Bhagwati (1978))—highlight this fact for many countries. These studies note that the justification for the above-mentioned allocation rule given by governments in these developing countries is to introduce some notion of “fairness” by allocating input quotas in some equitable, albeit arbitrary, manner.2 The consequence, as noted by Little, Scitovsky, and Scott (1970, p. 226) is that firms “would invest in additional capacity, not because this was needed to produce the additional output but because it provided a basis for obtaining a more generous allocation of imported inputs.”
Many markets in developing countries are also imperfectly competitive. Studies by Rodrik (1988) and Kirkpatrick, Lee, and Nixson (1984) based on four-firm concentration ratios suggest that imperfect competition is not only prevalent in manufacturing sectors of developing countries, but appears to be even more pervasive than in developed countries. In view of this evidence, an oligopoly model will be examined here in addition to the perfectly competitive model. Developing such a model will make it possible to analyze the strategic advantages that arise in input quota regimes in oligopolistic industries when compared to perfectly competitive ones.
The purpose of this paper is to show that quotas on imported inputs based on installed capacity can lead to capacity underutilization in manufacturing industries. This will be demonstrated for both perfectly and imperfectly competitive markets. In addition, the results show that excess capacity is less when strategic interaction between firms is taken into account, as in the imperfectly competitive case. The introduction of secondary markets for imported inputs in perfectly competitive markets illustrates the possibility of full-capacity utilization in manufacturing industries.
The policy implications of this paper are straightforward. A replacement of input quotas by input tariffs leads to full-capacity utilization in both perfectly competitive and imperfectly competitive markets, and eliminates strategic advantages arising in quota regimes for oligopolistic firms. This study, consequently, has a bearing on the sequencing theory of trade reforms, which is still in nascent form.3
This paper is organized as follows. Section I briefly reviews existing modeling approaches in the literature and compares them to the approaches taken in this paper. In Section II a theoretical model for analyzing tariffs and quotas on imported inputs is developed using a perfectly competitive framework. A comparison of input tariffs and input quotas in the perfectly competitive case reveals the existence of excess capacity only under the quota regime. It is shown that excess capacity may also be eliminated by the introduction of secondary markets. In Section III input quotas under oligopoly and perfect competition are compared. The last section contains concluding remarks.
I. An Overview of the Literature
Most studies on trade in developing countries have focused on trade in final goods. A very important aspect of the problem of trade deficits is ignored if imports of intermediate inputs are not taken into consideration. The reason is that a developing country facing balance of payments deficits may be willing to reduce its imports of final goods, especially luxury consumer goods (whose demand elasticity is high), but would be reluctant to forgo imports of intermediate inputs if the industries that use these inputs have already been established. Many industries have become so dependent on technology-embodied foreign inputs to sustain their production process that any decrease in these inputs is likely to have a direct adverse impact on gross domestic product (GDP). Given that imported inputs are considered “essentials,” devising optimal trade policies in intermediate inputs is important for those countries facing severe balance of payments problems.4
The literature on traded inputs in developing countries has been limited to neoclassical theories of effective protection (Corden (1971)), or immiserizing foreign (Brecher and Findlay (1983)) and domestic (Johnson (1967)) investment in the presence of domestic distortions. A basic limitation of conventional neoclassical theories is their inability to capture underutilization of factors of production without imposing price rigidities.
Abel (1981) develops a dynamic model of a firm with varying capacity utilization. He explains variations in utilization rates by assuming both capital and labor as quasi-fixed factors. At any instant these factors of production are fixed, and only utilization rates are varied in response to given demand conditions. In models developed below, an explicitly defined quota-allocation rule linking the two factors (capital and the intermediate input on which the quota is imposed) is sufficient to generate excess capacity. In other words, excess capacity is shown to emerge even if demand is unchanged.
The possibility of excess capacity as a means of deterring entry has been well documented in the literature on industrial organization. Spence (1977) and Bulow, Geanakopolos, and Klemperer(1985), among others, have shown that because investments in capital are irreversible decisions and represent pre-emptive commitments to the industry, they can be used to discourage entry.5 An empirical implication would be to expect concentrated industries to have lower capacity utilization. The model developed here predicts an inverse relationship in countries that have inputs linked to capacity creation. Even though installed capacity in these developing countries is used as a strategic variable, it is not used for deterring entry but for optimizing input quota allocations.
The theoretical equivalence of tariffs and quotas under the assumption of perfect competition has been proven by a number of writers (for example, Bhagwati (1969)). Nonequivalence, in contrast, has been shown in the presence of existing distortions; for example, under uncertainty (Fishelson and Flatters (1975)) and under monopoly in the domestic market (Panagariya (1980), among others). Krishna (1983) was among the first to demonstrate the nonequivalence of tariffs and quotas in a game-theoretic framework. She showed that when oligopolistic firms took government actions as given, a quota could be used as a facilitating instrument but a tariff could not. More recently, Reitzes (1989) demonstrated the nonequivalence when research and development behavior is used as a strategic variable. In the presence of rent-seeking activities, the equivalence can still be shown (Bhagwati and Srinivasan (1980)), although the welfare loss is greater than without rent-seeking activities. This welfare loss occurs because the act of lobbying for quotas or tariffs reduces the production possibility set for the economy as a whole (Krueger (1974) and Bhagwati and Srinivasan (1980)).
In the models developed in this paper, the nonequivalence of tariffs and quotas is demonstrated under both perfect and imperfect competition. In addition, excess capacity is shown to be less when strategic interactions between firms are taken into account. It is worth mentioning that the analysis demonstrates a special kind of rent-seeking activity, whereby the existence of controls on input quotas gives rise to rent-seeking behavior in the form of excess capacity creation.6
II. Quotas and Excess Capacity Under Perfect Competition
This section models the existence of excess capacity in manufacturing industries of developing countries. Quantitative restrictions on imported inputs are the key to explaining this phenomenon. This model enables one to, first, explain the existence of excess capacity despite unchanged demand for the final output; and, second, to illustrate the inefficiencies of quantitative versus price controls in a new perspective—quotas lead to capacity underutilization while tariffs do not.
The Basic Model
A one-period model is developed with many industries, each producing a final output with two kinds of inputs. There is an imported intermediate input, x, and a domestically produced input, capital k.
The principal idea is to study import-competing industries in developing countries that are protected by prohibitive tariffs on imports of final output, but face a quota on imports of scarce factor inputs. This is a realistic scenario for many developing countries protecting consumable manufacturing goods and importing industrial inputs. The justification is to encourage the growth of domestic industries by protecting their final output from foreign competition, but permitting limited imports of technology-embodied inputs. Thus, domestic demand-supply conditions affect the price of the final output in each industry.
The number of sellers in a typical industry, g, are fixed at a level ng. The sellers in each industry produce a standardized product, yg. To capture the “essentiality” of imported inputs in production, the case is studied in which there are no substitution possibilities for imported inputs.7 In some sense, this assumption is a limiting case of a situation where domestic inputs cannot compete with foreign inputs either on price or quality grounds; alternatively, foreign inputs are technology-embodied with no domestic substitutes.
The production function of each firm in each industry is
where yg is output of a representative firm in the gth industry, xg is the intermediate input of this firm in the gth industry, and kg is the capital stock of this firm in the gth industry. The input coefficients α and β carry the subscript g so that the technological coefficients can be assumed to be different across industries while retaining the assumption that they are the same for firms within the same industry. For notational convenience, the subscript g will be dropped and it will be assumed a typical industry is being observed. Henceforth
The industry’s inverse demand function is given by
where D’(Y) < 0, and
Three trade regimes are analyzed in the model of perfect competition: free trade, a tariff on the intermediate input, and a quota on the same input.
This is the case with no tariffs or quotas on the intermediate input. The firms solve the following problem:
From cost minimization the optimum levels of x and k used in producing any arbitrary level of y are determined. Firms minimize total costs, C, where
subject to the production constraint
Competitive equilibrium implies
where Y* is the equilibrium level of aggregate output in the industry. In other words, the price of the final output is simply a linear combination of the input prices adjusted for their respective technological coefficients. Equation (6) is the benchmark case against which comparisons of price and output levels in input tariff and input quota regimes will be made.
Figure 1.Free Trade and Import Tariff Cases
Tariff on Imported Intermediate Input
Let τ be the tariff levied on input x. Therefore, with ρx as the tariff-inclusive input price
The firms now face the following problem:
From cost minimization the optimal levels of x and k used for any arbitrary level of y are determined. Firms minimize total cost, C, where
subject to the production constraint in equation (3). In Figure 1, the appropriate cost curve, given ρx and Pk, that minimizes costs in the shaded region M, is CT. Equilibrium, once again, occurs at point A. Equation (2) (the free trade case) of course implies a lower cost than equation (8) (the input tariff case).
where Y*(T) is the equilibrium level of aggregate output in the industry under the tariff regime.
If the free trade case is compared with the tariff case, it is clear that the price of the final output is higher under the tariff regime by the extent of the tariff. That is, since τ > 0, D(Y*(T)) > D(Y*). Also, since D’(..) <0, Y*> Y*(T).
Moreover, regardless of input costs, cost minimization in both the free trade case and the input tariff case implies point A. At A, firms utilize their capacity fully. That is, if y* is the optimal level of output for each firm, then installed capacity (y*/β) equals used capacity (y*/β), and, therefore, capacity is fully utilized. Hence, there is no excess capacity under either the free trade case or the input tariff case.
Quotas on Imported Intermediate Input
The quota on input x is based on capacity creation. Let the quota allocation rule be
where θ is a policy parameter, exogenously set by the government, which determines the level of imports of x based on installed capacity. The case in which the quota is binding is analyzed here.9 The problem a typical firm faces is
Once again from cost minimization the optimal levels of x and k used for any arbitrary level of y are determined. The firms now minimize total cost (equation (2)). subject to two constraints: the production function (equation (3)) and the quota allocation rule (equation (10)).
In Figure 2, the shaded region N represents the constraint (10) when the input quota is binding. The intersection of region M with region N. which minimizes cost for
At point B in Figure 2
which is the same as in the cases of free trade and input tariffs for arbitrary levels of y (see equation (4)). However, the optimal k is different in the input quota case. Since the case in which the input quota is binding is being analyzed, at point B
Thus, in equilibrium
where Y*(Q) is the equilibrium level of aggregate output in the industry under the quota regime.
The existence of excess capacity under the quota regime is now fairly easy to establish. From equation (14), installed capacity = y/αθ, and from the production function (equation (3)). used capacity = y/β. Hence
Since the import quota is assumed to be binding, β > aθ (equation (12)). Given α, θ, β > 0, equation (16) shows that excess capacity is being created. Figure 2 illustrates the extent of excess capacity for any arbitrary level of y in the quota regime.
The level and price of the final output and capacity utilization under input quotas are now compared with the free trade and input tariff cases. Comparing equations (6) and (15), it is seen that since β > αθ, D(Y*) < D(Y*(Q)), and Y* > Y*(Q). In addition, since K* = y*/β and K*(Q) = Y*(Q)/αθ, the relative size of the installed capacity in the industry
Figure 2.Binding Quota on Input x
cannot be determined.10 This is because Y*(Q) < Y*, but β > αθ. Nevertheless, the quota regime leads to capacity underutilization, whereas the free trade regime does not.
In order to make a comparison between the input quota and the input tariff regimes that is meaningful in welfare terms, the tariff equivalent of the quota that implies the same level of final output and price can be determined. A comparison of equations (9) and (15) reveals that this condition is met when τ = (Pk/Px)[(β–αθ)/θβ]. In this case, even though the output price is the same under the two regimes, the quota regime alone creates idle capacity, which is clearly a waste in any developing economy. A replacement of input quotas by input tariffs, however, leads to full-capacity utilization in equilibrium.
In summary, capacity underutilization is unique to the input quota case. As expected, the output price is the lowest (and correspondingly, the level of output is the highest), under a free trade regime.
The Basic Model with Secondary Markets for Input Quotas
In this subsection it is shown that even if input quotas are binding, firms in some industries may utilize their installed capacity fully in equilibrium when secondary markets for input quotas exist.11
Define x = xf + xs, where x is the quantity of the intermediate input used in production, and xf is quantity purchased at the free trade price, . By definition, then, xf ≤ θk. Finally, xs is that part of x that is traded in the secondary market for quotas. It follows from the above definitions that if xs > 0, firms buy the input in the secondary market, and if xs < 0, firms sell this input.
Also, define k = kU + k1, where k is the capital installed by the firm, KU is capital utilized in production, and k1 is idle capacity. The representative firm now faces the following maximization problem:
subject to the following constraints:12
Given the production constraint, it is also known that for any arbitrary level of y
since x and kU are defined as the levels of intermediate input and capital utilized in production, respectively.
Substituting (17a)–(17f) in equation (17) and setting up the maximization problem with Kuhn-Tucker conditions yields
There are four solutions to this problem, of which the trivial one, y = 0 and k1 = 0, is ignored. The others are as follows.
Solution 1. y > 0 and k1 = 0. In this case
The two terms on the right-hand side of equation (18) reflect the shadow prices of the intermediate input and capital, respectively.
Using the relationship that
If quotas are binding, then β > αθ (from equation (12)). Firms buy the input quotas in the secondary market in addition to those available at the free trade price to meet their production needs. Moreover, a comparison of equations (6) and (19) reveals that, as expected, the output price is higher under the quota case than under the free trade case. However, if β < αθ and the quotas are not binding, firms would still buy inputs at the free trade price to the maximum limit imposed by the allocation rule—that is, xf = θk—and resell those quantities not used in production. Also, as is evident from equation (19), the equilibrium price of the final output is lower than the price under free trade (equation (6)). The reason for this surprising result is that by earning rents on the nonbinding quotas in the secondary markets, the firms in this particular industry are able to reduce their effective cost of capital.
Solution 2. y > 0 and k1 > 0. Now
Equation (20) may be rewritten as
since θRx = Px.
Equation (21) may be rearranged, by adding and subtracting Pk/β. to yield
A comparison of equations (6) and (22) reveals that the price under the quota regime will be higher (or lower) than the free trade price, depending on whether (Rx/α–Pk/β) is positive (or negative). Using the relationship that θRx = Pk, it can again be seen that when the quota binds (that is, β > αθ), the free trade price of the output is higher than the price of the output under the quota regime. Similarly, if the quota does not bind (that is, β < αθ), the reverse is true.
Solution 3. y = 0 and k1 > 0. In this case
This is the interesting case that is often cited in developing countries, in which firms simply install capacity without producing any final output. Their motivation is to get the input quotas based on installed capacity, so that they can resell in the secondary market at a premium.
It can clearly be seen that in each of the possible solutions, the profits made from selling input quotas in the secondary market can never exceed the cost of installing capital; that is, θRx ≤ Pk. This condition basically rules out the possibility of firms having infinite excess capacity in equilibrium. In addition, it is also evident that if the θ’s varied across all industries and if the input quotas were binding in each industry, the premium, Rx, on the input quotas would be determined by the industry with the highest θ. Since θRx ≤ Pk for all industries, and since all industries face the same price for capital and the same premium on input x, it must be the case that θRx, will equal Pk for the industry with the highest θ; that is, in the economy
For all other industries
The solution that will actually prevail in a particular economy with many industries will depend on the specific case in hand. For example, when θ’s bind and vary across all industries, the jth industry (the industry with the highest θ) will not only determine Rx, but will be the only industry with excess capacity in equilibrium (solution 2 or solution 3). Also, it will be the sole supplier of input x in the secondary market. The economic intuition for the jth industry to be the sole supplier is, of course, that firms in this industry have the lowest effective cost of installing capital.13 All other industries will satisfy their demand for additional units of x by buying in the secondary market instead of installing excess capacity, since it is cheaper to do so (solution l).14
If θ’s vary across industries and are not binding for some, then the sellers in the secondary market are those with nonbinding quotas and the rest are buyers. Capacity in all industries may be fully utilized in this situation (solution 1 for all industries).
When the θ’s are the same across all industries and the input quotas are binding in each industry, then in equilibrium
In this case, firms in all industries are indifferent between having excess capacity and buying inputs in the secondary market. In equilibrium, then, some will be suppliers while others will be buyers of input x in the secondary market (all solutions discussed above are now possible in the economy). If, however, quotas did not bind for firms in some industries but did for others, full utilization of capacity is a possibility in all industries.15 Finally, if quotas did not bind in each industry, then all industries utilize their capacity fully. In fact, this case (solution 1, equation (19), with Rx = 0) is the same as the free trade case (equation (6)).
In this section the existence of secondary markets for input quotas has been shown to create incentives for at least some industries to utilize their capacity fully. Consequently, an increase in the capacity utilization rate would be observed for the economy as a whole.
III. A Strategic Framework for Analyzing Input Quotas
So far only the perfectly competitive case has been considered. In view of the high concentration ratios in manufacturing industries in developing countries, it seems important to determine whether the results of the perfectly competitive case carry over to the imperfectly competitive case.16 This section shows that under similar conditions, capacity utilization under input quotas is higher in oligopolistic industries than in perfectly competitive ones.17
All assumptions made in the first section are retained. In addition, the total amount of the input quota available for the industry is fixed at a predetermined level . (This may occur when, for example, foreign exchange shortages in the economy set the limits on the intermediate imports of each industry.)
The Basic Model
Once again, three trade regimes are examined—free trade, a tariff on the intermediate input, and a quota on the same input. The production function is now defined as
where the subscript i now stands for the ith firm in a particular industry, yi is output of the ith firm, xi, is its intermediate input, and ki, the capital stock.
The total output in the industry concerned is denoted by
where the subscript j also stands for a firm and the summation is over all firms including the ith firm.18
In the case where an input quota is imposed, the quota allocation rule is
for each firm i, where
It is assumed that there are n firms in a typical industry that play a Cournot game; that is, each firm conjectures that when it changes its output the other firms will hold their output fixed.
This is the benchmark case with no tariff or quota on the imported input x:
Since cost minimization implies xi = yi/α and ki = yi/β,
where Y* and y* are the equilibrium output levels in the industry and firm, respectively.
A comparison of the free trade price under the perfectly competitive case and the oligopoly case reveals, as expected, that price is higher and output lower under oligopoly.
Tariff on Imported Intermediate Input
A tariff, τ, is imposed on input x. Then
The firms solve the following problem:
The net effect of imposing a tariff under oligopoly is to raise the output price over its free trade level, as was the case under perfect competition. However, the output price under oligopoly does not increase by the full extent of the tariff as it does in the perfectly competitive case.
Quotas on Imported Intermediate Input
In this case firms realize that the quota allocation rule gives them a strategic advantage. Since 6 depends on and K, and
The firms solve the following maximization problem:
Again, the case of the binding input quota is examined. Expressing xi, and yi in terms of ki yields
So the optimization problem is now
Since the firms know that
The first-order condition is
Letting k* be the optimal level of capital for a representative firm, and knowing that
Solving for θ
Moreover, since installed capacity = y*/αθ and used capacity = y* /β, capacity utilization = αθ/β.
The value of the quota allocation rule, θ, can now be compared under the two market structures. In the perfectly competitive case, each firm is too small to affect the value of θ, even though θ = ∑kj. Hence, for analytical purposes, θ is fixed.
From equation (15), in the perfectly competitive case.
where D(Y*(Q)) is the price of output under the quota regime. Moreover, since
Solving for θ from equation (28):
As n becomes very large, oligopoly (o) outcomes should tend to perfectly competitive (pc) outcomes. Indeed
Given θ, the capacity utilization (cu) under both market structures is the same; that is
Since equation (30) implies θo > θpc, the following can be concluded. Given a finite n, and given that an input quota is binding under a quota regime, capacity utilization is higher under oligopoly than under perfect competition.
This is a typical application of the theory of the second-best. In the presence of input quotas (a policy-induced distortion), the presence of oligopolistic markets (a market distortion) is a preferred alternative to perfectly competitive markets.
When the input quota is binding, or when x = θk, any firm can get more x when either θ or k increases. In the oligopoly case, since firms know that the quota allocation rule, θ, depends on the aggregate amount of capital, K, in the industry, and since each firm can affect K, these firms are able to increase θ by reducing
The introduction of quotas on imported inputs clearly illustrates new distortions that are created under certain quota allocation rules. Capacity underutilization becomes the natural outcome in equilibrium when quotas are based on installed capacity. In addition to generating excess capacity, a quota allocation rule based on installed capacity allows for strategic advantages to firms in oligopolistic competition as compared to those under perfect competition. As a result, there is less excess capacity in equilibrium for firms under oligopoly.
This paper also throws some light on a possible sequencing of trade reforms in developing countries. It is often argued that a developing country embarking on a trade liberalization process should, in the first stage, replace import quotas by tariffs. The analysis here indicates that such a policy directed at imported inputs will have the added benefit of eliminating excess capacity in the manufacturing sector. This may be accomplished even without changing the degree of restrictions on the imported intermediate input. Furthermore, a tariff in imperfectly competitive markets eliminates strategic advantages to oligopolistic firms of input quota regimes. If the purpose of quota allocation rules based on installed capacity is to incorporate some notion of “fairness,” then it is evident that governments are inadvertently favoring industries that are more oligopolistic. A replacement of input quotas by input tariffs would then ensure impartiality in the trade regime.
Some possible extensions of the modeling framework used here and the consequences of altering some of its basic assumptions can also be considered, such as the assumption of fixed technological coefficients. One way of generating excess capacity when there is some substitution between factors of production could be to assume “lumpiness” in installing capital; that is, to drop the conventional assumption of perfect divisibility of capital. Second, the quota allocation rule could be changed by making input quotas a function of past production instead of installed capacity. The intuition is that by introducing these dynamics it would be possible to generate industrial concentration over time, since the incumbents would then receive a larger share of input quotas. Finally, the models presented so far could be generalized by the introduction of quotas on many intermediate inputs. This would enrich the analysis by isolating those inputs that impose the greatest restriction on capacity utilization. Such extensions, while generalizing the results, would not necessarily alter the qualitative nature of the conclusions of this paper.
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Ratna Sahay, an Economist in the Asian Department, was an Economist in the Research Department when this paper was written. She holds a Ph.D. from New York University.
The author would like to thank Elhanan Helpman, Boyan Jovanovic, Peter Montiel, and Charles Wilson for their insightful comments, as well as the C. V, Starr Center for Applied Economies, and is especially grateful to Mohsin Khan and Carlos Végh for their valuable suggestions.
Other rules that are sometimes used are input quotas based on past performance or labor employed. If input quotas are linked to employment creation, the models presented below generate underutilization of labor in equilibrium.
Even though auctioning of import quotas is the most efficient way of allocating them, it is hardly ever used in developing countries.
“Sequencing” refers to the successive steps taken in the transition from a highly distortionary trade regime to a less distortionary one, with the final objective of complete liberalization.
Dixit (1980), however, among others, has argued for cases where it is possible to have a credible threat to deter entry without actually installing excess capacity. Nevertheless, if excess capacity is observed, it is likely to occur in oligopolistic industries for reasons put forward by Spence (1977).
Rent-seeking is defined as activities that represent ways of making profits but do not produce goods and services that enter a utility function directly or indirectly via increased production. This concept of rent-seeking was first introduced by Krueger (1974).
“Note that since is some arbitrary level of y, it holds for all possible y values. Hence, y is substituted for without any loss of meaning.
If the input quota were not binding, the solution would be the same as the free trade case.
K* and K*(Q) are defined as aggregate levels of capital installed in the industry in the free trade and input quota cases, respectively.
The author is grateful to Elhanan Helpman for suggesting the idea of introducing secondary markets in the input quota regime.
Equation (17a) will always hold with strict equality as long as Rx > 0. This occurs because even when the quotas do not bind, reselling of inputs in excess of production needs would always increase profits.
Recall from the economic interpretation of equation (18) that the shadow price of capital is (Pk–θRx).
There is also the possibility that some industries (excluding the jth industry) may not find it profitable to produce any output and to hold excess capacity; that is, y = 0,k1 = 0 for firms in this industry.
If θ’s are the same across industries, the reason quotas bind for some but not others is that the production technological coefficients (α’s and β’s) are different across industries.
The imperfectly competitive case, as noted in the introductory section, is probably the more realistic one to consider.
This section draws in part on Sahay (1989).
The need for this distinction will be obvious when the input quota case is discussed below.