Endogenous task-based technical change—factor scarcity and factor prices

Abstract This paper develops a static model of endogenous task-based technical progress to study how factor scarcity induces technological progress and changes in factor prices. The equilibrium technology is multi-dimensional and not strongly factor-saving in the sense of Acemoglu (2010). Nevertheless, labour scarcity induces labour productivity growth. There is a weak but no strong absolute equilibrium bias. This model provides a plausible interpretation of the famous contention of Hicks (1932) about the role of factor prices and factor endowments for induced innovations. It may serve as a microfoundation for canonical macro-economic models. Moreover, it accommodates features like endogenous factor supplies and a binding minimum wage.


Introduction
In a competitive environment, a process innovation allows firms to increase profits through a reduction of costs. A firm adopts such an innovation or attempts to invent it if the cost advantage due to an improved productivity of the factors of production outweighs the cost of making the associated innovation investment. At the level of the individual firm, a process innovation allows to produce the same amount of output with fewer resources. For the economy as a whole, aggregate output will increase if the factors of production set free by the process innovation either produce more in the industry where the innovation occurs or produce other commodities elsewhere. The question about the economic phenomena that explain when and why firms adopt a new process innovation or engage in its invention is therefore crucial for our understanding of the observed productivity differences across firms, industries, and countries.
The present paper addresses this question in a static version of the dynamic competitive economy with endogenous task-based technical progress devised in Irmen (2017) and Irmen and Tabaković (2017). This analytical framework formalizes a central idea of John Hicks' The theory of wages according to which (relative) factor endowments affect (relative) factor prices and induce firms to implement or invent new technologies that replace the more expensive factor (Hicks, 1932). 3 The analysis starts from the premise that a firm is an economic unit where tasks are performed to produce output. Accordingly, a firm's production function relates performed tasks to final output. These tasks are executed by two factors of production, capital and labour. New technologies are process innovations with the potential to increase the productivity of capital and labour in performing tasks. The factor productivity of both factors is endogenous and hinges on the firm's willingness to make innovation investments. Since factor markets clear, the equilibrium number of performed tasks depends on the productivity of each factor and on the economy's factor endowments. Moreover, technical change is factor augmenting at the macroeconomic level.
Two main sets of results are derived. The first concerns induced productivity growth and underlines the role of factor scarcity as an important determinant of technology choice. The second set of results deals with factor price biases due to technological change and hanging factor endowments. Two complementary analytical strategies are used to establish these findings. The first strategy is based on comparative statics of the competitive equilibrium. The second strategy relies on the notions of net output and net marginal product at given factor endowments. It establishes and exploits the fact that the equilibrium technology maximises net output at given factor endowments and that the equilibrium factor prices are equal to the respective net marginal product at given factor endowments.
As to induced productivity growth, the equilibrium technology is neither strongly factor saving nor strongly factor complementary in the sense of Acemoglu (2010), i.e., depending on the kind of process innovation the mar-ginal product of capital and labour may increase or fall. However, even without these regularity conditions the equilibrium productivity of a factor of production will be higher if this factor becomes scarcer. For instance, less labour increases the equilibrium incentives to substitute labour with technology and leads to a higher equilibrium labour productivity. At the same time, it weakens the incentives to substitute capital with technology. Accordingly, the equilibrium productivity of capital will be lower. It is in this sense that factor endowments determine the direction of technical change.
As to factor price biases-in the taxonomy of Acemoglu (2007)-technologies are shown to be absolutely and relatively biased towards the complementary factor, i.e., at given factor endowments a higher productivity of labour increases the real rental rate of capital and reduces the real wage. Moreover, there is neither a strong absolute nor a strong relative bias. 4 Hence, labour scarcity leads to a higher equilibrium real wage and a higher relative price of capital. The latter finding is driven by a partial and a general equilibrium effect of opposite sign. The partial equilibrium effect captures the effect of changes in a factor endowment for a given technology and is negative. The general equilibrium effect captures the effect of a change in factor endowments on factor prices through induced technical change. This effect is positive, i.e., there is a weak absolute and a weak relative equilibrium bias. Hence, labour scarcity induces technical change that increases the real wage and reduces the relative price of capital. Since the partial equilibrium effect dominates the general equilibrium effect, the long-run demand schedule of a factor is declining in its price. 5 Additional sets of new results are derived in the 'extensions' section. First, the link between the task-based model of this paper and some of the author's earlier work including Irmen (2011) and Hellwig and Irmen (2001a) is discussed. The former contribution studies a competitive three-sector economy. It is shown that the equilibrium of a static version of this multi-sector economy is isomorphic to the one derived in the present task-based model. The key is that the first-order condition determining the aggregate number of tasks performed in the task-based model coincides with the free-entry, zero profit condition of the intermediate-good sectors of the three-sector economy. As a consequence, the implications of factor scarcity for innovation incentives and factor prices derived in the present paper carry over to this multi-sector environment. The analysis of the link to a static version of the competitive growth model proposed by Hellwig and Irmen (2001a) reveals that this one-sector model has no weak absolute bias since the equilibrium technology maximises the real wage.
Second, the analysis turns to the role of endogenous factor supplies. Intuition suggests that the link between the scarcity of a factor, a higher factor price, and induced innovation may be counteracted by an increase in the aggregate supply of this factor. The analysis confirms this intuition for a scenario where either individuals supply more hours in response to a higher real wage or where the supply of labour increases in the rental rate of capital. However, this tendency does not invalidate the key predictions derived in the basic version of the model with inelastic factor supplies. It does however weaken the link between factor endowments, innovation incentives, and factor prices via a general equilibrium effect.
The third extension allows for one factor price to be exogenous. This turns the economy either into one with a minimum wage or into a small open economy. Both setups yield similar results concerning the role of changing factor endowments for the equilibrium technology, the remaining endogenous factor price, and employment levels. The analysis focusses on the case of a minimum wage. Then, the economy under scrutiny is similar to a static version of the one analysed in Hellwig and Irmen (2001b). A binding minimum wage is found to entirely determine the direction of technical change as well as the rental rate of capital. Compared to the equilibrium under laissez-faire it reinforces the incentive to save labour, reduces the incentive to raise the productivity of capital, and implies a lower rental rate of capital. Changing the economy's capital endowment leaves these variables unaffected but leads to adjustments of the level of employment.
The present paper builds on and contributes to at least two strands of the literature. First, it makes a contribution to the theory of endogenous capitaland labour-saving technical change that has its roots in the so-called "induced innovations" literature of the 1960s (see Fellner, 1961;von Weizsäcker, 1962von Weizsäcker, , 1966Kennedy, 1964;Samuelson, 1965;Drandakis & Phelps, 1966). A main focus of this literature is on the link between (relative) factor prices and induced technical change as envisaged by Hicks (1932). However, its lack of a sound micro-foundation has often been criticized (see e.g. Salter, 1966;Burmeister & Dobell, 1970, Chapter 3;Nordhaus, 1973;Funk, 2002;Acemoglu, 2003). It assumes competitive firms with access to a constant-returns-o-scale production function F(bK, aL) where K is capital, L is labour, and b and a are capital-and labour-augmenting technology terms. Obviously, profit-maximisation with respect to (b, a, K, L) is not well defined since F has increasing returns in all four variables. To circumvent this problem, firms maximise instead the current rate of cost reduction subject to some invention possibility frontier. While this ad hoc heuristic leads to results in support of Hicks' argument, the question remains open as to whether these findings would still hold under a sound micro-foundation.
The model developed in this paper provides such a micro-foundation. It gives rise to an endogenous 'technology frontier' along which b and a cannot simultaneously in-or decrease (see, Section 2.1), a property that is key to the exogenous invention possibility frontier of the 'induced innovations' literature (Burmeister & Dobell, 1970). Moreover, Section 4.2 establishes the close link between the comparative statics of my model and Hicks' famous contention. 6 Second, this paper complements the literature on a class of competitive models with endogenous technological change where the technology has a tendency to be strongly capital saving or strongly labour saving. As argued in Acemoglu (2010), models with this property include, e.g., Champernowne (1961) or Zeira (1998). However, the multidimensional technology in the model of the present paper does not comply with this regularity condition. 7 Nevertheless, labour (capital) scarcity induces a higher equilibrium productivity of labour (capital). From the results of the 'extensions' section, it is evident that this property is also shared by the competitive endogenous growth models proposed in Irmen (2011).
Another important dimension with respect to which the present setup differs from existing competitive models with endogenous technical change is that technical progress applies to tasks and requires an innovation investment for each of them. Therefore the gross marginal product of a factor exceeds its net marginal product which is equal to the respective equilibrium factor price. Nevertheless, most of the findings on absolute and relative factor price biases are consistent with those of Acemoglu (2007) where gross and net marginal products coincide. This paper is organized as follows. Section 1 presents the model with endogenous task-based technical progress. Section 2 establishes the existence of a general equilibrium, discusses its welfare properties, and introduces key concepts such as net output, equilibrium technology, and net marginal product. In Section 3, the link between factor scarcity, the equilibrium technology, and net output is discussed. The link between factor prices, technical progress and factor scarcity is the focus of Section 4. Section 5 has the above mentioned extensions. Section 5.1 establishes the equivalence between the model of Section 1 and a three-sector model of Irmen (2011). Moreover, it discusses the link to a static version of the one-sector model of Hellwig and Irmen (2001a). Section 5.2 deals with the role of an endogenous labour supply. Finally, Section 5.3 studies the effect of an exogenous factor price with a focus on a binding minimum wage. Proofs are relegated to the Appendix.

The model
Consider a static economy endowed with capital and labour. The respective factor endowments are denoted by K ∈  ++ and L ∈  ++ . These inputs are inelastically supplied. 8 There is a single manufactured final good that can be consumed or invested. If invested it may increase the productivity of capital and/ or labour in the performance of factor-specific tasks. The economy is perfectly competitive. All agents' preferences are defined over the consumption of the final good which also serves as numéraire.
Throughout this paper subscripts are often used to denote partial derivatives. For functions of one variable, it is the argument that appears as a subscript, for instance, f κ (κ) ≡ df (κ)/dκ. In the context of functions of several variables numbers are used as, for example, in F 12 (M, N) ≡ ∂ 2 F(M, N)/∂M∂N.

Technology
The production sector has a continuum [0,1] of competitive firms. Without loss of generality, their behavior may be analysed through the lens of a competitive representative firm. Two types of tasks have to be performed to produce output. The first type needs capital, the second labour as the only input. Let m ∈  + denote a task performed by capital and n ∈  + a task performed by labour. Then, m ∈ [0, M] and n ∈ [0, N] where M and N denotes the total 'number' of tasks of each type performed by the representative firm.
Tasks of the same type are identical. Therefore, total output depends only on M and N. Let F:  + 2 →  + denote the production function of the representative firm. It assigns the maximum output, Y, to each pair (M, N) ∈  + 2 , i.e., The function F is  2 with F 1 > 0 > F 11 and F 2 > 0 > F 22 for all (M, N) > 0. While tasks of each type are identical, they differ with respect to their marginal product. Moreover, F exhibits constant returns to scale (CRS) with respect to both task types. For further reference, let κ denote the task intensity of the firm, i.e., Then, the production function in intensive form is A task m requires k(m) = 1/b(m) units of capital, a task n needs l(n) = 1/a(n) units of labour. Hence, b(m) and a(n) denote the productivity of capital and labour, respectively. They are equal to b(m) = 1 + q b (m) and a(n) = 1 + q a (n) (1.3) where q b (m) ∈  + and q a (n) ∈  + are indicators of productivity growth associated with task m and n, respectively. These productivity levels require investments of i(q b (m)) ≥ 0 and i(q a (n)) ≥ 0 units of the final output. The investment cost function i:  + →  + is the same for all tasks,  2 , increasing and strictly convex. Hence, higher levels of productivity require larger investments. Moreover, it satisfies for all tasks and j = a, b One may think of an investment as a decision to adopt a new technology that is available in differing degrees of sophistication or as R&D outlays in the spirit of the lab-equipment model of Rivera-Batiz and Romer (1991). In both cases, q j measures the productivity gain that results from an investment that costs i(q j ) units of output.

Profit-maximisation
The representative firm takes the vector (R, w) of the real rental rate of capital and the real wage as given and chooses a plan comprising (q b (m), k(m)) for all m ∈ [0, M] and (q a (n), l(n)) for all n ∈ [0, N] as well as the choice of how many tasks (M, N) to perform. This plan is to maximise profits where C is the firm's cost reflecting factor and investment costs for each task, i.e., (1.6) With (1.3) one has k(m) = 1/(1 + q b (m)) and l(n) = 1/(1 + q a (n)). Accordingly, the firm's problem may be split up in two parts. First, for each m ∈ [0, M] and each n ∈ [0, N], the values (q b (m), q a (n)) ∈ R + 2 are to minimise C. This leads to the first-order (sufficient) conditions For each task of the respective type, these conditions equate the marginal reduction of the firm's capital cost/wage bill to the marginal increase in its investment costs. Assuming R > 0 and w > 0, the convexity of the investment cost function and the fact that lim q j → 0 i q (q j ) = 0, j = a, b, imply that these conditions determine a unique q b (m) = q b > 0 and q a (n) = q a > 0 for either task type. Accordingly, b(m) = b, a(n) = a, k = 1/b, and l = 1/a. Second, each performed task must be profitable, i.e., (1.10) Hence, for a task to be performed, its marginal value product must be at least as large as its cost. The former is equal to F 1 (m, N) and F 2 (M, n), respectively. The latter is the sum of the capital or wage cost and the investment outlays of the respective task. Since each task is associated with a strictly positive input requirement k(m) = 1/(1 + q b (m)) > 0 and a(n) = 1/(1 + q a (n)) > 0, M and N must be finite in equilibrium to exclude an excess demand for capital or labour. In other words, in equilibrium conditions (1.9) and (1.10) must hold as an equality. Since Π has CRS in (M, N) at (q b , q a ), this also implies that equilibrium profits are zero.
Finally, observe that conditions (1.9) and (1.10) will only pin down the task intensity κ = M/N since F has CRS in (M, N). The number of tasks will be determined by market clearing conditions.

Definition and characterization
An equilibrium consists of a plan Hence, at positive factor prices there must not be an excess demand, and equilibrium factor prices can only be strictly positive if there is full-employment of the respective factor.
Let θ ≡ K/L denote the capital intensity. Then the following holds.
The first step addresses the dependency of the equilibrium technology and the equilibrium factor prices on the task intensity κ as stated in (2.2) and (2.3). This property reflects the first-order conditions (1.7)-(1.10) and is illustrated in Figure 2.1 for tasks performed by capital. The left panel shows the marginal value product of the marginal task where use is made of the fact that F 1 (M, N) = f κ (κ). According to (1.7) and (1.9), this value product must be equal to the minimised cost of tasks performed by capital. The cost-minimisation is shown in the right panel. Hence, (κ, R, g b (κ)) is an admissible solution to these two equations. As f κκ (κ) < 0, increasing the task intensity from κ to κ' means that the cost minimum must fall. This requires a lower real rental rate of capital, R' < R, hence R κ (κ) < 0. Since a lower capital cost reduces the marginal advantage of a productivity enhancing investment, the new cost-minimum is reached at a lower level of q b , i.e., q b (κ') < q b (κ) and g κ b (κ) < 0. The same line of reasoning shows why w κ (κ) > 0 and g κ a (κ) > 0. The key difference here is that the marginal value product The second step concerns the determination of the equilibrium task intensity as stated in (2.4). From the first step, the equilibrium technology depends on the task intensity. However, the market clearing conditions (2.1) reveal that in equilibrium M = (1 + q b )K and N = (1 + q a )L, i.e., the task intensity depends on the equilibrium technology. Combining factor market clearing and (2.2) shows that the task intensity that performs both functions, κ, must be a solution to According to (2.4), there is a unique that satisfies this equation. Moreover, κ increases in the capital intensity, θ, since g κ a (κ) > 0 > g κ b (κ). For further reference, let me express this last result in terms of elasticities, i.e., Hence, due to induced technical change the response of the equilibrium task intensity to changes in the capital-labour ratio is less than proportionate.
Finally, observe that Theorem 1 implicitly defines a 'technology frontier' and a 'factor-price frontier' (Samuelson, 1960). The technology frontier links any pair (q b , q a ) > 0 that satisfies (2.2). It may be stated as where (g a ) -1 (q a ) is the inverse of q a (κ) and, accordingly, g:  ++ →  ++ . Since g κ a (κ) > 0, the slope of the inverse is also strictly positive. Hence, dq b /dq a < 0, i.e., q b and q a cannot increase simultaneously. Notice that unlike the exogenous invention possibility frontier stipulated by the 'induced innovations' literature of the 1960s, the technology frontier of the present model is the result of profit-maximising behavior. The factor-price frontier is defined for any pair of factor prices (R, w) > 0 that satisfies (2.3) with dR/dw < 0, i.e., R and w cannot increase simultaneously. I shall explore these properties in Sections 3 and 4.

Net output, equilibrium technology, and equilibrium factor prices
This section introduces the notions of net output and net marginal product at given factor endowments. These concepts are later used to establish key properties of the equilibrium technology and factor prices.

Net Output
Net output is aggregate output minus aggregate investment outlays. Let (q b , q a ) ∈  + 2 denote the vector of symmetric technology choices. It presumes a firm behavior where the same amount of investment is allocated to all tasks of the same type (though, not necessarily the profit-maximising amount of investment). Then, net output at symmetric technology choices is defined as (2.8) The argument (q b , q a , K, L) is used to study the effect of technical change at given factor endowments. Besides symmetric technology choices, it reflects the additional use of the market clearing conditions (2.1), i.e., M = (1 + q b )K and N = (1 + q a )L. Then, final output at given factor endowments is defined as 9 This reveals that i) technical change is factor augmenting, ii) a better technology means more output of the final good, i.e., Y 1 = KF 1 > 0 and Y 2 = LF 2 > 0, and iii) Y(q b , q a , K, L) is (strictly) super-modular in (q b , q a ), i.e., Y 12 = KLF 12 > 0. Here, super-modularity follows since F has positive, yet diminishing, marginal products, and CRS to scale in (M, N). Using (2.9) and the market clearing conditions (2.1) in (2.8) gives rise to the definition of net output at given factor endowments, i.e., are aggregate investment outlays given full employment of both factors of production.

Equilibrium technology
The following proposition derives an important property of the equilibrium technology. (2.11) Moreover, any (q b , q a ) ∈  + 2 that solves (2.11) is an equilibrium technology.
Hence, both the first and the second welfare theorem hold in this economy. 10 This finding confirms the claim that the static technology choice in competitive environments tends to be welfare maximising (see e.g. Acemoglu, 2007;or Zeira, 1998). However, in the present model there is a novel perspective on the equilibrium technology that will prove useful later. Indeed, the presence of the technology frontier (2.7) and the fact that the equilibrium technology is a global maximiser of V(q b , q a , K, L) leads immediately to the following corollary to Proposition 1.
(2.12) 10 At first sight, Proposition 1 may seem restrictive because it presumes a symmetric technology choice. However, this turns out to be a valid short cut since a planner, who chooses  For further reference, the maximum of net output at given factor endowments is henceforth referred to as equilibrium net output and denoted by V(K, L), i.e., ( , ) ( , , , )

Equilibrium factor prices and net marginal products
The net marginal product of capital at given factor endowments is the additional net output at (q b , q a , K, L) that results from a small increase in K. Analogously, for a small increase in L, one has the net marginal product of labour at given factor endowments. To develop an intuition for these concepts consider capital. If the economy's capital stock is fully employed then M = (1 + q b )K, and a small increase in capital means dM = (1 + q b )dK additional tasks. On the one hand, this implies an increase in the output of the final good equal to The net marginal product of capital at given factor endowments is then the difference between these two effects and equal to (2.14) Analogously, the net marginal product of labour at given factor endowments is This leads to the following result.
Proposition 2. The equilibrium factor prices satisfy (2.16) Hence, the equilibrium factor prices are equal to the respective net marginal products at given factor endowments evaluated at (q b , q a ). Intuitively, equilibrium factor prices adjust so that (1.9) and (1.10) hold as equality. This requires and to be equal to their respective net marginal products.

Factor scarcity, equilibrium technology, and net output
This section explores the role of factor scarcity for the equilibrium technology and for equilibrium net output.

Factor scarcity and equilibrium technology
The main result of this section is given in the following proposition.
Hence, a larger capital stock induces a lower q b and a higher q a and, mutatis mutandis, for a larger labour force. In other words, the equilibrium incentive to equip a factor with a better factor-augmenting technology declines if the factor becomes more abundant. At the same time, the equilibrium incentives for a better technology that augments the other factor increases. It is in this sense that a factor and 'its' technology are substitutes, whereas a factor and the 'other' technology are complements.
The intuition for these findings is closely linked to the one underlying Theorem 1 since changes in factor endowments increase or decrease the capital intensity, θ. For instance, a higher K increases θ so that the equilibrium task intensity, κ, shifts upwards and induces a lower q b and a higher q a in accordance with the technology frontier defined above. 11 To place Proposition 3 in a broader context recall from Proposition 1 that the equilibrium technology satisfies V 1 (q b , q a , K, L) = V 2 (q b , q a , K, L) = 0. Restricting attention to labour, total differentiation of these two conditions at (q b , q a , K, L) delivers In fact, Proposition 3 may also be expressed in terms of the relative scarcity of factors of production measured by θ. Then, it would state that dq b /dθ = g κ b (κ(θ)) κ θ (θ) < 0 and dq a /dθ = = g κ a (κ(θ))κ θ (θ) > 0 where the signs follow from Theorem 1 according to which q b = g b (κ(θ)) and q a = g a (κ(θ)). To the extent that changes in θ may result from simultaneous variations in capital and labour, rephrasing Proposition 3 in this way is slightly more general.
To sign these derivatives note the following. First, since (q b , q a ) maximises V(q b , q a , K, L), it holds that V 11 < 0, V 22 < 0 and V 11 V 22 -V 2 12 > 0. Second, (3.5) Hence, a higher q b increases the net marginal product of labour at given factor endowments whereas a higher q a reduces it. In other words, neither exhibits strictly decreasing nor strictly increasing differences in (q b , q a , L). 12 Therefore, in the taxonomy of Acemoglu (2010), the technology (q b , q a ) is neither strongly labour saving nor strongly labour complementary. 13 As a consequence, the products in the numerators of (3.2) and (3.3) are of the same sign. 14 Nevertheless, the overall sign of these numerators is unequivocal. The positive sign of dq b /dL follows since The negative sign of dq a /dL results since , is the slope of the minimised cost per task.
An alternative and insightful interpretation of the comparative statics stated in (3.2) and (3.3) can be gained from Corollary 1. Recall that the technologies q b and q a are linked via the technology frontier q b = g(q a ) introduced in (2.7). Along this frontier, the technology becomes effectively single-dimensional and 12 If a function f (x, t) defined on  n ×  is twice differentiable on some open set, then for each i = 1, …, n increasing (decreasing) differences means ∂f 2 (x, t)/∂x i ∂t ≥ 0(∂f 2 (x, t)/∂x i ∂t ≤ 0). 13 According to Acemoglu (2010), Definition 1, p. 1050, a technology is said to be strongly labour saving (strongly labour complementary) if improvements in the technology reduce (increase) the net marginal product of labour at (q b , q a , K, L). Analogously, it is strongly capital saving (strongly capital complementary) if improvements in the technology reduce (increase) the net marginal product of capital at (q b , q a , K, L). Here, improvements in the technology' refer to higher levels of both elements of the technology vector (q b , q a ).
14 Mutatis mutandis, the qualitative results of (3.2)-(3.5) and the ensuing interpretation are analogous for changes in the capital endowment. Hence, V has neither strictly decreasing nor strictly increasing differences in (q b , q a , K). Accordingly, the technology (q b , q a ) is neither strongly capital saving nor strongly capital complementary. Moreover, the logic behind the unequivocal signs of changes in K on the equilibrium technology is analogous to the one for changes in L. net output at given factor endowments can be stated as V(g(q a ), q a , K, L). Then, it is readily verified that 2 14 24 where V is evaluated at (q b , q a , K, L) and g q (q a ) < 0 is the slope of the technology frontier. In light of (3.4) and (3.5), both summands are negative. Hence, the technology q a is 'strongly labour saving along the technology frontier' . Moreover, total differentiation of the first-order condition associated with (2.12) and evaluation at (q b , q a , K, L) delivers ( ) which coincides with (3.2) and (3.3) but unequivocally reveals the sign of the comparative statics. Finally, observe that the qualitative results of Proposition 3 carry over to a world where firms have access to only one of the two technologies. For instance, without means to raise the productivity of capital, q b = 0, k = 1, and the equilibrium technology q a maximises net output at given factor endowments given by V(0, q a , K, L). Implicit differentiation of V 2 (0, q a , K, L) = 0 delivers again confirming the signs obtained in Proposition 3. More capital fosters innovation investments that increase the productivity of the complementary factor since V 23 = LF 12 > 0. Moreover, labour scarcity increases innovation incentives since, as in (3.5), V 24 = (1 + q a )LF 22 < 0. 15 Analogous results obtain if only the productivity of capital can be increased by means of innovation investments.

Factor scarcity and equilibrium net output
How does the equilibrium net output of (2.13) respond to changes in factor endowments? The answer is given by the equilibrium net marginal product of capital. Using Proposition 1-3, the latter is 97 A. Irmen, Endogenous task-based technical change-factor scarcity and factor prices 1 2 ( , ) , , , , The first line captures the effect of induced technical change on net equilibrium output. In light of Proposition 3 it holds that dq a /dK > 0 > dq b /dK, i.e., the productivity of labour increases whereas the one of capital falls. However, the effect of these incremental adjustments on equilibrium net output is negligible since, according to Proposition 1, the equilibrium technology has already been chosen to maximise net output at given factor endowments. Accordingly, the partial derivatives V 1 (q b , q a , K, L) and V 2 (q b , q a , K, L) are zero in equilibrium, and the first line of (3.8) vanishes. As a consequence, a small increase in capital augments equilibrium net output only to the extent that more tasks can be performed using the given technology (q b , q a ). According to Proposition 2, this effect is equal to the equilibrium real rental rate of capital. An analogous argument shows that the effect of changing labour on equilibrium net output is equal to w.

Factor prices, factor scarcity, and equilibrium technology
This section studies the role of factor scarcity and technical progress for the levels of absolute and relative factor prices.

Absolute factor prices
Denote the equilibrium factor prices of (2.16) by R(q b , q a , K, L) and w(q b , q a , K, L), respectively. The following proposition states the main result of this section.
Hence, a larger capital stock lowers the equilibrium rental rate of capital and increases the real wage, and, mutatis mutandis, for a larger labour force. To get the intuition consider dR/dK < 0. From Theorem 1 the equilibrium rental rate of capital declines in the task intensity which, in turn, increases in the capital-labour ratio. In other words, since a higher K increases θ, R must fall. Moreover, in accordance with the factor-price frontier, w increases. 16 To place Proposition 4 in a broader context note that the effects stated in (4.1) may be split up into a partial and a general equilibrium effect. Indeed, with Proposition 2 one finds , , Here, the first term of each expression captures the partial equilibrium effect of changing factor endowments for a given technology. The terms in brackets represent the general equilibrium effects due to induced technical progress.

Partial equilibrium effects
To understand the link between Proposition 4, (4.2), and (4.3) consider the real wage. From (2.15), the sign of the partial equilibrium effects are determined by diminishing returns to labour and the super-modularity of F as ∂w/∂L = (1 + q a ) 2 F 22 (q b , q a , K, L) < 0 and ∂w/∂K = (1 + q b )(1 + q a ) F 21 (q b , q a , K, L) > 0. An analogous argument applies to the equilibrium rental rate of capital.

General equilibrium effects
To provide an understanding of the general equilibrium effects, one needs to study first the partial effect of technical change on factor prices.
Since equilibrium factor prices are equal to the equilibrium net marginal products of the corresponding factor, the findings of Proposition 5 follow immediately from (3.4) and (3.5) and the corresponding expressions for capital V 23 > 0 > V 13 . In other words, they reflect the fact that the technology (q b , q a ) is neither strongly factor saving nor strongly factor complementary. 17 To highlight the importance of Proposition 5 it is worth contrasting the effects of technical change on factor prices with those of exogenous factor-augmenting technical change that arise in the neoclassical growth model. In this model and the present notation, final output equals ( ) Marginal cost pricing leads to an equilibrium real wage equal to 2 (1 ) Hence, technology q a may increase or decrease the price of labour. This reflects the tension between a positive productivity effect and a negative effect due to diminishing returns (see Irmen, 2014, for details). With endogenous technical change, the effect of q a on w(q b , q a , K, L) is derived from Proposition 2 as The sign is unequivocally negative since 2 2 , , ,  . This suggests that the ambiguity of (4.5) is due to an asymmetry in the analytical setup rather than to properties of the production function: if technical change is exogenous, then competitive firms compete in factor markets but not for their technology. If technical progress is endogenous, then firms compete for the resources that make technical progress happen. As a consequence, the positive productivity effect that appears in (4.5) is competed away. In other words, a higher q a cannot have a positive effect on the equilibrium real wage since the competitive equilibrium technology maximises net output. Mutatis mutandis, the same reasoning applies to the effect of q b on the equilibrium rental rate of capital.
Finally, observe that the models with and without endogenous technical change predict the same factor price movements for the cross-effects. Here, only the properties of F matter. More precisely, its super-modularity means that ∂R/∂q a = ∂R/∂q a = V 32 > 0 and ∂ŵ/∂q b = ∂w/∂q b = V 14 > 0.
In light of Proposition 3 and Proposition 5, it is now straightforward to sign the general equilibrium effects. For brevity, I denote those of (4.2) by ΔR K and Δw L , and the ones of (4.3) by ΔR L and Δw K .

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Economics and Business Review, Vol. 6 (20), No. 2, 2020 Proposition 6. It holds that ΔR K > 0, Δw L > 0, ΔR L < 0, Δw K < 0. (4.7) Hence, a larger capital stock leads to induced technical change that increases the price of capital and decreases the price of labour, and, mutatis mutandis, for labour. These results follow immediately from Propositions 3 and 5. The latter implies that all products that appear in (4.2) are strictly positive and those of (4.3) are strictly negative. As a result, induced technical change increases the price of the factor that has become more abundant and reduces the one of the factor that becomes scarcer.
To grasp the intuition consider ΔR K > 0. If K becomes more abundant, then there are two effects on R. On the one hand, the incentive to substitute capital with the capital-augmenting technology falls and q b declines. This diminishes the efficient amount of capital and increases R due to diminishing returns. On the other hand, a higher productivity of the complementary factor becomes more valuable. Therefore, q a increases which increases R, due to the super-modularity of F. 18 Mutatis mutandis, the intuition is analogous for the remaining three Δs.

Total effects
Summing up, the total effects shown in Proposition 4 reflect the tension between partial and general equilibrium effects of opposite sign. However, in all cases the partial effect dominates the general equilibrium effect. 19 In particular, (equilibrium) inverse factor demand functions are declining in the respective factor endowment. However, due to induced technical progress, the response of a factor price to a change in 'its' factor endowment is attenuated.
Finally, observe that Proposition 4 encompasses the cases where only one or none of the technologies are available. 20 If only one technology is available, 18 In the taxonomy of Acemoglu (2007), there is weak absolute equilibrium bias with respect to K and L since ΔR K > 0 and Δw L > 0. Therefore, Proposition 6 is in line with Theorem 2 in Acemoglu (2007), p. 1394 saying that under fairly mild conditions there is weak absolute equilibrium bias. However, here the intuition for this result is quite different from Acemoglu's given on page 1373. Moreover, it should be noted that the signs in (4.2) and (4.3) can only be different from zero if the equilibrium technology does not maximise equilibrium factor prices. See Section 5.1.2 for further discussion. 19 In the taxonomy of Acemoglu (2007), there is no strong absolute equilibrium bias since dR/dK < 0 and d w/dL < 0. This result may also be traced back to the fact that the Hessian of V(q b , q a , K, L) in (q b , q a , K) or (q b , q a , L) is negative definite. In fact, V is jointly strictly concave in both (q b , q a , K) and (q b , q a , L) as V 14 > 0 and V 23 > 0. 20 This is immediate from the proof of Proposition 4. Indeed, if only one technology is available, then one of the elasticities in the denominator of (2.6) is zero. If none of the technologies are available, then κ = θ and ε θ κ = 1. then one product representing a general equilibrium effect in (4.2) and (4.3) vanishes. If no technology is available, only the partial effect matters and we are back in the neoclassical growth model without technical change.

Relative factor prices
The symmetry of the results on absolute factor prices leads to clear-cut predictions for the effect of technical progress and factor scarcity on the relative factor price. Throughout, the findings of this section are expressed in terms of the relative price of capital R(q b , q a , K, L)/w(q b , q a , K, L). The main result is the following.
Proposition 7 establishes two distinct outcomes. First, it shows that the total effect of a larger capital stock on the relative price of capital is negative whereas the effect of a larger labour endowment is positive. This finding can be directly deduced from Proposition 4. For instance, since a larger capital stock reduces R and increases w, the relative price of capital must also fall.
Second, the derivatives on the left hand side of (4.8) represent total effects that may be decomposed into a partial and a general equilibrium effect. From the discussion of the partial effects on absolute factor prices, it is immediate that ∂(R/w)/∂K < 0 and ∂(R/w)/∂L > 0. To sign the general equilibrium effects, observe that Proposition 5 implies that ( ) It is in this sense that technology q b is biased towards labour whereas technology q a is biased towards capital.
In the taxonomy of Hicks (1932), p. 121-122, technical change associated with an increase in q b is called capital-saving since it decreases the ratio of the (net) marginal product of capital to that of labour. Technical change associated with an increase in q a is called labour-saving since it increases this ratio. Hence, 'laboursaving inventions' in the terminology of Hicks are those that exhibit a relative bias towards capital in the sense that ( ) Here q a , captures this type of invention.
Using these results in conjunction with Proposition 3 reveals that the general equilibrium effects satisfy 21 In light of Proposition 3 and Proposition 5 all products associated with dK are positive whereas those associated with dL are negative. Hence, the total effects shown in Proposition 7 reveal a tension between partial and general equilibrium effects. While their sign is determined by the partial effect, the general equilibrium effects weakens the response of the relative price of capital to changes in the respective factor endowment.
At this stage, the link between the results derived so far and the famous contention of John Hicks becomes apparent (Hicks, 1932). On page 124 he asserts that A change in the relative prices of the factors of production is itself a spur to invention, and to invention of a particular kind-directed to economising the use of a factor which has become relatively expensive. These incentives are implied by the first-order conditions of cost-minimisation (1.7) and (1.8). At (q b , q a ), they may be rearranged to (4.10) Since the numerator of the right-hand side increases in q b and the denominator increases in q a , a hike in R/w induced either by an increase in R or by a decrease in w implies a greater ratio q b /q a . In other words, the higher firms expect the relative price of capital to be, the more attractive is it for them to substitute capital with technology rather than labour with technology. Of course, factor prices and the technology are endogenous. Hicks suggests changing factor endowments as the driving force behind factor prices and technical change (Hicks, 1932, 124-125): The general tendency to a more rapid increase of capital than labour which has marked European history during the last few centuries has naturally provided a stimulus to labour-saving inventions.
Following the discussion of Propositions 3 and 4, it is indeed the case that a higher capital intensity, θ, means a lower q b , a larger q a , a lower R, and a higher w. Hence, when firms expect a larger capital intensity, they rightly anticipate the price of labour to increase and the price of capital to fall. The induced innovation investments attenuate these price movements. This is the role of the general equilibrium effects. However, in spite of induced innovations, the relatively scarcer factor becomes more expensive.

Alternative environments
This section studies two alternative economic environments and establishes the relationship to the model of endogenous task-based technical progress discussed so far. Section 5.1.1 establishes the equivalence between the model introduced in Section 1 and the three-sector economy studied in Irmen (2011) where tasks correspond to intermediate goods. Section 5.1.2 shows the link between the model of Section 1 and a static variant of the competitive one-sector growth model proposed in Hellwig and Irmen (2001a). Here, the key difference is the absence of diminishing returns of tasks in the production of the final good.

Competitive Three-Sector Economy
Consider a competitive economy with a final-good sector and two intermediate-good sectors, one producing the first intermediate with capital, the other producing the second intermediate with labour. Refer to these intermediates as the capital-intensive and the labour-intensive one. The final good serves as numéraire.

Technology and profit-maximisation
The representative final-good firm manufactures the final-good out of two intermediate inputs according to the production function (1.1). Now, M and N denote the respective aggregate amounts of the capital-intensive and the labourintensive intermediate inputs. The firm maximises profits equal to Y -p K M -p L N where p K and p L denote the real price of the respective intermediate. The respective first-order conditions are p K = F 1 (M, N ) and p L = F 2 (M, N ).
Intermediate-good firms either belong to the capital-or to the labour-intensive intermediate-good sector. Each sector is represented by the set  + with Lebesgue measure. All firms of a sector have access to the same sector-specific production function y k (m) = min{1, b(m)k(m)} and y l (n) = min{1, a(n)l(n)}, (5.1) where y k (m) and y l (n) is the output of firm m ∈  + or n ∈  + , respectively. There is a capacity limit equal to 1, 22 b(m) and a(n) denote a firm's capital and labour productivity, and k(m) and l(n) is the capital and labour input. The firms' respective capital and labour productivity is given by (1.3). Firms may increase their factor productivity by investing i(q b ) and i(q a ) units of the final good, where i has the same properties as stated in and before (1.4).
Intermediate-good firms maximise profits, i.e., y m k m q m y l n q n n and taking prices, (p K , p L , R, w) as given.
To derive the optimal production plan a firm reasons as follows. If it innovates, there will be an investment cost i(q j ) > 0. Such an innovation investment is only profit-maximising if the firm's profit margin is strictly positive, i.e., if p K > R/b(m) or p L > w/a(n). If this is the case then there is a positive scale effect, i.e., an innovating firm wants to apply the innovation to as large an output as possible and produces at the capacity limit, i.e., y k (m) = 1 or y l (n) = 1. The choice of ( ) ( ) The solution to this problem gives rise to first-order (sufficient) conditions that coincide with (1.7) and (1.8).
The aggregate capital demand is equal to of firms producing each one unit of the capital-intensive intermediate good.
Mutatis mutandis, the aggregate labour demand is 0 M ∫ l(n)dn. Accordingly, the factor market clearing conditions are given by (2.1). To prevent excess factor demands in both factor markets, M and N must be finite, i.e., in equilibrium some intermediate-good firms must not enter. Therefore, the maximum profit of any intermediate-good firm producing the capital-or the labour-intensive inter-mediate must be zero in equilibrium. 23 Using p K = F 1 (M, N) and p L = F 2 (M, N), these zero-profit conditions coincide with (1.9) and (1.10) as equalities.

Equilibrium
An equilibrium of the three-sector economy consists of production-cum-entry decisions {y k (m), k(m), q b (m)} of all firms m ∈ [0, M] and {y l (n), l(n), q a (n)} of all firms n ∈ [0, N], measures (M, N) of entering firms in both sectors producing one unit each, and prices (p K , p L , R, w). These variables solve the firms' first-order conditions for cost-minimisation, the zero-profit (free-entry) condition as well as the factor market clearing conditions (2.1) given (K, L) ∈  2 ++ . Proposition 8. Given (K, L) ∈  2 ++ , the competitive three-sector economy has a unique equilibrium. The equilibrium values for q b , q a , R, w, M and N coincide with those of Theorem 1. In addition, the equilibrium determines y k (m) = y k (n) = 1, p K = F 1 (M, N), and p L = F 2 (M, N).
Proposition 8 holds since the conditions for profit-maximisation and zeroprofits of the three-sector economy coincide with (1.7)-(1.10). Moreover, in both economies the factor market clearing conditions are given by (2.1). As a consequence, all concepts derived in Section 2 and the results that appear in Section 3 and Section 4 carry over to the three-sector economy.

Competitive One-Sector Economy
In the competitive one-sector economy studied in Hellwig and Irmen (2001a) firms produce a final good with the production function y l (n) of (5.1). Firms are represented by the set  + with Lebesgue measure, hence n ∈  + indexes firms. The final good is the numéraire. A new element is that innovation investments have to be undertaken and financed one period before they are used in production. Let R = 1 + r > 0 denote the exogenous real interest factor. 24 Firms maximise profits equal to y l (n) -wl(n) -Ri(q a (n)) where y l (n) is the respective firm's revenue from output sales, wl(n), its wage bill, and Ri(q a (n)), its outlays for the innovation investment.
Firms choose a production plan (y l (n), l(n), q a (n)) taking factor prices (R, w) as given. Let N denote the measure of firms that enter and produce output. Following the reasoning set out for the intermediate-good firms in the three-sector economy, this leads to symmetric profit-maximising choices for all n ∈ [0, N] firms satisfying l = 1/(1 + q a ) and -w/(1 + q a ) 2 + Ri q (q a ) = 0, which is the counterpart of (1.8). To exclude an excess demand for labour,

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Economics and Business Review, Vol. 6 (20), No. 2, 2020 N must be finite, i.e., in equilibrium some firms must stay out of the market. Therefore, entering and producing firms must earn zero-profits in equilibrium, i.e., 1 -w/(1 + q a ) -Ri(q a ) = 0. Finally, full employment of labour pins down the 'number' of entering firms as N = (1 + q a )L. These conditions determine unique equilibrium values (q a , l , w, N) ∈  4 ++ as functions of R, and y l (n) = 1. Interpreting y l (n) as the 'number' of tasks performed by firm n, the link to the model of Section 1 becomes obvious. In equilibrium, each firm performs one task at minimum costs. Moreover, aggregate output is equal to the 'number' of entering firms N, i.e., there are no diminishing returns associated with the number of performed tasks. Moreover, net output at given factor endowments is the difference between aggregate output, N = (1 + q a )L, and total investment outlays, (1 + q a )LRi(q a ), i.e., V(q a , L; R) = (1 + q a )[1 -Ri(q a )]L. This leads to the following results corresponding to Theorem 1, Proposition 1, and Proposition 2. Moreover, any q a ∈  2 + that solves (5.3) is an equilibrium technology.

Equilibrium Wage
(a) The equilibrium wage satisfies The equilibrium wage is independent of L. Moreover, there is a function w: According to Claim 1.(a), the equilibrium technology is independent of the economy's labour endowment. Hence, there is no direct analogy to Proposition 3. This is due to the absence of diminishing returns, i.e., each performed task adds the same amount to final output. Therefore, the equilibrium innovation incentives are the same for all firms. The equilibrium technology declines in the real interest factor reflecting higher marginal and total investment costs as R increases. Claim 1.(b) is the counterpart to Proposition 1 and confirms the validity of the two welfare theorems for this economy.
Claim 2.(a) states two findings. First, it confirms Proposition 2, that is, the equilibrium wage is equal to the net marginal product of labour at q a . Second, unlike the model of Section 1, the equilibrium technology also maximises the real wage. The reason is again the absence of diminishing returns. Therefore, V(q a , L; R) is linear in L, the equilibrium wage is equal to net output per worker, and q a maximises net output and the real wage. Claim 2.(b) reveals that the equilibrium wage necessarily falls in response to an increase in the price of the innovation investment. This reflects the direct price effect on investment outlays. By the envelope theorem, the indirect effect via induced innovation investments, (∂w/∂q a )(∂q a /∂R), is mute since the equilibrium technology maximises the real wage.
To establish a closer link between Claim 1.(a) and Claim 2.(b) and a potential role of factor endowments, one may want to think of R as being determined by (the world's) capital stock K with R K (K) < 0. Then, these claims imply that both q a and w increase with K. This confirms d q a /dK > 0 of Proposition 3 as well as d w/dK > 0 of Proposition 4. Moreover, the latter comparison leads to the interesting conclusion that there is no weak absolute bias (or weak relative) bias in models where the equilibrium technology maximises equilibrium factor prices.

Endogenous factor supplies
Factor supplies may respond to changing factor prices. Accordingly, the relative scarcity of employed factors of production becomes endogenous. This section allows for the labour supply to depend on the real wage and for the capital supply to depend on the real rental rate. The question is then how changing factor endowments affect the equilibrium technology and equilibrium factor prices. Conceptually, these two cases differ insofar as the real wage affects the intensive margin of the supply of labour whereas the real rental rate determines the extensive margin of the supply of capital.

Endogenous labour supply
In the short run, individuals may want to increase their labour supply in anticipation of a higher wage. Under full employment this behavior reduces, ceteris paribus, the ratio of capital to employed labour. In this sense, labour becomes more abundant. Then, Proposition 3 and Proposition 4 suggest that the productivity of labour and the wage decline in equilibrium.
To address this tension assume that the individual labour supply is a function of the real wage. To be precise, denote τ ∈ [0, 1] the fraction of an individual's time endowment that she supplies to the labour market and normalize this endowment to unity. Assume further that τ = τ(w) where τ:  ++ → (0, 1) with τ w (w) > 0. From Theorem 1, w = w(κ) which results from firms' first-order conditions. Accordingly, τ = τ(w(κ)), and is the elasticity of τ(w(κ)) with respect to κ. To find the equilibrium task intensity note that the aggregate labour supply is equal to Lτ(w(κ)) and, for any strictly positive real wage, the labour market clearing condition (2.1) delivers N = ((1 + g a (κ))Lτ(w(κ)).

Proposition 10.
There is a unique equilibrium task intensity, that solves In addition, where all terms are evaluated at κ.
Proposition 10 extends equations (2.5) and (2.6) to the case of an endogenous labour supply. It is readily verified that τ w (w) > 0 is sufficient for the existence of a unique κ > 0 that satisfies (5.6). Equation (5.7) states the elasticity of κ to changes in θ which is positive. Hence, κ increases in θ. However, compared to (2.6), the responsiveness is weaker since ε κ τ > 0. This is due to an adjustment in the individual supply of labour and leads to the main result of this section: if individuals increase their labour supply at the intensive margin in response to a higher wage, then the effect of changing θ of the equilibrium task intensity weakens but does not change its direction. Therefore, Proposition 3 and Proposition 4 remain valid. 25

Endogenous supply of capital
Denote the supply of capital by K = K(R) where K:  ++ →  ++ with K R (R) > 0. Theorem 1 implies R = R(κ) with R κ (κ) < 0. Hence, the equilibrium capital supply satisfies = + ( g κ K R κ and 25 Proposition 3 and Proposition 4 may also remain valid if one allows for τ w (w) < 0 for some w > 0. This means that ε κ τ (κ) is negative for some κ. As long as a unique κ exists and ε κ τ (κ) is not too negative, ε κ τ of (2.6) exists and remains positive. However, in this case a higher wage reduces the supply of labour at the intensive margin which strengthens the responsiveness of κ to changes in θ relative to the case of exogenous labour supply.
The capital market clearing condition (5.8) delivers = + ( The same line of reasoning as used in Proposition 10 leads to the conclusion that (5.8) pins down a unique κ > 0. Moreover, with ε L κ = d ln κ/d ln L one readily verifies that As expected, increasing the supply of labour reduces the equilibrium task intensity. Therefore, the qualitative predictions made in Proposition 3 and Proposition 4 concerning changes is the labour endowment remain valid. However, the elastic supply of capital weakens this link. The intuition for this is straightforward. Given K an increase in L reduces the task intensity. This shifts the rental rate of capital upwards, leads to an increased supply of capital, and, therefore, to a greater task intensity. This general equilibrium effect weakens but does not dominate the effect of L on the equilibrium task intensity. 26

Exogenous factor prices: The case of a minimum wage
What is the role of factor scarcity for technology and factor prices if one factor price is fixed above its equilibrium value determined in Theorem 1? To answer this question I introduce an exogenous minimum wage into the model of Section 1. 27 Accordingly, the equilibrium factor market clearing condition for labour stated in (2.1) must be extended. With w min > 0 denoting the real minimum wage this condition becomes 26 Following similar arguments as in Footnote 24 one readily verifies that Proposition 3 and Proposition 4 may also remain valid if one allows for K R (R) < 0 for some R > 0. 27 Alternatively, one could address this topic in a small open economy (SOE) facing an exogenous rental rate of capital under perfect international capital mobility. If the rental rate paid in the worldwide capital market exceeds R of Theorem 1, then there will be strictly positive net capital exports in equilibrium. This is the counterpart to the equilibrium level of unemployment of labour under a binding minimum wage in the closed economy under scrutiny here. Moreover, in the SOE the effect of changing the labour endowment on the equilibrium values of technology, factor prices, and net capital exports mimics the effect of changing the capital endowment as stated in Proposition 11 below.

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Economics and Business Review,Vol. 6 (20) Hence, the actual wage must not be lower than the minimum wage, and, if it is equal to the minimum wage an excess supply of labour may occur in equilibrium. The equilibrium of the economy is then defined as in Section 2 with (5.10) replacing the respective condition in (2.1). 28 Clearly, what matters is whether the minimum wage is binding or not. If it is not binding then w > w min and there must be full employment. In other words, the equilibrium is as described in Theorem 1. However, if it is binding then w = w min and the character of the equilibrium changes drastically. Firms adjust their investment behavior, the rental rate of capital falls, and there is unemployment of labour. To see why the remainder of this section assumes that the minimum wage exceeds its equilibrium level under laissez-faire, i.e., w min > w. Moreover, the subscript min is used to denote the equilibrium values associated with w min .
The analysis starts with the cost-minimising choice of q a (n) given by (1.8). Now, this equation directly determines q a min . To make this more precise, let q a (w) denote the functional relationship between q a and w defined by (1.8). Then, q a :  ++ →  ++ and, with c q (q a ) ≡ 2i q (q a ) + (1 + q a )i qq (q a ) >0, one has q w a (w) = [(1 + q a )c q (q a )] -1 > 0. Hence, q a min = q a (w min ) > q a = q a (w). (5.11) Intuitively, competitive firms must raise the productivity of labour to meet the challenge of an excessive real wage.
Upon combining (1.8) and (1.10) for a given wage one finds as the equilibrium condition that equates the value product of the marginal task N to the minimised cost of tasks performed by labour. Since w pins down q a , (5.12) implicitly defines a functional relationship κ(w) where κ:  ++ →  ++ and ( ) . Hence, the equilibrium task intensity under a binding minimum wage satisfies κ min = κ(w min ) > κ = κ(w).
(5.13) Intuitively, the cost minimum of tasks performed by labour is higher under a binding minimum wage. Therefore, the value product of the marginal task must also increase. Accordingly, the equilibrium task N intensity increases.
A higher task intensity reduces the value product of the marginal task M. Therefore, the equilibrium incentive to invest in the capital-augmenting technology will fall. To confirm this intuition formally, combine (1.7) with (1.9), and use κ(w) as defined above. This gives the equilibrium condition that equates the value product of the marginal task M to the minimised cost of tasks performed by capital as (5.14) The latter defines a functional relationship q b (w), where q: As to the rental rate of capital, let R(q b ) denote the functional relationship between R and q b implied by (1.7). Then, R:  ++ →  ++ with R q (q b ) = [(1 + q b ) c q (q b )] -1 > 0. Now, the function q b (w) defined above provides the link between the rental rate of capital and the wage. Indeed, one has < . Accordingly, the equilibrium rental rate of capital under a binding minimum wage satisfies (5.16) Intuitively, the rental rate of capital falls below its laissez-faire level so that the cost-minimum is attained at q b min < q b . Hence, under a binding minimum wage the equilibrium technology as well as the equilibrium rental rate of capital is fully determined by w min . In addition, these variables also depend on the functional forms chosen for f and i but, unlike under laissez-faire, not on factor endowments. However, it is important to see that the capital stock is a determinant of the level of equilibrium employment.
Under a binding minimum wage, the level of employment is equal to the demand for labour, At the same time full employment of capital requires ( ) Then, the equilibrium level of employment satisfies κ(w) = M/N, or Since q w a (w) > 0 > q w b (w), and κ w (w) > 0, the latter implicitly defines a function L d (w, K) where L d : R 2 ++ → R ++ . A higher (minimum) wage reduces the level of employment, i.e., ∂L d (w, K)/∂w < 0. Intuitively, this reflects two reinforcing channels. First, the number of tasks performed by labour declines since N min = M min /κ min = (1 + q b min )K/κ min < N. Second, each of the N min tasks requires less labour as q a min > q a . Hence, induced labour-saving technical change reinforces the employment reducing effect of a binding minimum wage. As a consequence, L min = L(w min , K) < L. (5.18) A higher capital endowment unequivocally increases the level of employment, i.e., ∂L d (w, K)/∂K > 0. Intuitively, for a given equilibrium technology, a higher K implies a proportionate increase in M. To keep κ(w min ) constant, this requires a proportionate increase in N, hence also in L d . The following proposition summarizes the results derived in this section.
Proposition 11. Consider a binding minimum wage in the economy of Section 1, i.e., w min > w. Then Finally, observe that net output under a binding minimum wage is strictly smaller than under laissez-faire. To see this, consider net output of (2.10) evaluated at q b (w), q a (w), and L d (w, K) as defined above. This gives ( ) In light of (5.12) and (5.14), one finds i.e., the adjustments of the equilibrium technology induced by a higher real wage have no first-order effect on net output. Intuitively, one may think of the competitive technology choice as maximising net output in an economy where factor endowments are given by K and L d min . Accordingly, a higher minimum wage affects net output only because it reduces employment. g a :  ++ →  ++ such that q b = q b (κ) > 0 with g κ b (κ) < 0, and q a = q a (κ) > 0 with g κ a (κ) > 0. Using these findings in (1.7) and (1.8) reveals that the factor prices satisfy ( 1 κ L. Hence, the equilibrium task intensity is indeed determined by (2.5). There is a unique κ > 0 that solves the latter equation. To see this, denote its right-hand side by RHS(κ), which is a continuous function RHS:  ++ →  ++ with RHS κ (κ) < 0 since g κ a (κ) > 0 > g κ b (κ). Moreover, it satisfies lim κ → 0 R HS(κ) > 0. Hence, there is a unique κ > 0 that satisfies κ = RHS(κ). Implicit differentiation reveals that κ = κ(θ) with κ θ (θ) > 0 and θ ≡ K/L. 

Proof of Proposition 1
(⇒) By construction, the equilibrium technology (q b , q a ) satisfies the first-order conditions (1.7)-(1.10) as equalities and the full employment conditions (2.1). It is shown that the solution to ( , ) ( , , , ) max b a b a q q V q q K L + ∈ coincides with (2.2). This establishes the first part of the proposition. Consider V(q b , q a , K, L) of (2.10) and recall that, by definition, net output at given factor endowments includes the full employment conditions (2.1) for symmetric technology choices, i.e., M = (1 + q b )K and N = (1 + q a )L. Then, with ( ) As F has constant returns to scale, (5.21) may be written as (5.19). Hence, (5.21) gives rise to the same functions as stated in (2.2).

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Proof of Corollary 1
From Theorem 1 any technology (q b , q a ) that qualifies as an equilibrium technology satisfies q b = g(q a ) of (2.7). From Proposition 1, the equilibrium technology is a global maximum of V(q b , q a , K, L) on  + 2 . Hence, the equilibrium technology also solves (2.12). 

Proof of Proposition 2
The first-order conditions (1.7) and (1.8) deliver q b (m) = q b and q a (n) = q a . Market clearing (2.1) means that M = (1 + q b )K and N = (1 + q a )L. Using this information in (1.9) and (1.10) reveals that equilibrium factor prices are equal to the net marginal products of (2.14) and (2.15), respectively, where (q b , q a ) = (q b , q a ). 

Proof of Proposition 6
Recall the partial effects derived in the proof of Proposition 5. Then, with (2.2) and (2.4) of Theorem 1 the proposition follows immediately. 