Equilibrium in Factors Market: Properties Ram Singh Microeconomic Theory Lecture 12 Ram Singh: (DSE) Factor Prices Lecture 12 1 / 17
Questions What is the relationship between output prices and the wage rate for the relevant FOPs? Does competitive market provide fair wage to all FOPs? How does distribution of wealth affect the wage rates for different FOPs? What market factors affect the wage rates for different FOPs? What non-market factors affect the wage rates for different FOPs? Ram Singh: (DSE) Factor Prices Lecture 12 2 / 17
Firms and FOPs I Readings: MWG. Assume N M Economy. Further, There are no intermediate goods; There are pure inputs (factors of production) and pure consumptions goods; Pure inputs/fops are l = 1,..., L. Set of FOPs is L = {1,.., L} total endowments of factors is z = ( z 1,..., z L ) >> 0 and is initially owned by consumers. Consumers do not directly consume these endowments. A production plan of kth firm can be written as y k R L+M y k = ( z k 1,..., z k L, y k k,..., y k L+M) Ram Singh: (DSE) Factor Prices Lecture 12 3 / 17
Firms and FOPs II For simplicity, assume One firm produces only one good; good j is produced by firm j. That is, k = j, and j = 1,..., M. Set of firms and also the consumption goods becomes J = {1,.., M} So a production plan of k = jth firm can be written as y j R L+M y j = ( z j 1,..., zj j L, 0,..., yj,..., 0) Alternatively, a production plan of j = kth firm can be written as y j R L+1 where j = 1,..., M. y j = ( z j 1,..., zj L, y j j ), Ram Singh: (DSE) Factor Prices Lecture 12 4 / 17
Firms and FOPs III Take any given price vector p = ( p 1,..., p M ) for outputs and w = ( w 1,..., w L ) for inputs. Firm j will choose ȳ j Y L+1 that solves where f j (z j ) = y j j, and { max y j Y L+M { max y j Y L+1 L k=1 p j f j (z j ) w k.z j k + p j.y j j } } L w k.z j k k=1 z j = (z j 1,..., zj L ) is input vector used by firm j., i.e., Ram Singh: (DSE) Factor Prices Lecture 12 5 / 17
Production Equilibrium I Assume Assume small open economy output price vector p is given for the economy Now, equilibrium in the factors market is a price vector w = (w 1,..., w L ), and factor allocation for each firm z 1 = (z 1 1,..., z 1 L). =. z M = (z M 1,..., z M L ) such that M z j l = z l Ram Singh: (DSE) Factor Prices Lecture 12 6 / 17
Production Equilibrium II and z j solves Assume max z j { p j f j (z j ) L k=1 w l.z j l, } i.e., max { pj f j (z j ) w.z j. } (1) z j f j (.) is strictly increasing and strictly concave for all j = 1,..., M Ram Singh: (DSE) Factor Prices Lecture 12 7 / 17
Production Equilibrium III Now, the equilibrium is characterized by the following FOCs: p j f j (z j ) z j l = w l for all l = 1,..., L, & j = 1,..., M (2) M z j l = z l for all l = 1,..., L. (3) Remark We have assumed p = ( p 1,..., p M ) and z = (z 1,..., z L ) to be given (parameters) (w 1,..., w M) and (z 1,..., z M ) depend on p = ( p 1,..., p M ) and z = (z 1,..., z L ). Therefore, in a competitive equilibrium, (w 1,..., w M) and p = ( p 1,..., p M ) will be determined simultaneously. Ram Singh: (DSE) Factor Prices Lecture 12 8 / 17
Maximizing the Cake-size I Theorem The equilibrium factor allocation, (z 1,..., z M ), maximizes the aggregate/total revenue for the economy. Let the output price vector be p = ( p 1,..., p M ) the input price vector be w = (w 1,..., w L ) Z = (z 1,..., z M ) any arbitrary allocation of FOPs across firms Ram Singh: (DSE) Factor Prices Lecture 12 9 / 17
Maximizing the Cake-size II At Z profit for the entire economy is: M ( pj f j (z j ) w.z j) = ( p 1 f 1 (z 1 ) w.z 1) +... + ( p j f j (z j ) w.z j) +... + ( p M f J (z M ) w.z M) (4) where w.z j = L l=1 w l.z j l, for all j = 1,..., M. By re-writing the RHS, (4) becomes M ( pj f j (z j ) w.z j) M M = p j f j (z j ) w.z j (5) From (1) z j solves: max z j solves max z 1,...,z J { p j f j (z j ) } L l=1 w l.z j l. Therefore, (z 1,..., z M ) M ( pj f j (z j ) w.z j), i.e., Ram Singh: (DSE) Factor Prices Lecture 12 10 / 17
Maximizing the Cake-size III max z 1,...,z J M M p j f j (z j ) w.z j (6) Next, note that in equilibrium, j zj = z must hold. Therefore, (z 1,..., z M ) also solves: M M p j f j (z j ) w.z j, (7) max z 1,...,z J subject to constraint that j zj = z. However, when j zj = z must hold, we have M M w.z j = w. z j = w. z. Ram Singh: (DSE) Factor Prices Lecture 12 11 / 17
Maximizing the Cake-size IV That is, (z 1,..., z M ) essentially solves the following optimization problem: p j f j (z j ), (8) such that j zj = z. Remark max z 1,...,z J j The Competitive equilibrium allocation is also Revenue maximizing allocation. Therefore, The optimum factor allocation can be determined without determining factor prices. Ram Singh: (DSE) Factor Prices Lecture 12 12 / 17
Maximizing the Cake-size V Question Should a country focus on Revenue (GDP) maximizing allocation of FOPs? While deciding on allocation of FOPs, can we ignore the issue of equity in distribution of gains from growth? Ram Singh: (DSE) Factor Prices Lecture 12 13 / 17
Wage-rates I Question Are wage-rates fair under a competitive economy? Let Note that: z = M z j and z = j M F (z) = p 1 f 1 (z 1 ) +... + p M f M (z M ) F (z) = M p jf j (z j ), i.e., F(z) denotes the total aggregate revenue for the entire economy. Therefore, given output price vector p = ( p 1,..., p M ), in view of (6) we know that z = (z 1,..., z M ) solves: j z j Ram Singh: (DSE) Factor Prices Lecture 12 14 / 17
Wage-rates II max {F (z) z 0 w.z} This optimization problem has the following FOCs: w 1 = F (z) z 1 Since, in equilibrium z = z, we get. =. wl = F (z) z L ( l L) [ wl = F( z) ] z l Ram Singh: (DSE) Factor Prices Lecture 12 15 / 17
Wage-rates III That is, w 1 = F ( z) z 1. =. wl = F ( z) z L Remark In a competitive setting Each FOP is paid equal to its marginal social productivity (in money terms) A Ceteris Paribus increase in supply of a FOP decrease its market price (wage). Why? Ram Singh: (DSE) Factor Prices Lecture 12 16 / 17
Wage-rates IV Question How will distribution of wealth affects the market prices for FOPs? How will an increase in supply of a FOP affect its market price? Ram Singh: (DSE) Factor Prices Lecture 12 17 / 17