Development Economics

Transcription

1 Development Economics Slides 3 Debraj Ray Columbia, Fall 213 Development traps and multiple equilibrium, contd.

2 Complementarities via the Production Process The division of labor is limited by the size of the market. Smith (who else?), but the deeper insight is Allyn Young s: The division of labor determines the size of the market. Combine for a theory of production complementarities (Ciccone and Matsuyama 1996) Idea: roundaboutness in production (e.g., construction) Intermediate goods have setup costs, so need market. But can benefit production via increased variety. This generates increasing returns in a natural way.

3 Final output: X = [ n ] σ/(σ 1) x(i) 1 σ 1 di where n is variety, and σ > 1 is rate of substitution across inputs. No one machine is necessary in the production process. Capital K produces known intermediate inputs. Concavity divide equally: K = xn, where x is amount of each machine. Therefore X = {n(k/n) (σ 1)/σ } σ/(σ 1) = n 1/(σ 1) K. It is in this sense that variety is TFP. Complementarities arise from the determination of n.

4 The demand for machines: Given n, producers demand x(i) of machine i, chosen to maxf ( [ ) n σ/(σ 1) x(i) di] (σ 1)/σ,L wl Optimal control problem. FOC: n p(i)x(i)di. F X σ σ 1 [ n ] 1/(σ 1) σ 1 x(i) (σ 1)/σ di σ x(i) 1/σ = p(i), x(i) = F σ X X p(i) σ Note: p(i)x(i) = F X σ X, so demand is elastic. p(i) σ 1

5 The supply of machines: Each sector has setup cost of S (in labor units). After that a units of labor one unit of machine. So MC = wa. Each sector monopolistic (but different sectors are substitutes). Producers in machine sector i choose p(i) to max F σ X X p(i) σ 1 waf σ X X p(i) σ, Because each machine is of measure, equivalent to: max p(i) 1 p(i) σ 1 wa p(i) σ. Necessary and sufficient first-order conditions yield p(i) = waσ σ 1 w [choice of labor units]

6 We can use this to price the composite X. First: X = [ n = n σ/(σ 1) x, ] σ/(σ 1) x(i) (σ 1)/σ di because the same amount of every machine is bought. So the effective price of X is P = cost of buying X X = pnx n σ/(σ 1) x = w n 1/(σ 1). P /w, the relative factor price, is declining in n. Let α F X (X,L)X/F (X,L) be relative factor share of X; then α = α(n 1/(1 σ) ) A(n). is a nondecreasing function (flat if F is Cobb-Douglas).

7 So the operating profit for a machine producer is given by π = (p aw)x = px(1 a) = px σ = αy σn = A(n) n where the third equality is from the choice of labor units, and Y σ, the fourth equality from αy = npx, where Y is final output. An increase in variety, then, has three effects on profit: [1] Decreases share to each variety (1/n term). [2] Increases factor share of intermediates (A(n) term). [3] Via any effect on national income Y. Free-entry condition pins down equilibrium variety n: π = A(n) n Y σ = startup costs = ws,

8 Manipulate the condition from previous slide: nσ A(n) = Y ws = w(l + anx) + nπ ws. Now observe that π = ws, and L + anx = (T ns), where T is the total labor endowment in the economy. So nσ A(n) = T S. Potential for multiplicity is intimately linked to n/a(n). E.g., in Cobb-Douglas, A(n) is constant so unique solution. But if A(n) increases with n, then multiplicity possible.

9 Implications Complementarities may result in multiple equilibria. When they do, the equilibria are typically Pareto-ranked. Two fundamentally identical societies can behave differently. Complementarities change the way we think about policy. Temporary versus permanent interventions. amnesties, minimum wage legislation, temporary fines, family planning programs, affirmative action program... Warning: delicate task. Implementing equilibrium-tipping policies may be a