Complementarities via the Production Process

Transcription

1 Development Economics Slides 3 Debraj Ray Columbia, Fall 2013 Development traps and multiple equilibrium, contd. 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.

2 Final output: applez n X = x(i) di /( 1) where n is variety, and > 1 is rate of substitution across inputs. No one machine is necessary in the production process. A capital K produces known intermediate inputs. Concavity ) divide equally: K = xn, where x is amount of each machine. Therefore X = {n (K/n) ( 1)/ } /( 1) = n 1/( 1) K. It is in this sense that variety is TFP. Complementarities arise from the determination of n. The demand for machines: Given n, producers demand x(i) of machine i, chosen to applez! n maxf x(i) ( /( 1) Z n 1)/ di,l wl p(i)x(i)di. 0 Optimal control problem. FOC: 0 ) applez n F X x(i) ( 1)/ di 1 0 1/( 1) 1 x(i) 1/ = p(i), x(i)= F X X p(i) [elastic demand] Note: p(i)x(i)= F X X, so demand is elastic. p(i) 1

3 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 di erent sectors are substitutes). Producers in machine sector i choose p(i) to max F X X p(i) 1 wa F X X p(i), Because each machine is of measure 0, equivalent to: max p(i) 1 p(i) 1 wa p(i). Necessary and su cient first-order conditions yield p(i)= wa 1 w [choice of labor units] We can use this to price the composite X. First: X = applez n x(i) ( 1)/ di 0 = n /( 1) x, /( 1) because the same amount of every machine is bought. So the e ective price of X is P = cost of buying X X = pnx n /( 1) x = w n 1/( 1). ) P /w, the relative factor price, is declining in n. Let F X (X,L)X/F (X,L) be relative factor share of X; then = (n 1/(1 ) ) A(n). is a nondecreasing function (flat if F is Cobb-Douglas).

4 So the operating profit for a machine producer is given by =(p aw)x = px(1 a)= px = Y n = A(n) n where the third equality is from the choice of labor units, and the fourth equality from Y = npx, where Y is final output. An increase in variety, then, has three e ects on profit: [1] Decreases share to each variety (1/n term). Y, [2] Increases factor share of intermediates (A(n) term). [3] Via any e ect on national income Y. Free-entry condition pins down equilibrium variety n: = A(n) n Y = startup costs = ws, Manipulate the condition from previous slide: n A(n) = Y ws w(l + anx)+n =. ws Now observe that = ws, and L + anx =(T ns), where T is the total labor endowment in the economy. So n 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.

5 Implications Complementarities may result in multiple equilibria. When they do, the equilibria are typically Pareto-ranked. Two fundamentally identical societies can behave di erently. Complementarities change the way we think about policy. Temporary versus permanent interventions. amnesties, minimum wage legislation, temporary fines, family planning programs, a rmative action program... Warning: delicate task. Implementing equilibrium-tipping policies may be a Questions of Transition Problem: these theories are way too nondeterministic. Why does yesterday s state a ect today s state? Why is QWERTY stickier than fashion? Why does a transition follow a logistic path? Percentage of adopters (x) x t Time

6 Lagged Externalities r N, r O r N r N (1) r O r N (0) x y % in the new sector QWERTY vs fashion? Logistic curve? Beliefs About Beliefs Hold or sell? For good states, hold; for bad, sell. Otherwise, multiple equilibria(?) Morris and Shin (AER 1998) r r hold (x, g) r hold (x, s) r sell (x, s) r hold (x, b) % selling the peso (x)

7 Equilibrium Transition? Fertility Decline in Bangladesh Munshi and Myaux (JDE 2006) : Total fertility rate goes from 4.5 to 2.9. This is a huge drop. Norms governing fertility use and contraception. Contraception went from 40% in 1983 to 63% in 1993.

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9 (&'3,'"#$%&'()$*+(,8($3"7C,/&$0&1,?DDE,!!! /($3"7, ";!W*)2*!!! D0##! T9]HX! FH;a! 6-$23BW$)20-)!7)50,)! FIF! F;X! 6-^),$0?#)*! a9gfz! XGZ! a;h! <'?3#!#0(3$0&-! X]a! Z;F! S3*),$&/4! FX! G;F! \$%)2*! TYI! T;a!!!! =;!c&-b'*)2*! a9fyx! ]];H!!!! <00,?FCGHF,?II, Khan-Bairagi (1998) 0.7 Hindus 0.65 Full Sample Contraceptive Prevalence Muslims Year Fig. 1. Contraceptive prevalence over time.

10 Table 1 Inter-quartile transition probabilities 0.25 quantile Distribution mean 0.55 std.dev quantile quantile Distribution mean 0.27 std.dev quantile quantile quantile Table 2 Descriptive statistics Full sample Hindus Muslims Illiterate Literate (1) (2) (3) (4) (5) Panel A: Individual characteristics Age (8.01) (8.00) (8.01) (8.18) (7.44) Number of children 2.41 (1.99) 2.18 (1.79) 2.45 (2.03) 2.57 (2.05) 2.14 (1.86) Education 2.12 (3.12) 1.48 (2.68) 2.26 (3.19) (2.55) Husband s education 3.21 (4.00) 3.07 (3.81) 3.24 (4.04) 1.53 (2.62) 5.91 (4.34) Panel B: Occupation of household head (%) Farming Fishing Business Housework Other Total Panel C: Asset ownership Land (hectares) 1.00 (2.55) 0.72 (1.39) 1.06 (2.74) 0.82 (2.41) 1.29 (2.74) Cows 1.06 (1.57) 0.81 (1.42) 1.11 (1.59) 0.91 (1.46) 1.28 (1.70) Boats 0.55 (0.61) 0.63 (0.76) 0.54 (0.57) 0.55 (0.61) 0.56 (0.60) No. of Observations 21, ,723 13, Panel D: Contraceptive prevalence Probability of using 0.55 (0.50) 0.59 (0.49) 0.54 (0.50) 0.53 (0.50) 0.57 (0.50) contraceptives No. of Observations 144,186 26, ,772 91,727 52,459 Means (standard deviations) in panel A, panel C and panel D. The individual is the unit of observation in panels A C. The individual-year is the unit of observation in panel D. All statistics in this table are computed over the full sample period.

11 Is the fertility decline a multiple equilibrium phenomenon? Idea: regress current contraception use on overall contraception But there is a problem here with identification (Manski critique) Omitted variable that correlates individual and village-level use? Or peer e ects that are needed to generate multiple equilibria? y it = A + y i,t 1 + x v(i) t 1 + Z it + C v(i) t + ivt y i is 0-1 for contraceptive use by couple i, t is time, x is aggregate village-level use, v(i) is the village of person i, Z a vector of individual characteristics (such as age), A is a constant and C v t is unobserved omitted variable for village v at date t. C v t C v t can be decomposed into three parts. First component only depends on the village: C1 v. Second component only depends on time: C t2. Third varies in a village-specific way over time. Components 1 and 2 dealt with by village and time fixed e ects. The last one screws everything up: identification problem.

12 Main Idea in Munshi-Myaux Paper Inter-religion communication low. So include own-group and cross-group use separately. If own-e ect strong, then pushes back the Manski critique: For critique to work, there has to be an omitted variable which is village-, time- and group-specific. y it = A + m y i,t 1 + mm x v(i),m t 1 ++ mh x v(i),h t 1 + m Z it + C v(i),m t + ivt where i is m-household, and m and h labels self-explanatory. Can get spurious e ects only if C v(i),m t and C v(i),h t orthogonal. Table 3 Partitioning the village by religion Dependent variable: contraception All villages More than 5% Hindus/Muslims More than 15% Hindus/Muslims Annual data Muslims Hindus Muslims Hindus Muslims Hindus Muslims Hindus (1) (2) (3) (4) (5) (6) (7) (8) Lagged contraceptive prevalence (own group) (0.013) (0.014) (0.016) (0.017) (0.018) (0.020) (0.023) (0.023) Lagged contraceptive prevalence (other group) (0.006) (0.007) (0.011) (0.016) (0.013) (0.024) (0.011) (0.012) Lagged contraception (0.003) (0.005) (0.004) (0.005) (0.004) (0.006) (0.005) (0.008) R Number of observations 139,875 43,101 79,927 29,771 49,730 20,756 70,787 21,419 Box Pearson Q statistic Standard errors in parentheses. Standard errors are robust to heteroskedasticity and correlated residuals within each village-period. 2