Stolper and Samuelson Theorem

Transcription

1 Stolper and Samuelson Theorem Ram Singh Course 001 October 27, 2015 Ram Singh: (DSE) Factor Price Equalization October 27, / 13

2 International Trade: Basic Set-up I Optional Readings: MWG, Stolper and Samuelson, (1941), Review of Economic Studies; and Samuelson, P. A. (1948), Economic Journal Consider a small open economy Two goods are produced; food and cloth/car, f and c Two FOPs are used; labour and capital, l and t Production technology; constant returns to scale (?) Production functions; strictly quasi-concave (?) Food : y f = y f (z f l, z f t ) Car/cloth : y c = y c (z c l, z c t ) In view of constant returns assumption, Ram Singh: (DSE) Factor Price Equalization October 27, / 13

3 International Trade: Basic Set-up II the optimum mix of inputs (cost minimizing combination of FOPs) does not depend on the level of operation. Therefore, to search for optimum mix of FOPs, we can take output level to be one: That is, we can focus on the following optimization: Given vector of factor prices (w, r), min{wâ j l + râ j t } (1) y j (â j l, âj t ) = 1, where j = f, c. In view of the assumption on y j, (1) has a unique solution. Let a f l, af t solve (1) for food sector a c l, ac t solve (1) for car/cloth sector Ram Singh: (DSE) Factor Price Equalization October 27, / 13

4 International Trade: Basic Set-up III That is, 1 = y f (al f, at f ) 1 = y c (al c, ac t ) Clearly, for j = f, c we have al f = zl f (w/r) at f = zt f (w/r) al c = zl c (w/r) at c = zt c (w/r) Ram Singh: (DSE) Factor Price Equalization October 27, / 13

5 International Trade: Basic Set-up IV Now, the production process is characterized by: al f = zl f (w/r) (2) at f = zt f (w/r) (3) al c = zl c (w/r) (4) at c = zt c (w/r) (5) l = y f al f + y c al c = y f zl f (w/r) + y c zl c (w/r) (6) t = y f at f + y c at c = y f zt f (w/r) + y c zt c (w/r) (7) p f = wal f + rat f = wzl f (w/r) + rzt f (w/r) (8) p c = wal c + rat c = wzl c (w/r) + rzt c (w/r) (9) Ram Singh: (DSE) Factor Price Equalization October 27, / 13

6 Factor Intensity Definition Food sector is labour intensive if, Assumption ( (w, r) >> (0, 0)) [ a f l a f t ] > ac l at c Suppose for all possible pair of input prices we have al f at f > ac l a c t WLOG, assume F sector is labour intensive (??): That is, [ ] a f ( (w, r) >> (0, 0)) l > ac l at f at c Ram Singh: (DSE) Factor Price Equalization October 27, / 13

7 Factor Prices Across Countries I If, Production technology is the same across countries There is free flow of goods across countries So, the output prices are the same across countries then, Free trade is a substitute for free flow of FOP. Theorem Factor Price Equalization Theorem. Suppose, the factor intensity assumption holds. For any given output price vector and technology, the factor prices will be the same across countries. Ram Singh: (DSE) Factor Price Equalization October 27, / 13

8 Factor Prices Across Countries II Let, s = w r. So, p = pf p c = waf l + ra f t wa c l + ra c t = saf l + a f t sa c l + a c t (10) Note sign of dp ds = sign of (ac t a f l a f t a c l ) > 0. Question What is the ceteris paribus effect of change in factor endowments on output levels? What is the ceteris paribus effect of change in output prices on factor prices? Ram Singh: (DSE) Factor Price Equalization October 27, / 13

9 Stolper-Samuelson Theorem I Theorem Stolper-Samuelson Theorem. Given the factor intensity assumption, an increase in relative price of good c leads to increase in relative price of t, and vice-versa. Differentiating (8) and (9) we get Let θ f l = waf l p f dp f = a f l dw + a f t dr + wda f l + rda f t (11) dp c = a c l dw + ac t dr + wda c l + rda c t (12), the share of labour in food sector θ f t = raf t p f, the share of capital in food sector θ c l = wac l p, the share of labour in cloth sector c Ram Singh: (DSE) Factor Price Equalization October 27, / 13

10 Stolper-Samuelson Theorem II θ c t = rac t p c, the share of capital in cloth sector Clearly, θ f l + θ f t = 1 and θ c l + θ c t = 1. Now, we have w.r p f.p c (ac t al f at f al c ) = w.af l r.at f p f p f w.ac l p c r.a c t p c = θ f l θ c t θ f t θ c t = θl f (1 θl c ) (1 θf l )θl c = θl f θl c > 0. (13) Ram Singh: (DSE) Factor Price Equalization October 27, / 13

11 Stolper-Samuelson Theorem III That is, the share of l is higher in l intensive sector, and vice-versa. The least cost condition gives us where âl i da i l, etc. al i From above, we get wda j l + rda j t = 0, i.e., (14) wa j da j l l a j + ra j da j t l l a j l = 0, i.e., (15) θl i âi l + θ tâi i t = 0, (16) Ram Singh: (DSE) Factor Price Equalization October 27, / 13

12 Stolper-Samuelson Theorem IV dp f = a f l dw + a f t dr dp c = a c l dw + ac t dr. That is, That is, dp f = af l dw + af t dr p f p f p f dp c p c = ac l dw p c + ac t dr p c = af l w p f = ac l w p c ˆp f = θ f l ŵ + θ f t ˆr ˆp c = θ c l ŵ + θc t ˆr. dw w + af t r p f dw w + ac t r p c dr r dr r. Ram Singh: (DSE) Factor Price Equalization October 27, / 13

13 Stolper-Samuelson Theorem V That is, That is, ˆp c ˆp f = (θl f θl c )(ˆr ŵ) (17) ˆr ŵ = 1 θ (ˆpc ˆp f ) (18) where θ = θ f l θ c l < 1. Theorem Given the factor intensity assumption, an increase in relative price of good c leads to more than proportionate increase in relative price of t, and vice-versa. Ram Singh: (DSE) Factor Price Equalization October 27, / 13