Factor Price Equalization and Stolper-Samuelson Theorem

Factor Price Equalization and Stolper-Samuelson Theorem

Ram Singh

Course 001

October 28, 2016

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 andc Two FOPs are used; labour and capital,landt Production technology; constant returns to scale (?) Production functions; strictly quasi-concave (?)

Food : y^f =y^f(z_l^f,z_t^f) Car/cloth : y^c =y^c(z_l^c,z_t^c) In view of constant returns assumption,

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{waˆ^j_l+rˆa^j_t} (1)

y^j(ˆa^j_l,aˆ^j_t) = 1, wherej =f,c.

In view of the assumption ony^j, (1) has a unique solution. Let a^f_l,a_t^f solve (1) for food sector

a^c,a^c solve (1) for car/cloth sector

International Trade: Basic Set-up III

That is,

1 = y^f(a_l^f,a^f_t) 1 = y^c(a^c_l,a^c_t) Clearly, forj =f,c we have

a^f_l = z_l^f(w/r) a^f_t = z_t^f(w/r) a^c_l = z_l^c(w/r) a^c_t = z_t^c(w/r)

International Trade: Basic Set-up IV

Now, the production process is characterized by:

a^f_l = z_l^f(w/r) (2)

a^f_t = z_t^f(w/r) (3)

a^c_l = z_l^c(w/r) (4)

a^c_t = z_t^c(w/r) (5)

¯l = y^fa^f_l +y^ca^c_l =y^fz_l^f(w/r) +y^cz_l^c(w/r) (6)

¯t = y^fa^f_t+y^ca^c_t =y^fz_t^f(w/r) +y^cz_t^c(w/r) (7) p^f = wa^f_l +ra^f_t =wz_l^f(w/r) +rz_t^f(w/r) (8) p^c = wa^c_l +ra^c_t =wz_l^c(w/r) +rz_t^c(w/r) (9)

Factor Intensity

Definition

Food sector is labour intensive if,

(∀(w,r)>>(0,0)) a^f_l

a^f_t >a_l^c a_t^c

Assumption

Suppose for all possible pair of input prices we have a^f_l

a^f_t > a^c_l a^c_t or a^f_l

a^f_t < a^c_l a^c_t

WLOG, assume F sector is labour intensive (??): That is, (∀(w,r)>>(0,0))

a^f_l a^f_t >a^c_l

a^c_t

,i.e.,(a_t^ca^f_l −a^f_ta^c_l)>0

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.

Factor Prices Across Countries II

Let,s= ^w_r. So,

p= p^f

p^c = wa^f_l +ra^f_t wa^c_l +ra^c_t

= sa^f_l +a^f_t

sa^c_l +a^c_t (10)

p = g(s) (11)

Sign ofdp

ds =sign of(a_t^ca^f_l −a^f_ta^c_l)>0.

Thereforeg⁰(s)>0. Lets=h(p), where h() =g⁻¹(.)

Clearly,h⁰()>0

Factor Prices Across Countries III

Question

What is theceteris paribuseffect of change in factor endowments on output levels?

What is theceteris paribuseffect of change in output prices on factor prices?

Stolper-Samuelson Theorem I

Theorem

Stolper-Samuelson Theorem. Given the factor intensity assumption, an increase in relative price of c leads to increase in relative price of t, and vice-versa.

Differentiating (8) and (9) we get

dp^f = a^f_ldw+a^f_tdr +wda^f_l +rda^f_t (12) dp^c = a^c_ldw+a^c_tdr+wda^c_l +rda^c_t (13) Let

θ_l^f =^wafl

p^f , the share of labour in food sector θ_t^f =^raft

p^f, the share of capital in food sector θ_l^c= ^wa_pc^cl, the share of labour in cloth sector

Stolper-Samuelson Theorem II

θ_t^c= ^ra_pc^ct, the share of capital in cloth sector Clearly,θ^f_l +θ^f_t =1 andθ^c_l +θ^c_t =1.

Recall(a^c_ta^f_l −a^f_ta^c_l)>0. So, we have

w.r

p^f.p^c(a^c_ta^f_l −a^f_ta^c_l) = w.a^f_l p^f

r.a^f_t

p^f −w.a^c_l p^c

r.a^c_t p^c

= θ^f_lθ_t^c−θ^f_tθ^c_t

= θ^f_l(1−θ^c_l)−(1−θ_l^f)θ_l^c

= θ^f_l −θ^c_l >0. (14)

Stolper-Samuelson Theorem III

That is, the share ofl is higher inlintensive sector, and vice-versa. The least cost condition gives us

wda^j_l+rda^j_t = 0,i.e., (15) wa^j_lda^j_l

a^j_l +ra^j_lda^j_t

a^j_l = 0,i.e., (16)

θⁱ_laˆⁱ_l+θⁱ_tˆaⁱ_t = 0, (17)

whereaˆⁱ_l =^dail

aⁱ_l , etc.

From (11) and (12), we get

Stolper-Samuelson Theorem IV

dp^f = a^f_ldw+a^f_tdr

dp^c = a^c_ldw+a^c_tdr. That is,

dp^f

p^f = a^f_ldw p^f +a^f_tdr

p^f =a^f_lw p^f

dw w +a^f_tr

p^f dr

r dp^c

p^c = a^c_ldw

p^c +a^c_tdr

p^c =a^c_lw p^c

dw w +a^c_tr

p^c dr

r . That is,

ˆp^f = θ^f_lwˆ +θ^f_tˆr pˆ^c = θ^c_lwˆ +θ_t^cˆr.

Stolper-Samuelson Theorem V

That is,

pˆ^c−pˆ^f = (θ^f_l −θ^c_l)(ˆr−wˆ) (18) That is,

ˆr−wˆ = 1

θ(ˆp^c−pˆ^f) (19)

whereθ=θ^f_l −θ_l^c<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.

Effect of Labour Supply I

Differentiating (6) and (7) we get

d¯l = a^f_ldy^f+a^c_ldy^c (20) d¯t = a^f_tdy^f+a^c_tdy^c (21) Note,

a^f_ldy^f = dy^f y^f

a^f_ly^f

¯l

= yˆ^fλ^f_l,

whereλ^f_l = ^afl_¯^yf

l . Clearly,

λ^f_l +λ^c_l = 1 λ^f_t +λ^c_t = 1

Effect of Labour Supply II

Now, (21) and (22) will give us

yˆ^f−yˆ^c = 1 λ(ˆ¯l−ˆ¯t) whereλ=λ^f_l −λ^f_t >0