Effect of Change in P on W and r (Stolper-Samuelson)

Effect of Change in P on W and r (Stolper-Samuelson)

Formal analysis: effect of change in P on w and r. Y is numeraire.

A. Equilibrium conditions (P=MC conditions)

(1) a_Lx W + a_Kx r = P (cost of producing X)

(2) a_Ly w + a_Ky r = 1 (cost of producing Y)

B. Totally differentiate the 2 above conditions:

a_Lx dw + a_Kx dr = dP - ((wd a_Lx + r a_Kx )/P).

Dividing by P, etc. gives the elasticity form:

S_lx dw/w + S_Kx dr/r = dP/P - ((wd a_Lx + rd a_Kx )/P) .

These a’s are not constants: a_Lx = a_Lx (X, r/w) = K*/L* where * is equilibrium level. W and r are functions, implicitly, of X* and Y* and s_Lx (Elasticity form) = w a_Lx /P

Similarly for Y:

W a_Ly dw/w + r a_Ky dr/r = 0 - (wd a_Ly + rd a_Ky ).

Now a_Lx and a_Kx change only due to technology or w/r ratio. If w/r ratio rises, a_Lx falls and a_Kx rises.

C. What is slope of isoquant? Remember, CRS is assumed. Otherwise, a change in P has a scale effect as well as a substitution effect on s. That is, a_Lx is not only a function of a_K.x .

This is the equation of the isoquant. In equilibrium (under cost minimization for the firm, which is given P, w and r) the isoquant is tangent to the isocost line. Min cost over a_Lx ,a_Kx = min a_Lx (a_Kx )w + a_Kx r over a_{K x}

==>( d a_Lx /d a_Kx ) w + r = 0

Now dL/dK given X=X0 corresponds to d a_Lx /d a_Kx = -r/w

Around equilibrium. Envelope theorem. Thus,

rd a_Kx + wd a_Lx = 0

(Hence usefulness of this for distortions, when you’re not around the equilibrium and during growth.) Thus

s_Lx dw/w + s_kx dr/r = dP/P, or

Define Q_lx = s_Lx = w a_Lx /P. Then

(3) Q_lx w^ + Q_Kx r^ = P^ and

(4) Q_Ly w^ + Q_Ky r^ = 0

Or, in matrix notation,

Q_Lx Q_Kx W^ P

Q_Ly Q_Ky R^ = 0

C. Solve matrix: w^ = (Q_ky ‘/detQ )P^ and

R^ = (a_Lx ‘/det(Q))P^

Det(Q) = Q_Lx Q_Ky - Q_Kx Q_ly and Q is the matrix of Qij ‘s.

D. Since Y is K-intensive by definition, denominator is positive. That is, P rising means w rises and r falls. Denominator is a fraction < numerator; in elasticity form, w^/P^ > 1. Magnification effect: wage rate goes up more than price. And r^/P^ < 0. So

W^ > P^ > r^

The change in price is a weighted average of changes in factor prices.

E. What happens to w/r? (W/r)^ = {1/det(Q)}P^ This also reflects the magnification effect via the factor intensities. If Q_kx and Q_ky are close together, large change in w due to small change in P. (3) and (4) give this result also.

F. There is a 1:1 relationship between the factor rentals ratio and commodity prices. This is the Stolper-Samuelson Theorem. Suppose we impose a tariff on an imported good. What happens to the MPK/MPL? Should laborers favor the tariff? Previously, producers of that good were assumed to benefit and consumers of it to lose out. Thus, there were no unambiguous answers. But Stolper-Samuelson gives unambiguous answer.