IS-OPEN1

Derivation of IS, LM and B/P=0 Curves

The equilibrium in the product market

Y = c + i + g + x - m

c = c(y) 0 £ c_y£ 1

i = I(r) i_r < 0

g = g

x = x(q, e) x_q > 0; x_e > 0

m = m(y, q, e) 0 £ m_y £ 1; m_q < 0; m_e < 0

where

y = income

c = consumption expenditure

i = investment expenditure

g = government expenditure

x = exports

m = imports

q = price of foreign goods in foreign currency

e = price of foreign exchange

Thus

y = c(y) + i(r) + g + x(q, e) – m(y, q, e)

Taking total differential

dy = c_ydy + i_rdr + dg + x_qdq + x_ede - m_ydy - m_qdq - m_ede

dr = [1/i_r][(1 - c_y + m_y)dy - dg - (x_q - m_q)dq - (x_e - m_e)de]

which is an IS curve whose slope is 1/ir(1 - c_y + m_y) and is negative.

The curve shifts to right with dg > 0 as -1/i_r > 0,

and with dq > 0 if (x_q - m_q) > 0,

and with de > 0 if (x_e - m_e) > 0.

Note:

dy = [1/(1 - c_y + m_y )][i_rdr + dg + (x_q - m_q)dq + (x_e - m_e)de]

The equilibrium in the money market

M^d = L(y, r) L_y >0, L_r < 0

M^s = D + R

M^s = P*Md

Where

M^d = Demand to hold money

M^s = Supply of money

y = Real income

r = interest rate

D = Domestic component of money supply

R = International component of money supply

Thus

M^s = PL(y, r)

Taking total differential

dM^s = PdL + LdP

= P[L_ydy + L_rdr] + LdP

dr = [-1/PL_r][ PL_ydy + LdP - dM^s]

= [-1/PL_r][ PL_ydy + LdP - (dD + dR)]

which is the LM curve whose slope is -PL_y/PL_r > 0

The curve may shift down if dP < 0 or dD, dR > 0

However,

P = aP_d + (1-a)P_f

The general price level (implicit price deflator or GDP deflator) is the weighted average of price of domestic goods, P_d, and price of foreign goods, P_f = eq.

Assume for simplicity that there is one foreign price level. Assume also that our domestically produced goods are all tradable, ie there is not one goods that are neither exported to nor imported from abroad, then

P_d = P_f

Thus

P = eq

So that

dP = qde + edq

Thus, the LM curve would shift if de, dq ¹ 0.

The External Equilibrium

B/P = 0

= Px(q, e) - eqm(y, q, e) + F(r)

where F is net capital inflows such that F_r > 0

Taking total differential

dB/P = Pdx + xdP - eqdm - medq - mqde + dF

= P[x_ede + x_qdq] + xdP -eq[m_ydy + m_ede + m_qdq] - m_edq - m_qde + F_rdr

= [Px_e - eqm_e - mq]de + [Px_q - eqm_q - me)dq - eqm_ydy + F_rdr + xdP

= 0

dr = [1/F_r][eqm_ydy - xdP - (Px_e - eqm_e-mq)de - (Px_q - eqm_q - me)dq]

As P = eq or dP = qde + edq

Then

dr = [1/F_r][eqm_ydy - (Px_e - eqm_e- (x – m)q)de - (Px_q - eqm_q - (x – m)e)dq]

which represents a relation between r and y such that the external equilibrium is achieved.

The relation has a slope of eqm_y/F_r > 0.

That is, if income increases, imports rise leading to current account deficit. To maintain equilibrium in the external sector, capital inflows are needed to offset the current account deficit. To induce capital inflows, interest rate must rise.

Points below the B/P curve indicate a deficit in the external account. This is because at the interest rate given at the point, income is too high. Thus more imports.

Also, at the income level indicated by the point, interest rate is too low, leading to an inadequate capital inflow.

The steepness of the B/P curve depends on the value of F_r. If F_r is low the curve is steep. If the capital flows do not depend on interest rate, F_r = 0, then the curve is a vertical line.

If F_r is very large, the curve is fairly flat. If it goes to infinity, ie a small change in interest rate would induce infinite flows of capital, then the curve is horizontal.

Shift of the B/P=0 curve:

If devalue, de > 0, then the curve shifts down, provided that the Marshall–Lerner condition is satisfied.

The curve also shifts down if dq > 0.

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