International Parity Conditions in a Two-Country OLG Model Under Free Trade

Abstract

This chapter extends the basic OLG growth model of Chap. 2 into a two-country OLG model with free trade of two commodities and government bonds but internationally immobile labor and capital. In the original version of the model set-up young households in both countries hold national fiat money for transaction purposes. The intertemporal choice problem of young households is split into two parts: first, the determination of utility maximizing asset holding and consumption expenditures and second, the expenditure minimizing allocation of the consumption basket on the two commodities produced domestically and abroad. From the latter the purchasing power parity in its absolute and relative version is derived, while from the former the uncovered interest parity condition is obtained. Finally, the basic neoclassical (Heckscher-Ohlin) model of inter-sectoral trade is presented as a special case of the general two-country model.

Mathematical Appendix

Not in every case an optimization problem has a so-called interior solution, where all optimization variables are strictly positive. If some variables take the value zero at the optimum, the solution is called a corner solution. When the solution to an optimization problem is a corner solution, the classical optimization conditions in the appendix of Chap. 2 cannot be applied. Therefore, we need the conditions of Kuhn and Tucker (1951) for solving a non-linear optimization problem (= Kuhn-Tucker conditions).

It would be too time-consuming to derive these conditions in a mathematically exact way. The interested reader is referred to Takayama (1974, 86 ff.). We will simply apply these first-order conditions to solve the optimization problem of the young household in Sect. 9.4.

Like in classical optimization (see the appendix to Chap. 2) we transform the maximization problem under constraints by means of a Lagrangian into a saddle point problem. The Lagrangian at hand is more complicated than that in Chap. 2, because two constraints (rather than one constraint) have to be considered. Consequently we have to distinguish between two Lagrange multipliers, ${\lambda_t}\ \mathrm{ and}\ {\lambda_{\mathrm{ t}+1}}$.

The Lagrangian of the domestic young household’s choice problem is:

$$ {\varLambda^I}\left( {c_t^{II,1 },c_t^{AI,1 },c_{t+1}^{II,2 },c_{t+1}^{AI,2 },I_t^{I,I },I_t^{A,I },\lambda_t^I,\lambda_{t+1}^I} \right) $$

$$ \equiv \ln \left( {c_t^{II,1 }+c_t^{AI,1 }} \right)+\beta\ln \left( {c_{t+1}^{II,2 }+c_{t+1}^{AI,2 }} \right) $$

$$ +\lambda_t^I\left( {W_t^I-P_t^{x,I }c_t^{II,1 }-P_t^{x,A }{E_t}c_t^{AI,1 }-P_t^{y,I}\left( {\frac{{I_t^{I,I }}}{{{L^I}}}+\left( {1-\delta } \right)\frac{{K_t^I}}{{{L^I}}}} \right)} \right.\left. {-P_t^{y,A }{E_t}\frac{{I_t^{A,I }}}{{{L^I}}}} \right) $$

$$ {+\lambda_{t+1}^I\left[ {\left( {Q_{t+1}^I+(1-\delta )P_{t+1}^{y,I }} \right)\left( {\frac{{I_t^{I,I }}}{{{L^I}}}+\frac{{I_t^{A,I }}}{{{L^I}}}+(1-\delta )\frac{{K_t^I}}{{{L^I}}}} \right)-P_{t+1}^{x,I }c_{t+1}^{II,2 }-P_{t+1}^{x,A }{E_{t+1 }}c_{t+1}^{AI,2 }} \right]}. $$

(9.26)

Next, we calculate all first partial derivatives of the Lagrangian Eq. 9.26 and set up the following Kuhn-Tucker conditions:

$$ \frac{{\partial {\varLambda^I}}}{{\partial c_t^{II,1 }}}=\frac{1}{{c_t^{II,1 }+c_t^{AI,1 }}}-\lambda_t^IP_t^{x,I}\le 0,\ \frac{{\partial {\varLambda^I}}}{{\partial c_t^{II,1 }}}c_t^{II,1 }=0,\;c_t^{II,1}\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial c_t^{AI,1 }}}=\frac{1}{{c_t^{II,1 }+c_t^{AI,1 }}}-\lambda_t^I{E_t}P_t^{x,A}\le 0,\ \frac{{\partial {\varLambda^I}}}{{\partial c_t^{AI,1 }}}c_t^{AI,1 }=0,\;c_t^{AI,1}\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial I_t^{I,I }}}=-\lambda_t^I\frac{{P_t^{y,I }}}{{{L^I}}}+\lambda_{t+1}^I\frac{{\left( {Q_{t+1}^I+\left( {1-\delta } \right)P_{t+1}^{y,I }} \right)}}{{{L^I}}}\le 0,\ \frac{{\partial {\varLambda^I}}}{{\partial I_t^{I,I }}}I_t^{I,I }=0,\;I_t^{I,I}\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial I_t^{A,I }}}=-\lambda_t^I\frac{{P_t^{y,A }E{}_t}}{{{L^I}}}+\lambda_{t+1}^I\frac{{\left( {Q_{t+1}^I+\left( {1-\delta } \right)P_{t+1}^{y,I }} \right)}}{{{L^I}}}\le 0,\ \frac{{\partial {\varLambda^I}}}{{\partial I_t^{A,I }}}I_t^{A,I }=0,\;I_t^{A,I}\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial c_{t+1}^{II,2 }}}=\frac{\beta }{{c_{t+1}^{II,2 }+c_{t+1}^{AI,2 }}}-\lambda_{t+1}^IP_{t+1}^{x,I}\le 0,\ \frac{{\partial {\varLambda^I}}}{{\partial c_{t+1}^{II,2 }}}c_{t+1}^{II,2 }=0,\;c_{t+1}^{II,2}\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial c_{t+1}^{AI,2 }}}=\frac{\beta }{{c_{t+1}^{II,2 }+c_{t+1}^{AI,2 }}}-\lambda_{t+1}^IP_{t+1}^{x,A }{E_{t+1 }}\le 0,\ \frac{{\partial {\varLambda^I}}}{{\partial c_{t+1}^{AI,2 }}}c_{t+1}^{AI,2 }=0,\;c_{t+1}^{AI,2}\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial \lambda_t^I}}=\left( {\mathrm{ active}\text{--}\mathrm{ period}\ \mathrm{ budget}\ \mathrm{ constraint}} \right)\ge 0,\ \frac{{\partial {\varLambda^I}}}{{\partial \lambda_t^I}}\lambda_t^I=0,\;\lambda_t^I\ge 0, $$

$$ \frac{{\partial {\varLambda^I}}}{{\partial \lambda_{t+1}^I}}=\left( {\mathrm{ retirement}\text{--}\mathrm{ period}\ \mathrm{ budget}\ \mathrm{ constraint}} \right)\ge 0,\ \frac{{\partial {\varLambda^I}}}{{\partial \lambda_{t+1}^I}}\lambda_{t+1}^I=0,\;\lambda_{t+1}^I\ge 0. $$

Since the log-linear utility function is concave, the saddle-point conditions are sufficient for the solution of the optimization problem in 9.5.

To illustrate the mechanics of the Kuhn-Tucker conditions, we assume for demonstration purposes that the no-arbitrage condition for the x-good is not fulfilled, i.e. $P_t^{x,I }<{E_t}P_t^{x,A }$. The young household wants to consume a positive amount of the x-good and due to the price constellation $c_t^{II,1 }>0$ is reasonable. According to the first line of the Kuhn-Tucker conditions $1/(c_t^{II,1 }+c_t^{AI,1 })=\lambda_t^IP_t^{x,I }$. This implies, together with the assumed international price constellation for x, $1/(c_t^{II,1 }+c_t^{AI,1 })-\lambda_t^IP_t^{x,A }{E_t}<0$ and $c_t^{{AI\rm{,}1}}=0$ in the second row of the Kuhn-Tucker conditions. This is in contradiction to the assumption that all level variables are strictly positive at the utility maximizing point. Thus, $P_t^{x,I }<{E_t}P_t^{x,A }$ cannot be compatible with a utility maximizing situation and the no-arbitrage condition holds.

At the optimum all variables of the Lagrangian are strictly positive. Therefore, the Kuhn-Tucker conditions collapse to the classical first-order conditions. All inequalities of the Kuhn-Tucker conditions become equalities. Using the equations of the first two lines ${1 \left/ {{(c_t^{II,1 }+c_t^{AI,1 })}} \right.}=\lambda_t^IP_t^{x,I }$ and ${1 \left/ {{(c_t^{II,1 }+c_t^{AI,1 })}} \right.}=\lambda_t^IP_t^{x,A }{E_t}$, we obtain immediately that:

$$ P_t^{x,I }={E_t}P_t^{x,A }. $$

(9.27)

The equations of the 3rd and 4th as well as of the 5th and 6th lines imply:

$$ P_t^{y,I }={E_t}P_t^{y,A }, $$

(9.28)

$$ P_{t+1}^{x,I }={E_{t+1 }}P_{t+1}^{x,A }. $$

(9.29)

The budget constraints are the consequence of the strict positivity of the Lagrangian multipliers. Considering Eqs. 9.27, 9.28 and 9.29 in the budget constraints, these can be equivalently written as:

$$ P_t^{x,I }c_t^{I,1 }+P_t^{y,I}\left( {\frac{{I_t^I}}{{{L^I}}}+\left( {1-\delta } \right)\frac{{K_t^I}}{{{L^I}}}} \right)=W_t^I, $$

(9.30)

$$ P_{t+1}^{x,I }c_{t+1}^{I,2 }=\left( {Q_{t+1}^I+\left( {1-\delta } \right)P_{t+1}^{y,I }} \right)\left( {\frac{{I_t^I}}{{{L^I}}}+\left( {1-\delta } \right)\frac{{K_t^I}}{{{L^I}}}} \right), $$

(9.31)

where:

$$ c_t^{I,1}\equiv c_t^{II,1 }+c_t^{AI,1 },\ c_{t+1}^{I,2}\equiv c_{t+1}^{II,2 }+c_{t+1}^{AI,2 },\ \frac{{I_t^I}}{{{L^I}}}\equiv \frac{{I_t^{I,I }}}{{{L^I}}}+\frac{{I_t^{A,I }}}{{{L^I}}}. $$

Dividing both sides of Eq. 9.30 by $P_t^{y,I }$ and both sides of Eq. 9.31 by $P_{t+1}^{y,I }$ and labeling price ratios by lower case letters, guides us to the transformed choice problem of young households as mentioned in the main text.