1) Closed Economy

Assume we have a closed economy. Being closed, it cannot borrow, and therefore investment = domestic saving.

In period 1 it has capital \(K_1\) and a Cobb-Douglas production function, so that output in period 1 is fixed and given by: \[Y_1 = AK_1^{\alpha}...(1)\]

where A is a constant due to the fixed labour supply and technology.

In addition, the economy has a social utility function of the form:

\[U(C_1, C_2) = \frac{C_1^{1-\theta}-1}{1-\theta} + \beta \frac{C_2^{1-\theta}-1}{1-\theta}...(2)\] where \(\beta = \frac{1}{1+\rho}\) is the social impatience factor.

In period 1 the economy maximises utility by choosing consumption \(C_1\) and hence investment \(I_1\): \[I_1 = Y_1 - C_1...(3)\] Therefore, in period 2 the capital stock is: \[K_2 = (1-\delta)K_1 + I_1\] Therefore, output in period 2 is:

\[Y_2 = A\left[ (1-\delta)K_1 + I_1 \right]^{\alpha}...(4)\] where \(\delta\) is the depreciation rate.

In addition, we assume that there is no saving in period 2, so all output is consumed: \[C_2 = Y_2\]

Therefore:

\[C_2 = A\left[ (1-\delta)K_1 + Y_1 - C_1 \right]^{\alpha}...(5)\] This equation is the inter-temporal budget constraint or PPF. See the diagram. We assume that capital cannot be consumed; hence maximum consumption in period 1 is \(Y_1\) and therefore the PPF does not reach the x-axis. In general, the economy maximises utility by consuming \(C_1\) and thus investing \(I_1 = Y_1 - C_1\), consuming \(C_2\) in period 2. The PPF is tangent to an indifference curve \(U_C\) at the point \((C_1, C_2)\).

To find \(I_1\) that maximises U, the optimisation problem is: \[\max_{C_1} \frac{C_1^{1-\theta}-1}{1-\theta} + \beta \frac{C_2^{1-\theta}-1}{1-\theta}\] \[s.t. C_2 = A\left[ (1-\delta)K_1 + Y_1 - C_1 \right]^{\alpha}\] We find \(C_1\) that maximises U and then obtain \(I_1 = Y_1 - C_1\).

U is maximised at the point where the PPF is tangent to the indifference curve, i.e.: \[\frac{dC_2}{dC_1} = - \frac{\partial U}{\partial C_1} / \frac{\partial U}{\partial C_2}\] Deriving equations (5) and (2): \[-A \alpha \left[ (1-\delta)K_1 + Y_1 - C_1 \right]^{\alpha-1} = - \frac{C_1^{-\theta}}{\beta C_2^{-\theta}}\] \[\therefore A \alpha C_2^{-\theta} \left[ (1-\delta)K_1 + Y_1 - C_1 \right]^{\alpha-1} = \frac{C_1^{-\theta}}{\beta }\]

Substitute \(C_2\) from equation (5): \[A^{1-\theta} \alpha \left[ (1-\delta)K_1 + Y_1 - C_1 \right]^{\alpha-1-\theta\alpha} = \frac{C_1^{-\theta}}{\beta }\] Rearranging: \[\left[ (1-\delta)K_1 + Y_1 - C_1 \right]^{\alpha-1-\theta\alpha} = \frac{C_1^{-\theta}}{A^{1-\theta} \alpha\beta}\] \[\therefore (1-\delta)K_1 + Y_1 - C_1 = \left[ \frac{C_1^{-\theta}}{A^{1-\theta} \alpha\beta} \right]^{\frac{1}{\alpha-1-\theta\alpha}}\] \[\therefore C_1 + \left[ \frac{C_1^{-\theta}}{A^{1-\theta} \alpha\beta} \right]^{\frac{1}{\alpha-1-\theta\alpha}} - (1-\delta)K_1 - Y_1 = 0...(6)\]

We now have an equation with one unknown, \(C_1\), but for which there is no simple analytical solution. However, a solution can be found by using a numerical method run on a computer, finding the value of \(C_1\), where \(0 \le C_1 \le Y_1 + K_1\), that makes equation (6) closest to 0.

For example, for the following values:
\(K_1\) = 100
\(A\) = 50
\(\alpha\) = 0.5
\(\therefore Y_1\) = 500
And:
\(\beta\) = 0.8
\(\theta\) = 0.8
\(\delta\) = 0.1

We get \(C_1\) = 409.2 and therefore \(I_1\) = 90.8.

2) Open Economy

Now the economy becomes open. It can thus borrow an amount B, up to a maximum of \(\frac{(1-\delta)^{\alpha}Y_1}{1+r}\), where r is the international interest rate, thus attaining a maximum total expenditure in period 1 of \(E_1 = Y_1 + B\). The economy maximises utility by allocating this total expenditure between consumption and investment, so that:

\[I_1 = Y_1 + B - C_1...(7)\] where \(0 \le B \le \frac{(1-\delta)^{\alpha}Y_1}{1+r}\).

Indeed, both investment and consumption in period 1 can be larger in the open economy due to borrowing.

Therefore, in period 2 the capital stock of the open economy is now: \[K_2 = (1-\delta)K_1 + Y_1 + B - C_1...(8)\] Therefore: \[Y_2 = A\left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha}...(9)\] In period 2 the loan must be repaid. The repayment amount is \((1+r)B\) which is thus subtracted from \(Y_2\) hence yielding \(C_2\):

\[C_2 = A\left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha} - (1+r)B...(10)\]

Equation (10) is the inter-temporal budget constraint or PPF of the open economy. The diagram shows the original closed economy PPF and the open economy PPF in green. The open economy borrows B thus attaining a total expenditure of \(Y_1 + B\) in the first period. Out of this total, it consumes \(C_1^*\) and invests the difference: \(I_1^* = Y_1 + B - C_1^*\). Due to this investment, the economy grows and in the second period it consumes \(C_2^*\). Thanks to borrowing, it can both consume and invest more in the first period and still consume more in the second. In the diagram, the point \((C_1^*, C_2^*)\) is on a higher indifference curve than the closed economy at \((C_1, C_2)\).

To find \(I_1\) that maximises U in the open economy, the optimisation problem is now: \[\max_{C_1} \frac{C_1^{1-\theta}-1}{1-\theta} + \beta \frac{C_2^{1-\theta}-1}{1-\theta}\] \[s.t. C_2 = A\left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha} - (1+r)B\] By applying the same method of equating the slopes as for the closed economy, we get: \[-A \alpha \left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha-1} = - \frac{C_1^{-\theta}}{\beta C_2^{-\theta}}\] \[\therefore A \alpha C_2^{-\theta} \left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha-1} = \frac{C_1^{-\theta}}{\beta }\]

Substitute \(C_2\) from equation (10):

\[A \alpha \left[ A\left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha} - (1+r)B \right]^{-\theta} \left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha-1} = \frac{C_1^{-\theta}}{\beta }\]

Rearranging: \[A \alpha \left[ A\left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha} - (1+r)B \right]^{-\theta} \left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha-1} - \frac{C_1^{-\theta}}{\beta } = 0...(11)\] Once again, we find a solution for this equation with a computational method, finding the value of \(C_1\) that minimises equation (11).

For example, for interest rate, r=20%, we get \(C_1^*\) = 566 and \(I_1^*\) = 329.28, both larger than in the closed economy.

3) Open Economy with Lending

The upper portions of the open economy PPFs may not be realistic. Due to the diminishing marginal returns on investment, if the interest rate is high enough, there is a point where the economy can achieve higher consumption in period 2 by lending instead of further investment. This occurs where the lending budget constraint (with slope -(1+r)) is tangent to the PPF. It only makes sense to invest while the magnitude of the slope of the PPF is greater than the magnitude of the slope of the lending budget constraint, 1+r. When the the slope (magnitude) of the PPF is less than 1+r, more future consumption can be obtained by lending than by investing.

In Fig.3, the lending budget constraint is the red line. To the left of its point of tangency to the original PPF, it enlarges the PPF along itself. To the right, it is not defined, because it would imply non-feasible bundles of \(C_1\) and \(C_2\). (The lending budget constraint extrapolated to the right intersects the \(C_1\) axis at a point greater than the maximum amount that the international banks are willing to lend this economy, \(\frac{(1-\delta)^{\alpha}Y_1}{1+r}\)). If it is tangent to the PPF to the left of the point of tangency of the indifference curve, as in Fig.3, the optimum consumption and investment remain the same. However, if it touches to the right of the point of tangency of the indifference curve, the economy can attain a higher indifference curve that is tangent to the lending budget constraint, as in Fig.4.

To fully express the optimisation problem, we need to find the point where the lending budget constraint touches the PPF, \((c_{1L}, c_{2L})\). We do this by equating the slopes: \[-A \alpha \left[ (1-\delta)K_1 + Y_1 + B - c_{1L} \right]^{\alpha-1} = - (1+r)\] where \(c_{1L}\) is the value of \(C_1\) where the lending budget constraint touches the PPF.

Rearranging: \[c_{1L} = (1-\delta)K_1 + Y_1 + B - \left( \frac{A \alpha}{1+r} \right)^{\frac{1}{1-\alpha}}...(12)\] Substituting \(c_{1L}\) into equation (10): \[c_{2L} = A\left[ (1-\delta)K_1 + Y_1 + B - c_{1L} \right]^{\alpha} - (1+r)B...(13)\] Hence the equation of the lending budget constraint is: \[C_2 = c_{2L} + (1+r)c_{1L} - (1+r)C_1...(14)\] The optimisation problem with this combined restriction is: \[\max_{C_1} \frac{C_1^{1-\theta}-1}{1-\theta} + \beta \frac{C_2^{1-\theta}-1}{1-\theta}\] \[s.t. C_2 = \left\{ \begin{matrix} c_{2L} + (1+r)c_{1L} - (1+r)C_1 & when \ C_1 \le c_{1L} \\ A\left[ (1-\delta)K_1 + Y_1 + B - C_1 \right]^{\alpha} - (1+r)B & when \ C_1 > c_{1L} \end{matrix} \right\}\] When \(C_1 > c>{1L}\), the solution is the same as above, given by equation (11). When \(C_1 \le c_{1L}\), we equate the slopes:

\[\ - \frac{\partial U}{\partial C_1} / \frac{\partial U}{\partial C_2} = -(1+r)\] \[\therefore \frac{C_1^{-\theta}}{\beta C_2^{-\theta}} = 1+r\] Rearranging: \[C_2 = \frac{1}{\beta^{\frac{1}{\theta}}(1+r)^{\frac{1}{\theta}}}C_1\] Substitute for \(C_2\) from equation (14): \[c_{2L} + (1+r)c_{1L} - (1+r)C_1 = \frac{1}{\beta^{\frac{1}{\theta}}(1+r)^{\frac{1}{\theta}}}C_1\] Rearranging: \[C_1 = \frac{\beta^{\frac{1}{\theta}}(1+r)^{\frac{1}{\theta}}[c_{2L} + (1+r)c_{1L}]}{1 + \beta^{\frac{1}{\theta}}(1+r)^{1+\frac{1}{\theta}}}...(15)\]

In all cases, once consumption \(C_1\) is known, investment \(I_1\) can easily be derived from: \[I_1 = Y_1 - B - C_1\]

To summarise, an open economy can increase consumption both in the present and in the future by borrowing in the present and investing more in the present than is feasible for a closed economy.

Of course, the open economy will only be better off if it makes sound investments that produce a return that in the long run is greater than the amount borrowed and invested. If the investments amount to sunk costs with little return, then the economy will be worse off, as it will then have to repay the loan from a similar GDP.