Derive the intertemporal equilibrium allocation of consumption assets trade balance and current account and give an economic interpretation to your results Explain how your results change if instead ?

Assuming a small open economy with perfect capital mobility, the intertemporal equilibrium can be derived as follows:

First, we need to define the household's intertemporal budget constraint, which states that the present value of consumption must be equal to the present value of income:

C + PV(B) = PV(Y)

Where C is current consumption, PV(B) is the present value of the household's assets (which can be positive or negative), and PV(Y) is the present value of the household's income (which includes both domestic and foreign sources).

Second, we need to define the economy's current account balance, which is the difference between exports and imports:

CA = X - M

Where X is exports and M is imports.

Third, we need to define the economy's asset market equilibrium, which states that the supply of and demand for assets must be equal:

PV(B) = S - I + CA

Where S is national saving, I is investment, and CA is the current account balance.

Finally, we can combine these three equations to derive the intertemporal equilibrium allocation of consumption, assets, trade balance, and current account:

C = PV(Y) - PV(B) PV(B) = S - I + CA CA = X - M

Intuitively, this equilibrium allocation implies that households consume the present value of their income minus the present value of their assets, while the economy's asset market clears by equating national saving, investment, and the current account balance. The current account balance, in turn, is determined by the difference between exports and imports.

If 𝜆<1, this implies that households have a higher discount rate and therefore value present consumption more than future consumption. As a result, they will save less and consume more in the present, which will increase imports and worsen the current account balance. This will lead to a higher interest rate, which will attract more foreign capital and increase the supply of assets, thereby reducing their price and increasing the current account balance. The final equilibrium allocation will therefore involve higher consumption, lower saving, higher imports, lower exports, and a larger current account deficit