## Introduction

The balance of payments is a systematic record of all economic transactions between residents of one country and the rest of the world over a specified period, typically a year. It can be affected by factors such as changes in exchange rates, trade policies, and global economic conditions. The balance of payments is an essential indicator of a country's economic health and can influence its currency's value and ability to borrow from international markets.

## Components of BoP

BoP is divided into three main categories: the current account, the capital account, and the financial account.

The current account includes transactions related to the trade of goods and services, income from investments, and unilateral transfers such as foreign aid. The balance of trade, which represents the difference between a country's exports and imports, is a major component of the current account. A surplus on the current account means that a country is a net lender to the rest of the world, while a deficit means it is a net borrower.

The capital account includes transactions related to the purchase and sale of non-produced, non-financial assets such as patents, copyrights, and trademarks. It also includes capital transfers, such as debt forgiveness and inheritances.

The financial account includes transactions related to the purchase and sale of financial assets, such as stocks, bonds, and foreign currency. It also includes foreign direct investment, which is the ownership of a company or assets in a foreign country, and portfolio investment, which is the purchase of stocks and bonds in foreign companies. A surplus on the financial account means that a country is investing more abroad than it is receiving in foreign investment, while a deficit means the opposite.

## Mathematical derivation

The mathematical derivation rests on few assumptions, such as current accounts incorporate imports and exports of goods only, capital accounts incorporate capital inflows and outflows only, and the capital account is fully convertible.

**Capital outflows**

$H = \alpha - \beta r ---(i)$

**Current accounts balance**

$NX=X-M$

$NX=\bar{X} - \gamma - \lambda Y$ [$\theta = \bar{X}-\gamma$]

$NX =\theta - \lambda Y ---(ii)$

**External sector equilibrium**

$H=NX---(iii)$

Here,

H, NX, Y, and r represent capital outflows, net exports, national income, and rate of interest respectively. $\beta$ represents the interest elasticity of capital outflows and $\lambda$ is income elasticity of imports. Exports depend upon foreign countries' economies, so it is exogenously determined.

Now,

From Equation (iii),

$H=NX$

$\alpha - \beta r = \theta - \lambda Y$

$r = \frac{\alpha - \theta}{\beta} + \frac{\lambda}{\beta}Y---(1)$

Equation (1) represents the upward-sloping Balance of Payments curve.

The slope of BoP curve: $\frac{\lambda}{\beta}$

Intercept of BoP curve: $\frac{\alpha - \theta}{\beta}$

**The slope of BoP curve**

The slope of BoP curve depends upon the slope of the capital outflow function and net exports function. If the slope of the net exports function, that is marginal propensity to imports, increases, then the slope of the BoP curve also increases thereby steepening the BoP curve. Likewise, the increase in interest elasticity of capital will decrease both slope and intercept. Consequently, the new BoP curve will be flatter and lie below the original BoP curve.

**The shift in BoP curve**

The shift in BoP curve depends upon $\alpha, \beta$, and $\theta$. An increase in either $\theta$ or $\beta$ or both will decrease the intercept thereby shifting the BoP curve downward. While an increase in $\alpha$ will increase the intercept thereby shifting the BoP curve upward. This intercept of BoP curve depends upon the slope and intercept of capital function, and the slope of the net exports function.

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