Microeconomic Analysis of Export Credit Program

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1. Introduction

It is important for creating a multi-pillar economy to increase an export of final products absorbed with science, technology, innovation and knowledge.

There are both financial and non-financial instruments to diversify and support exports and one of the non-financial methods is to develop an export credit insurance.

The study found that 80 - 90 percent of the world’s trade was based on loans and insurance guarantees [1] . From here, export credit insurance is an important instrument for managing risks, and the company has an opportunity to expand its foreign trade.

The “export credit insurance” is the guarantee of repayment of trade *recei**vables** *in the form of insurance when exporter sells its goods on short-term loans.

The primary purpose of export credit insurance is to make commerce more accessible, mutually beneficial and less risky for businesses.

In our theoretical model, we consider a multi-country trade model where one of countries imports its good from other countries. We assume that there is one tradable good, namely good x. The import demand and export supply functions are analyzed based on utility and profit maximization problems. We assume that there is no production in the importing country and no consumption in the exporting countries with regarding to the tradable good.

2. Import Demand and Export Supply Functions

Assume that the utility function of the importing country is represented by Cobb-Douglas function:

$U\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)=A{x}_{1}^{{\alpha}_{1}}{x}_{2}^{{\alpha}_{2}}\cdots {x}_{n}^{{\alpha}_{n}}$

where, ${\alpha}_{i}\ge 0,{\displaystyle {\sum}_{i=1}^{n}{\alpha}_{i}}\le 1,i=1,2,\cdots ,n$ .

${x}_{i}$ ―quantity of export good x from country i, $i=1,2,\cdots ,n$ .

Consider the utility maximization problem subject to cost and demand constraints.

$\mathrm{max}U\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)=A{x}_{1}^{{\alpha}_{1}}{x}_{2}^{{\alpha}_{2}}\cdots {x}_{n}^{{\alpha}_{n}}$ (1)

${\sum}_{i=1}^{n}{p}_{i}{x}_{i}}\le C$ (2)

${\sum}_{i=1}^{n}{x}_{i}}\le A$ (3)

${p}_{i}$ ―is a price per unit good for country i, where C is budget, A is a demand of the importing country.

Problem (1)-(3) is concave maximization problem for which Kuhn-Tucker optimality conditions [2] can be written in the following way.

$\{\begin{array}{l}\frac{\partial L}{\partial {x}_{i}}=-\frac{\partial U}{\partial {x}_{i}}+{\lambda}_{1}{p}_{i}+{\lambda}_{2}=0,i=1,2,\cdots ,n.\\ {\lambda}_{1}\left({\displaystyle {\sum}_{i=1}^{n}{p}_{i}{x}_{i}}-C\right)=0,\\ {\lambda}_{2}\left({\displaystyle {\sum}_{i=1}^{n}{x}_{i}}-A\right)=0,\\ {\lambda}_{1}\ge 0,{\lambda}_{2}\ge 0,{\lambda}_{1}+{\lambda}_{2}>0,\end{array}$ (4)

where L is the Lagrange function, $L=-U\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)+{\lambda}_{1}\left({{\displaystyle \sum}}^{\text{}}{p}_{i}{x}_{i}-C\right)+{\lambda}_{2}\left({{\displaystyle \sum}}^{\text{}}{x}_{i}-A\right)$ .

It is hard to solve the system (4) of nonlinear equations and inequalities. That is why in order to solve problem (1)-(3), we need to apply an optimization method, for instance, the conditional gradient method. Before we write the algorithm of this method, we introduce the following notation. Denote by D a set of feasible solutions of problem (1)-(3).

$D=\left\{x\in {R}^{n}|{\displaystyle {\sum}_{i=1}^{n}{p}_{i}{x}_{i}}\le C,{\displaystyle {\sum}_{i=1}^{n}{x}_{i}}\le A,{x}_{i}\ge 0,i=\stackrel{\xaf}{1,n}\right\}.$

It is clear that the set D is convex. Also, the gradient of the function U is as follows:

$U\left(x\right)=\frac{\partial U}{\partial x}=\left(\frac{\partial U}{\partial {x}_{1}},\frac{\partial U}{\partial {x}_{2}},\cdots ,\frac{\partial U}{\partial {x}_{n}}\right),$

where

$\frac{\partial U}{\partial {x}_{i}}=A{x}_{1}^{{\alpha}_{1}}{x}_{2}^{{\alpha}_{2}}\cdots \left({\alpha}_{i}{x}_{i}^{{\alpha}_{i}-1}\right)\cdots {x}_{n}^{{\alpha}_{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=\text{1},\text{2},\cdots $ (5)

Now we are ready to formulate the algorithm of the conditional gradient method [2] .

Algorithm CGM

Step 1. Choose a feasible point ${x}^{k}$ in D, $k:=0$ .

Step 2. Compute the gradient of the function U at ${x}^{k}$ by Formula (5).

Step 3. Solve the following linear programming problem

$\underset{x\in D}{\mathrm{max}}\langle {U}^{\prime}\left({x}^{k}\right),x\rangle $ .

Let ${\stackrel{\xaf}{x}}^{k}$ be a solution to this problem, i.e.

$\langle {U}^{\prime}\left({x}^{k}\right),{\stackrel{\xaf}{x}}^{k}\rangle ={\mathrm{max}}_{x\in D}\langle {U}^{\prime}\left({x}^{k}\right),x\rangle $ .

Step 4. Denote by ${\eta}_{k}$ the value

${\eta}_{k}=\langle {U}^{\prime}\left({x}^{k}\right),{\stackrel{\xaf}{x}}^{k}-{x}^{k}\rangle $ .

Step 5. If ${\eta}_{k}=0$ . Then ${x}^{k}$ is a solution to problem (1)-(3) and stop. Otherwise, go to the next step.

Step 6. Construct the direction ${h}^{k}$ :

${h}^{k}={\stackrel{\xaf}{x}}^{k}-{x}^{k}$ .

Step 7. Choose ${\alpha}_{k}$ from the condition

$U\left({x}^{k}+{\alpha}_{k}{h}^{k}\right)>U\left({x}^{k}\right)$ ,

$0<{\alpha}^{k}<1$ .

Step 8. Construct a next approximation point ${x}^{k+1}$ as

${x}^{k+1}={x}^{k}+{\alpha}_{k}{h}^{k}$ and set $k:=k+1$ , go to step 2.

A convergence of the algorithm is given by the following statement.

Theorem [2] . Assume that the gradient ${U}^{\prime}\left(x\right)$ satisfies the Lipschitz condition on D with a constant L, i.e.

$\Vert {U}^{\prime}\left(x\right)-{U}^{\prime}\left(y\right)\Vert \le L\Vert x-y\Vert ,L>0,\forall x,y\in D$

Let ${x}^{0}$ be an arbitrary feasible point of D. Then a sequence $\left\{{x}^{k}\right\}$ constructed by Algorithm CGM converges to a solution of problem (1)-(3), that is

$\underset{k\to \infty}{\mathrm{lim}}U\left({x}^{k}\right)=\underset{x\in D}{\mathrm{max}}U\left(x\right).$

3. Export Supply Function

In order to obtain the supply function of the exporting country, we need to solve the profit maximization problem.

Assume that the firm of exporting country produces a single output ${x}_{i}$ using several inputs. Suppose that their short-run variable costs are defined as a cubic function of the form.

${c}_{i}\left({x}_{i}\right)={a}_{i}{x}_{i}^{3}+{b}_{i}{x}_{i}^{2}+{c}_{i}{x}_{i},i=1,2,\cdots ,n$

Let $F{C}_{i}$ be a short run fixed cost of the i-th firm.

Then profit maximization problem of the i-th firm is formulated as:

${\mathrm{max}}_{{x}_{i}}\left\{{\pi}_{i}\left({x}_{i}\right)={p}_{i}{x}_{i}-{c}_{i}\left({x}_{i}\right)-F{C}_{i}\right\}$ (6)

We write down optimality conditions for problem (6).

${{\pi}^{\prime}}_{i}\left({x}_{i}\right)={p}_{i}-3{a}_{i}{x}_{i}^{2}-2{b}_{i}{x}_{i}-{c}_{i}=0$

Solving this equation, we get the short run supply function of the i-th firm:

${x}_{i}\left({p}_{i}\right)=\frac{-{b}_{i}+\sqrt{{b}_{i}^{2}-3{a}_{i}\left({c}_{i}-{p}_{i}\right)}}{3{a}_{i}}$ (7)

Taking into account demand and supply of the importing and exporting countries, we conclude that

$D\le {\displaystyle {\sum}_{i=1}^{n}\left(\frac{-{b}_{i}+\sqrt{{b}_{i}^{2}-3{a}_{i}\left({c}_{i}-{p}_{i}\right)}}{3{a}_{i}}\right)}$ .

4. Impact of an Export Credit Program (ECP)

There are many works devoted to subsidy values ECPs. The present approach was developed in Baricell and Vercarmen (1994), Baron (1983), Hybery et al. [3] .

A cost-benefit analysis was conducted by Fleid and Hill (1984). The option-pricing method was used by Dahl, Wilson, and Gustafson (1999), Dierson and Scherrick (1999), and Schich (1997) [3] .

These research papers consider subsidy values provided by government-supported export credit programs. In these works, subsidy is considered as a cost savings to an importer which is a price discount on imports. Therefore, an importing country has benefits from an export credit program due to a cost savings. As a result, his decision how much to import is influenced by the cost savings. On the other hand, the cost saving may be considered as an additional income to the importing country.

It is assumed that the indirect benefits of an ECP, received by the consumer (importing country), can be considered as a fixed discount “d” on his important payment [3] . The parameter “d” is a measure to capture the indirect benefits arising from of the potential policy parameters of export credit programs. It is shown that “d” is the difference of two present value streams [3] [4] [5] [6] . The first present value stream (PV_{1}) is calculated under the scenario that there is no subsidy offered to the consumer. The second present value stream (PV_{2}) deals with a subsidy offered to the consumer through an export credit program. Thus, the fixed discount rate “d” is defined for each exporting country as:

${d}_{i}=\frac{P{V}_{1}^{i}-P{V}_{2}^{i}}{P{V}_{1}}100,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n$ (8)

It is clear that it the consumer receives benefits from an EPC, his budget constraint is affected by the cost saving on the import payment. Thus, budget constraint of the consumer (importing country) has the following form:

$\underset{i=1}{\overset{n}{{\displaystyle \sum}}}\left(1-{d}_{i}\right){p}_{i}{x}_{i}\le C$

The discount rate ${d}_{i}$ is assumed to be

$0\le {d}_{i}<1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n$

If ${d}_{i}=0$ , it means that there is no discount on the import payment.

If ${d}_{i}=1$ , then there is a full discount which means that variable ${x}_{i}$ does not play any role in the utility maximization problem. That is why it is important to consider the case for $0<{d}_{i}<1$ .

We formulate again the utility maximization problem for the importing country (consumers) under ECP.

$\mathrm{max}{U}_{i}\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$ (9)

${\sum}_{i=1}^{n}\left(1-{d}_{i}\right){p}_{i}{x}_{i}}\le C$ (10)

${\sum}_{i=1}^{n}{x}_{i}}\le A$ (11)

The Marshallian demand functions for the consumer can be found by solving problem (9)-(11) by algorithm CGM.

5. Export Supply and Non-Payment Risk under ECP

We claim that the country’s export supply is influenced by the country’s export credit programs. We assume that the risk of not getting paid depends on the price ( ${p}_{i}$ ) of the exporting good. The price ( ${p}_{i}$ ) of getting paid is assumed to be a random variable. If ${p}_{i}$ is a negotiated price then ${p}_{i}$ is not a random variable. Introduce the following variables and notations

${\theta}_{i}$ : credit score that explains non-payment of the export sales, $i=1,2,\cdots ,n$ .

${F}_{i}\left({\theta}_{i}\right)$ : cumulative probability distribution function of getting paid, $i=1,2,\cdots ,n$ .

${z}_{i}$ : coverage level from an export insurance or guarantee policy such that $0\le {z}_{i}\le 1$ .

${\sigma}_{i}$ : loading factor that reflects the administrate cost of providing the public guarantee or insurance scheme, $\stackrel{\xaf}{z}=1,2,\cdots ,n$ .

${E}_{i}$ : expected value, $i=1,2,\cdots ,n$ .

${W}_{i}$ : guarantee fee (or insurance premium rate) per unit of the exporting good ${x}_{i}$ .

${W}_{i}={W}_{i}\left({z}_{i},{\stackrel{\xaf}{F}}_{i},{p}_{i}\right)$

Compute the following statistics parameters.

${E}_{i}\left({p}_{i}{F}_{i}\left({\theta}_{i}\right)\right)={p}_{i}{E}_{i}\left[{F}_{i}\left({\theta}_{i}\right)\right]={p}_{i}{\stackrel{\xaf}{F}}_{i},$

$Var\left[{p}_{i}{F}_{i}\left({\theta}_{i}\right)\right]=E\left[{\left\{{p}_{i}{F}_{i}\left({\theta}_{i}\right)-{E}_{i}\left({p}_{i}{F}_{i}\left({\theta}_{i}\right)\right)\right\}}^{2}\right]={p}_{i}^{2}{\sigma}_{{\theta}_{i}}^{2}$

Stochastic profit functions of the exporting countries are:

$\begin{array}{c}{\pi}_{i}={p}_{i}{F}_{i}\left({\theta}_{i}\right){x}_{i}+{z}_{i}{x}_{i}{p}_{i}\left(1-{F}_{i}\left({\theta}_{i}\right)\right)-{\sigma}_{i}W\left({z}_{i},{\stackrel{\xaf}{F}}_{i},{p}_{i}\right){x}_{i}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-a{x}_{i}^{3}-{b}_{i}{x}_{i}^{2}-{c}_{i}{x}_{i}-F{C}_{i},\end{array}$

The expected value and variance of the stochastic profit is:

${E}_{i}\left({\pi}_{i}\right)={p}_{i}{\stackrel{\xaf}{F}}_{i}{x}_{i}+{z}_{i}{p}_{i}\left(1-{\stackrel{\xaf}{F}}_{i}\right){x}_{i}-{\sigma}_{i}{W}_{i}-{a}_{i}{x}_{i}^{3}-{b}_{i}{x}_{i}^{2}-{c}_{i}{x}_{i}-F{C}_{i},$

$Va{r}_{i}\left({\pi}_{i}\right)={\sigma}_{{\pi}_{i}}^{2}={\left(1-{z}_{i}\right)}^{2}{\sigma}_{{\theta}_{i}}^{2}{x}_{i}^{2}{p}_{i}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n$

Assume that the firm is constant absolute risk aversion so its profit function [1] is

${\mathrm{max}}_{{x}_{i}}\left\{{\pi}_{i}=E\left({\pi}_{i}\right)-\frac{{\lambda}_{i}}{2}{\sigma}_{{\pi}_{i}}^{2}\right\}$ , (12)

where ${\lambda}_{i}$ is a constant factor that measures the risk attitude of the representative firm. Write down problem (12) in the form:

$\begin{array}{l}{\mathrm{max}}_{{x}_{i}}\{{\pi}_{i}={p}_{i}{\stackrel{\xaf}{F}}_{i}{x}_{i}+{z}_{i}{p}_{i}\left(1-{\stackrel{\xaf}{F}}_{i}\right){x}_{i}-{\sigma}_{i}{W}_{i}{x}_{i}-{a}_{i}{x}_{i}^{3}-{b}_{i}{x}_{i}^{2}\begin{array}{c}\\ \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{c}_{i}{x}_{i}-F{C}_{i}-\frac{{\lambda}_{i}}{2}{\left(1-{z}_{i}\right)}^{2}{\sigma}_{{\theta}_{i}}^{2}{x}_{i}^{2}{p}_{i}^{2}\},\end{array}$ (13)

The optimality condition for problem (13) is

$\frac{\text{d}{\pi}_{i}}{\text{d}{x}_{i}}=0$ (14)

or equivalently,

${p}_{i}{\stackrel{\xaf}{F}}_{i}{x}_{i}+{z}_{i}{p}_{i}\left(1-{\stackrel{\xaf}{F}}_{i}\right)-{\sigma}_{i}{W}_{i}-3{a}_{i}{x}_{i}^{2}-2{b}_{i}{x}_{i}-{c}_{i}-{\lambda}_{i}{\left(1-{z}_{i}\right)}^{2}{\sigma}_{{\theta}_{i}}^{2}{x}_{i}^{2}{p}_{i}^{2}=0,$

$3{a}_{i}{\lambda}_{i}^{2}+\left(2{b}_{i}+{\lambda}_{i}{\left(1-{z}_{i}\right)}^{2}{\sigma}_{{\theta}_{i}}^{2}{p}_{i}^{2}\right){x}_{i}+\left[{c}_{i}-{p}_{i}{\stackrel{\xaf}{F}}_{i}-{z}_{i}{p}_{i}\left(1-{\stackrel{\xaf}{F}}_{i}\right)+{\sigma}_{i}{W}_{i}\right]=0,$

Solving this quadratic equation, we find

${x}_{i}=\frac{-\left(2{b}_{i}+{\lambda}_{i}{\left(1-{z}_{i}\right)}^{2}{\sigma}_{{\theta}_{i}}^{2}{x}_{i}^{2}{p}_{i}^{2}\right)+\sqrt{{\left(2{b}_{i}+{\lambda}_{i}{(1-{z}_{i})}^{2}{\sigma}_{{\theta}_{i}}^{2}{p}_{i}^{2}\right)}^{2}-12{a}_{i}\left({c}_{i}-{p}_{i}{\stackrel{\xaf}{F}}_{i}-{z}_{i}{p}_{i}\left(1-{\stackrel{\xaf}{F}}_{i}\right)+{\sigma}_{i}{W}_{i}\right)}}{6{a}_{i}}$ (15)

${x}_{i}$ is an optimal export quantity of exporting country under risk of non-payment, $i=1,2,\cdots ,n$ .

We consider the following cases for Formula (15).

Case 1. ${z}_{i}=0$ and ${\sigma}_{i}=0$ .

It means that self-insurer can be obtained by setting both coverage level and loading factor cost equal to zero. In this case Formula (15) has a form

${x}_{1}=\frac{-\left(2{b}_{i}+{\lambda}_{i}{\sigma}_{{\theta}_{i}}^{2}{p}_{i}^{2}\right)+\sqrt{{\left({\sigma}_{{\theta}_{i}}^{2}+{\lambda}_{i}{\sigma}_{{\theta}_{i}}^{2}{p}_{i}^{2}\right)}^{2}+12{a}_{i}\left({p}_{i}{\stackrel{\xaf}{F}}_{i}-{c}_{i}\right)}}{6{a}_{i}}$

Case 2. ${\stackrel{\xaf}{F}}_{i}=1$ and ${\sigma}_{{\theta}_{i}}^{2}=0$ . It means that there is the full guarantee (insurance) coverage of assuring a certainly payment. Then Formula (15) has the form:

${x}_{i}=\frac{-{\sigma}_{{\theta}_{i}}^{2}+\sqrt{4{b}_{i}^{2}+12{a}_{i}\left({p}_{i}-{c}_{i}-{\sigma}_{i}{W}_{i}\right)}}{6{a}_{i}}$

In general, ${\stackrel{\xaf}{F}}_{i}\ne 1$ and ${\sigma}_{{\theta}_{i}}^{2}\ne 0$ since a default of the export payment is arisen from the importing country and the government agency of the exporting country cannot control the default.

Case 3: If there is no risk of export payment then we can set ${z}_{i}=1$ . The optimal export quantity will be:

${x}_{i}=\frac{-2{b}_{i}+\sqrt{4{b}_{i}^{2}+12{a}_{i}\left({p}_{i}-{c}_{i}-{\sigma}_{i}{W}_{i}\right)}}{6{a}_{i}}$

Case 4: If there is a full guarantee (insurance) and no premium or loading cost then we have

${\sigma}_{i}=0$ .

Then formula (15) has the form:

${x}_{i}=\frac{-\left(2{b}_{i}+{\lambda}_{i}{\left(1-{z}_{i}\right)}^{2}{\sigma}_{\theta}^{2}{p}_{1}^{2}\right)+\sqrt{2{b}_{i}+{\lambda}_{i}{\left(1-{z}_{i}\right)}^{2}{\sigma}_{{\theta}_{i}}^{2}{p}_{i}^{2}-12{a}_{i}\left({c}_{i}-{p}_{i}{\stackrel{\xaf}{F}}_{i}-{z}_{i}{p}_{i}\left(1-{\stackrel{\xaf}{F}}_{i}\right)\right)}}{6{a}_{i}}$

6. Computational Results

For computational simulation we considered six countries trade model with one importer for different discount rates for each country.

The utility function of the importing country was

$U\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}\right)=A{x}_{1}^{0.012}{x}_{2}^{0.024}{x}_{3}^{0.028}{x}_{4}^{0.017}{x}_{5}^{0.015}$

with C = 300000, A = 70000, ${p}_{1}=5$ , ${p}_{2}=20$ , ${p}_{3}=35$ , ${p}_{4}=16$ , ${p}_{5}=30$ and ${d}_{1}=0.12$ , ${d}_{2}=0.15$ , ${d}_{3}=0.18$ , ${d}_{4}=0,16$ , ${d}_{4}=0,16$ .

7. Conclusion

We considered economic impact of export credit insurance for multi countries with one importer extending the papers [3] by Paul Pienstra-Munnicha, Calum Turkey and Wan W. Koo and [7] by Enkhbat R. and Tungalag N. Importer’s [8] [9] utility function was Cobb-Douglas type and exporter’s cost was a cubic function. The multi-counties partial trade models were developed to analyze the economic impact of the export credit insurance on trade flows. The export credit program helps to increase the exported quantity under certain assumptions. We have also shown that under stochastic factors the optimal export quantities can be computed analytically. In general, the paper aims to analyze economic impacts of ECP for multi-countries from a view of point microeconomic analysis. Also a computational experiment was implemented for five countries trade model.

Acknowledgements

This paper was supported by the project titled “Study on development of export credit program in Mongolia” of School of Business, National University of Mongolia.

References

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