Formula Sheet

This appendix summarizes the main formulas used in NREC4410: International Agricultural Trade. Use it as a quick reference. The formula is only useful if you can explain the economic meaning behind it.

Trade basics

Trade balance

\[ TB = X - M \]

where $X$ is exports and $M$ is imports.

Trade openness

\[ \text{Trade openness} = \frac{X + M}{GDP} \]

In percentage form:

\[ \text{Trade openness (\%)} = \frac{X + M}{GDP} \times 100 \]

Total trade value from trade openness

\[ X + M = \frac{\text{Trade openness (\%)}}{100} \times GDP \]

Ricardian model

Labor constraint

For two goods, food $F$ and cloth $C$:

\[ a_{LF}Q_F + a_{LC}Q_C \leq L \]

where $a_{LF}$ and $a_{LC}$ are unit labor requirements and $L$ is total labor.

Production possibility frontier

Solving for cloth:

\[ Q_C = \frac{L}{a_{LC}} - \frac{a_{LF}}{a_{LC}}Q_F \]

Maximum production of each good

\[ Q_F^{\max} = \frac{L}{a_{LF}} \]

\[ Q_C^{\max} = \frac{L}{a_{LC}} \]

Opportunity cost of food in terms of cloth

\[ OC_F = \frac{a_{LF}}{a_{LC}} \]

Opportunity cost of cloth in terms of food

\[ OC_C = \frac{a_{LC}}{a_{LF}} \]

A country has comparative advantage in the good with the lower opportunity cost.

Trade possibility frontier

If a country specializes in food and the world relative price of food is $P_F/P_C$:

\[ Q_C = \left(\frac{P_F}{P_C}\right)(Q_F^{\max} - Q_F) \]

If a country specializes in cloth:

\[ Q_F = \left(\frac{P_C}{P_F}\right)(Q_C^{\max} - Q_C) \]

Heckscher-Ohlin model

Labor constraint

\[ a_{LC}Q_C + a_{LF}Q_F \leq L \]

Capital constraint

\[ a_{KC}Q_C + a_{KF}Q_F \leq K \]

where $L$ is labor and $K$ is capital.

Factor intensity

Good $C$ is labor-intensive relative to good $F$ if:

\[ \frac{a_{LC}}{a_{KC}} > \frac{a_{LF}}{a_{KF}} \]

Good $F$ is capital-intensive relative to good $C$ if:

\[ \frac{a_{KF}}{a_{LF}} > \frac{a_{KC}}{a_{LC}} \]

Factor abundance

Home is labor abundant relative to Foreign if:

\[ \frac{L}{K} > \frac{L^*}{K^*} \]

Home is capital abundant relative to Foreign if:

\[ \frac{K}{L} > \frac{K^*}{L^*} \]

Zero-profit pricing equations

For cloth:

\[ P_C = a_{LC}w + a_{KC}r \]

For food:

\[ P_F = a_{LF}w + a_{KF}r \]

where $w$ is the wage and $r$ is the rental rate of capital.

Consumer surplus, producer surplus, and total surplus

For linear demand and supply:

\[ Q_d = a - bP \]

\[ Q_s = c + dP \]

Autarky equilibrium

\[ Q_d = Q_s \]

\[ a - bP = c + dP \]

\[ P^A = \frac{a-c}{b+d} \]

\[ Q^A = c + dP^A \]

Demand choke price

\[ P_{\max} = \frac{a}{b} \]

Supply price intercept

\[ P_{\min} = -\frac{c}{d} \]

Consumer surplus

\[ CS = \frac{1}{2}(P_{\max} - P)Q_d \]

Producer surplus

\[ PS = \frac{1}{2}(P - P_{\min})Q_s \]

If supply passes through the origin, $P_{\min}=0$, so:

\[ PS = \frac{1}{2}PQ_s \]

Total surplus

\[ TS = CS + PS \]

Free trade in partial equilibrium

Export supply

\[ ES(P) = Q_s(P) - Q_d(P) \]

Import demand

\[ ED(P) = Q_d(P) - Q_s(P) \]

World equilibrium

\[ ES(P^W) = ED(P^W) \]

At the world price $P^W$:

\[ \text{Exports} = Q_s(P^W) - Q_d(P^W) \]

\[ \text{Imports} = Q_d(P^W) - Q_s(P^W) \]

Import tariff

For a small importing country:

\[ P_d = P_w + t \]

where $P_d$ is the domestic price, $P_w$ is the world price, and $t$ is the tariff per unit.

Government tariff revenue

\[ TR = t \times M_t \]

where $M_t$ is imports after the tariff.

Deadweight loss

\[ DWL = \text{production distortion} + \text{consumption distortion} \]

For linear diagrams, each distortion is usually a triangle.

Import quota

A quota fixes the maximum import quantity:

\[ M \leq \bar{M} \]

Quota rent per unit

\[ \text{Quota rent per unit} = P_d - P_w \]

Total quota rent

\[ \text{Quota rent} = (P_d - P_w)\bar{M} \]

The national welfare effect depends on who receives the quota rent.

Tariff-rate quota

A tariff-rate quota applies a lower tariff inside the quota and a higher tariff above it:

\[ t = \begin{cases} t_1, & M \leq \bar{M} \\ t_2, & M > \bar{M} \end{cases} \]

where $t_1 < t_2$.

Domestic subsidy

A production subsidy raises the price received by producers without necessarily raising the consumer price.

\[ P_p = P_c + s \]

where $P_p$ is the producer price, $P_c$ is the consumer price, and $s$ is the subsidy per unit.

Government cost

\[ GC = s \times Q_s \]

Export subsidy

An export subsidy raises the price received by exporters.

\[ P_d = P_w + s \]

where $s$ is the export subsidy per unit.

Government cost

\[ GC = s \times X_s \]

where $X_s$ is the subsidized export volume.

In a small exporting country, an export subsidy is welfare reducing because the producer gain is smaller than the combined consumer loss and government cost.

Dumping and antidumping

Dumping margin

\[ \text{Dumping margin} = NV - P_x \]

where $NV$ is normal value and $P_x$ is the export price.

If dumping causes injury to domestic producers, an importing country may impose an antidumping duty.

Effective rate of protection

Value added at world prices

\[ VA_w = P_w^o - \sum_i P_w^i a_i \]

Value added at domestic protected prices

\[ VA_d = P_d^o - \sum_i P_d^i a_i \]

Effective rate of protection

\[ ERP = \frac{VA_d - VA_w}{VA_w} \times 100 \]

where $P^o$ is the output price, $P^i$ is the input price, and $a_i$ is the input requirement.

Regional trade agreements

Consumer surplus gain from a price fall

If price falls from $P_0$ to $P_1$ and quantity rises from $Q_0$ to $Q_1$:

\[ \Delta CS = (P_0 - P_1)Q_0 + \frac{1}{2}(P_0 - P_1)(Q_1 - Q_0) \]

Tariff revenue loss

\[ \Delta TR = -tQ_0 \]

if all imports switch from a tariff-paying non-member to a duty-free FTA partner.

Net welfare effect

\[ \Delta W = \Delta CS + \Delta TR \]

If $\Delta W > 0$, trade creation dominates. If $\Delta W < 0$, trade diversion dominates.

Foreign exchange

Let the exchange rate be domestic currency per unit of foreign currency.

Domestic price of an imported good

\[ P_d = E \times P_f \]

where $E$ is the exchange rate and $P_f$ is the foreign currency price.

If $E$ rises, the domestic currency depreciates and imports become more expensive.

Gravity model

A simple gravity model of trade is:

\[ Trade_{ij} = A \frac{GDP_i^{\alpha}GDP_j^{\beta}}{Distance_{ij}^{\gamma}} \]

A log-linear version is:

\[ \ln Trade_{ij} = \alpha_0 + \alpha_1\ln GDP_i + \alpha_2\ln GDP_j - \alpha_3\ln Distance_{ij} + u_{ij} \]

Expected signs:

\[ \alpha_1 > 0, \quad \alpha_2 > 0, \quad \alpha_3 > 0 \]

Distance enters with a negative effect because higher distance usually means higher trade costs.

TINA and partial-equilibrium simulation

A tariff reduction changes the tariff-inclusive import price:

\[ \frac{\Delta P_{ij}^k}{P_{ij}^k} = \frac{\Delta \tau_{ij}^k}{1 + \tau_{ij}^k} \]

Import demand responds according to elasticity:

\[ \Delta M_{ij}^k = \epsilon_{ij}^k \times \frac{\Delta \tau_{ij}^k}{1 + \tau_{ij}^k} \times M_{ij}^k \]

where:

$M_{ij}^k$ is imports of product $k$ by country $i$ from country $j$
$\tau_{ij}^k$ is the tariff rate
$\epsilon_{ij}^k$ is import demand elasticity

A tariff cut reduces price. If import demand elasticity is negative, the quantity imported from the partner rises.

--- title: "Formula Sheet" execute: echo: true warning: false message: false --- This appendix summarizes the main formulas used in **NREC4410: International Agricultural Trade**. Use it as a quick reference. The formula is only useful if you can explain the economic meaning behind it. ## Trade basics **Trade balance** $$ TB = X - M $$ where $X$ is exports and $M$ is imports. **Trade openness** $$ \text{Trade openness} = \frac{X + M}{GDP} $$ In percentage form: $$ \text{Trade openness (\%)} = \frac{X + M}{GDP} \times 100 $$ **Total trade value from trade openness** $$ X + M = \frac{\text{Trade openness (\%)}}{100} \times GDP $$ ## Ricardian model **Labor constraint** For two goods, food $F$ and cloth $C$: $$ a_{LF}Q_F + a_{LC}Q_C \leq L $$ where $a_{LF}$ and $a_{LC}$ are unit labor requirements and $L$ is total labor. **Production possibility frontier** Solving for cloth: $$ Q_C = \frac{L}{a_{LC}} - \frac{a_{LF}}{a_{LC}}Q_F $$ **Maximum production of each good** $$ Q_F^{\max} = \frac{L}{a_{LF}} $$ $$ Q_C^{\max} = \frac{L}{a_{LC}} $$ **Opportunity cost of food in terms of cloth** $$ OC_F = \frac{a_{LF}}{a_{LC}} $$ **Opportunity cost of cloth in terms of food** $$ OC_C = \frac{a_{LC}}{a_{LF}} $$ A country has comparative advantage in the good with the lower opportunity cost. **Trade possibility frontier** If a country specializes in food and the world relative price of food is $P_F/P_C$: $$ Q_C = \left(\frac{P_F}{P_C}\right)(Q_F^{\max} - Q_F) $$ If a country specializes in cloth: $$ Q_F = \left(\frac{P_C}{P_F}\right)(Q_C^{\max} - Q_C) $$ ## Heckscher-Ohlin model **Labor constraint** $$ a_{LC}Q_C + a_{LF}Q_F \leq L $$ **Capital constraint** $$ a_{KC}Q_C + a_{KF}Q_F \leq K $$ where $L$ is labor and $K$ is capital. **Factor intensity** Good $C$ is labor-intensive relative to good $F$ if: $$ \frac{a_{LC}}{a_{KC}} > \frac{a_{LF}}{a_{KF}} $$ Good $F$ is capital-intensive relative to good $C$ if: $$ \frac{a_{KF}}{a_{LF}} > \frac{a_{KC}}{a_{LC}} $$ **Factor abundance** Home is labor abundant relative to Foreign if: $$ \frac{L}{K} > \frac{L^*}{K^*} $$ Home is capital abundant relative to Foreign if: $$ \frac{K}{L} > \frac{K^*}{L^*} $$ **Zero-profit pricing equations** For cloth: $$ P_C = a_{LC}w + a_{KC}r $$ For food: $$ P_F = a_{LF}w + a_{KF}r $$ where $w$ is the wage and $r$ is the rental rate of capital. ## Consumer surplus, producer surplus, and total surplus For linear demand and supply: $$ Q_d = a - bP $$ $$ Q_s = c + dP $$ **Autarky equilibrium** $$ Q_d = Q_s $$ $$ a - bP = c + dP $$ $$ P^A = \frac{a-c}{b+d} $$ $$ Q^A = c + dP^A $$ **Demand choke price** $$ P_{\max} = \frac{a}{b} $$ **Supply price intercept** $$ P_{\min} = -\frac{c}{d} $$ **Consumer surplus** $$ CS = \frac{1}{2}(P_{\max} - P)Q_d $$ **Producer surplus** $$ PS = \frac{1}{2}(P - P_{\min})Q_s $$ If supply passes through the origin, $P_{\min}=0$, so: $$ PS = \frac{1}{2}PQ_s $$ **Total surplus** $$ TS = CS + PS $$ ## Free trade in partial equilibrium **Export supply** $$ ES(P) = Q_s(P) - Q_d(P) $$ **Import demand** $$ ED(P) = Q_d(P) - Q_s(P) $$ **World equilibrium** $$ ES(P^W) = ED(P^W) $$ At the world price $P^W$: $$ \text{Exports} = Q_s(P^W) - Q_d(P^W) $$ $$ \text{Imports} = Q_d(P^W) - Q_s(P^W) $$ ## Import tariff For a small importing country: $$ P_d = P_w + t $$ where $P_d$ is the domestic price, $P_w$ is the world price, and $t$ is the tariff per unit. **Government tariff revenue** $$ TR = t \times M_t $$ where $M_t$ is imports after the tariff. **Deadweight loss** $$ DWL = \text{production distortion} + \text{consumption distortion} $$ For linear diagrams, each distortion is usually a triangle. ## Import quota A quota fixes the maximum import quantity: $$ M \leq \bar{M} $$ **Quota rent per unit** $$ \text{Quota rent per unit} = P_d - P_w $$ **Total quota rent** $$ \text{Quota rent} = (P_d - P_w)\bar{M} $$ The national welfare effect depends on who receives the quota rent. ## Tariff-rate quota A tariff-rate quota applies a lower tariff inside the quota and a higher tariff above it: $$ t = \begin{cases} t_1, & M \leq \bar{M} \\ t_2, & M > \bar{M} \end{cases} $$ where $t_1 < t_2$. ## Domestic subsidy A production subsidy raises the price received by producers without necessarily raising the consumer price. $$ P_p = P_c + s $$ where $P_p$ is the producer price, $P_c$ is the consumer price, and $s$ is the subsidy per unit. **Government cost** $$ GC = s \times Q_s $$ ## Export subsidy An export subsidy raises the price received by exporters. $$ P_d = P_w + s $$ where $s$ is the export subsidy per unit. **Government cost** $$ GC = s \times X_s $$ where $X_s$ is the subsidized export volume. In a small exporting country, an export subsidy is welfare reducing because the producer gain is smaller than the combined consumer loss and government cost. ## Dumping and antidumping **Dumping margin** $$ \text{Dumping margin} = NV - P_x $$ where $NV$ is normal value and $P_x$ is the export price. If dumping causes injury to domestic producers, an importing country may impose an antidumping duty. ## Effective rate of protection **Value added at world prices** $$ VA_w = P_w^o - \sum_i P_w^i a_i $$ **Value added at domestic protected prices** $$ VA_d = P_d^o - \sum_i P_d^i a_i $$ **Effective rate of protection** $$ ERP = \frac{VA_d - VA_w}{VA_w} \times 100 $$ where $P^o$ is the output price, $P^i$ is the input price, and $a_i$ is the input requirement. ## Regional trade agreements **Consumer surplus gain from a price fall** If price falls from $P_0$ to $P_1$ and quantity rises from $Q_0$ to $Q_1$: $$ \Delta CS = (P_0 - P_1)Q_0 + \frac{1}{2}(P_0 - P_1)(Q_1 - Q_0) $$ **Tariff revenue loss** $$ \Delta TR = -tQ_0 $$ if all imports switch from a tariff-paying non-member to a duty-free FTA partner. **Net welfare effect** $$ \Delta W = \Delta CS + \Delta TR $$ If $\Delta W > 0$, trade creation dominates. If $\Delta W < 0$, trade diversion dominates. ## Foreign exchange Let the exchange rate be domestic currency per unit of foreign currency. **Domestic price of an imported good** $$ P_d = E \times P_f $$ where $E$ is the exchange rate and $P_f$ is the foreign currency price. If $E$ rises, the domestic currency depreciates and imports become more expensive. ## Gravity model A simple gravity model of trade is: $$ Trade_{ij} = A \frac{GDP_i^{\alpha}GDP_j^{\beta}}{Distance_{ij}^{\gamma}} $$ A log-linear version is: $$ \ln Trade_{ij} = \alpha_0 + \alpha_1\ln GDP_i + \alpha_2\ln GDP_j - \alpha_3\ln Distance_{ij} + u_{ij} $$ Expected signs: $$ \alpha_1 > 0, \quad \alpha_2 > 0, \quad \alpha_3 > 0 $$ Distance enters with a negative effect because higher distance usually means higher trade costs. ## TINA and partial-equilibrium simulation A tariff reduction changes the tariff-inclusive import price: $$ \frac{\Delta P_{ij}^k}{P_{ij}^k} = \frac{\Delta \tau_{ij}^k}{1 + \tau_{ij}^k} $$ Import demand responds according to elasticity: $$ \Delta M_{ij}^k = \epsilon_{ij}^k \times \frac{\Delta \tau_{ij}^k}{1 + \tau_{ij}^k} \times M_{ij}^k $$ where: - $M_{ij}^k$ is imports of product $k$ by country $i$ from country $j$ - $\tau_{ij}^k$ is the tariff rate - $\epsilon_{ij}^k$ is import demand elasticity A tariff cut reduces price. If import demand elasticity is negative, the quantity imported from the partner rises.