1.6 — The Standard Trade Model I

# 1.6 — The Standard Trade Model

## ECON 324 • International Trade • Spring 2023

### Ryan Safner<br> Associate Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner\@hood.edu</a> <br> <a href="https://github.com/ryansafner/tradeS23"><i class="fa fa-github fa-fw"></i>ryansafner/tradeS23</a><br> <a href="https://tradeS23.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>tradeS23.classes.ryansafner.com</a><br>

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# Outline

## [From Ricardian to Neoclassical Model](#3)
## [PPF: Increasing Costs](#7)
## [Indifference Curves](#28)
## [Autarky Optimum](#40)
## [Global Market for x](#52)
## [The Complete Picture](#62)

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# From Ricardian to Neoclassical Model

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# The Standard Trade Model

.pull-left[
.smaller[
- The .hi[standard] (or .hi[neoclassical]) .hi[trade model] is a more general model
  - Ricardian one-factor model: *special case*
  - Same with H-O (next) model

- We will extend the concepts we learned from the Ricardian model
  - more traditional neoclassical assumptions

- .hi-purple[A straightfoward neoclassical story about relative prices changing]
]
]

---

# What We're Adding to Ricardo

]

- Other factors of production with diminishing returns
  - Increasing opportunity costs of production

- Determination of global equilibrium relative prices via supply & demand

- Effects of the terms of trade changing

- Effects of countries' economies development & trade policy
]

---

# Tools for the Standard Model

.pull-left[
.smallest[
- We will do everything with graphs rather than equations
  - I expect you to understand and be able to interpret, if not be able to draw own graphs

- I will break today up into separate tools we will then combine
  1. PPF with increasing costs
  2. Indifference curves
  3. Comparative advantage in autarky
  4. Global market relative demand and relative supply
  5. International trade equilibrium
  6. Terms of trade changes (next class)
]
]

---

# PPF: Increasing Costs

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# Factors of Production I

`$$q=Af(t,l,k)$$`

.pull-left[
.smaller[
- Economists typically classify inputs, known as .hi[factors of production (FOP)]:

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Factor </th>
   <th style="text-align:left;"> Owned By </th>
   <th style="text-align:left;"> Earns </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Land (t) </td>
   <td style="text-align:left;"> Landowners </td>
   <td style="text-align:left;"> Rent </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Labor (l) </td>
   <td style="text-align:left;"> Laborers </td>
   <td style="text-align:left;"> Wages </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Capital (k) </td>
   <td style="text-align:left;"> Capitalists </td>
   <td style="text-align:left;"> Interest </td>
  </tr>
</tbody>
</table>
]
.smallest[
- `$A$`: "total factor productivity" (ideas/knowledge/institutions)
- and Entrepreneurs/Owners who earn Profit

]
]
.pull-right[
.center[
![](https://www.dropbox.com/s/cg5lbqrxsjij5iu/factors.jpg?raw=1)
]

]

---

# Factors of Production II

`$$q=f(l,k)$$`

- We often assume just two inputs: labor `$l$` and capital `$k$`

]

]

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# Marginal Product of Labor

- .hi[Marginal product of labor `\$(MP_l)\$`]: additional output produced by adding one more unit of labor (holding `$k$` constant)
`$$MP_l = \frac{\Delta q}{\Delta l}$$`

- `$MP_l$` is slope of `$TP$` at each value of `$l$`!

- Note: via calculus: `$\frac{\partial q}{\partial l}$`
]

]

---

# Marginal Product of Capital

- .hi[Marginal product of capital `\$(MP_k)\$`]: additional output produced by adding one more unit of capital (holding `$l$` constant)
`$$MP_k = \frac{\Delta q}{\Delta k}$$`

- `$MP_k$` is slope of `$TP$` at each value of `$k$`!

- Note: via calculus: `$\frac{\partial q}{\partial k}$`

]

]

---

# Diminishing Returns

- .hi-purple[Law of Diminishing Returns]: adding more of one factor of production **holding all others constant** will result in successively lower increases in output

- In order to increase output, need to increase use of *all* factors!

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Diminishing Returns

- .hi-purple[Law of Diminishing Returns]: adding more of one factor of production **holding all others constant** will result in successively lower increases in output

- In order to increase output, need to increase use of *all* factors!

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Competitive Markets and Factor Switching

.smallest[
- We still assume output markets and factor markets (for land, labor, capital) are perfectly competitive

- Firms hire resources up to the point where marginal cost of one more unit of `$l$` or `$k$` is equal to its marginal benefit in production ("marginal revenue product")

- Implies that in equilibrium, each factor of production is paid its marginal revenue product:
`$$\begin{align*}
p_l&=p_{y} * MP_l\\
p_k&=p_{y} * MP_k \\
\end{align*}$$`
  - Where `$p_l$` and `$p_k$` are prices of labor and capital, and `$p_y$` is the price of some output
  
- If you want to remember why, see my slides on [Factor Markets](https://ios20.classes.ryansafner.com/slides/1.8-slides.html#1)

- Multiple combinations of `$l$` and `$k$` can produce equivalent output `$y$`

- .hi-purple[Takeaway: producers will substitute between labor and capital depending on relative prices and technology]
]

---

# PPF: Increasing Costs

- .hi[Marginal rate of transformation (MRT)] .hi-purple[*increases*] as we produce more of a good
  - Again: .b[“slope”], .b[“relative price of x”], .b[“opportunity cost of x”]
  - Amount of y given up to get 1 more x

`$$-\frac{p_x}{p_y}$$`

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" style="display: block; margin: auto;" />

]

---

# PPF: Increasing Costs

`$$-\frac{p_x}{p_y}$$`

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" />

]

---

# PPF: Increasing Costs

`$$-\frac{p_x}{p_y}$$`

- `$A \rightarrow B$` raises opportunity cost of producing x
]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" />

]

---

# PPF: Increasing Costs

`$$-\frac{p_x}{p_y}$$`

- `$A \rightarrow B$` raises opportunity cost of producing x

- `$A \leftarrow B$` raises opportunity cost of producing y

]

.pull-right[
<img src="1.6-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" />

]

---

# What Causes a Curved PPF?

.pull-left[
.smallest[
- Diminishing returns to each factor of production `$(\downarrow MP_L, MP_K, MP_T)$` (holding others constant)

- Substitution of factors of production and combinations based on relative factor prices

- Moving Left/Right `$\implies$` changes in relative prices between `$x$` and `$y$`

`$$\bigg(\displaystyle\frac{p_x}{p_y}\bigg)^1 \rightarrow \bigg(\displaystyle\frac{p_x}{p_y}\bigg)^2$$`

- We dive deeper into these issues in the next model
]
]

]

---

# Optimal Production Choice in Autarky

- A country begins in .hi[autarky] with no international trade

- Where on its PPF should it produce? It should find an .hi-purple[optimum] combination of (x,y)

- Every point on its PPF is determined by relative prices `$\frac{p_x}{p_y}$`
  - As a curve, each point has a different slope (derivative)
]

]

---

# Optimal Production Choice in Autarky

- Assume: country will produce to maximize the market value of its production

1. **Choose:** .hi-purple[ < a production & consumption bundle >]

2. **In order to maximize:** .hi-green[< market value >]

3. **Subject to:** .hi-red[< technology and market prices >]

]

]

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# Optimal Production Choice in Autarky

`$$p_xx+p_yy=V$$`

- Describes the equation of .hi-purple["isovalue lines"]
  - Each line: set of combinations of `$x$` and `$y$` worth the same total market value
  - Higher lines `$\implies$` higher market value
]
]

]

---

# Optimal Production Choice in Autarky

`$$p_xx+p_yy=V$$`

- Solved for `$y$` to graph:
`$$y=\frac{V}{p_y}-\frac{p_x}{p_y}x$$`

]
]

]

---

# Optimal Production Choice in Autarky

`$$y=\frac{V}{p_y}-\color{magenta}{\frac{p_x}{p_y}}x$$`

- Again, .hi[slope] is the .hi[relative price of x]

]

]

---

# Optimal Production Choice in Autarky

`$$y=\frac{V}{p_y}-\color{magenta}{\frac{p_x}{p_y}}x$$`

- Again, .hi[slope] is the .hi[relative price of x]

- Given `$p_x$` and `$p_y$`, pick the point on PPF .hi-purple[tangent] to .hi-purple[highest] line

- .b[Point A]: maximized market value of output under current constraints
]

]

---

# Isovalue Lines depend on Relative Prices in Autarky

- If relative prices were to .hi-turquoise[change] (in autarky)

`$$\left(\frac{p_x}{p_y}\right)^1 \rightarrow \left(\frac{p_x}{p_y}\right)^2$$`

there would be a new set of isovalue lines with a .hi-turquoise[different slope].
]
]
.pull-right[

]

---

# Isovalue Lines depend on Relative Prices in Autarky

- If relative prices were to .hi-turquoise[change] (in autarky)

`$$\left(\frac{p_x}{p_y}\right)^1 \rightarrow \left(\frac{p_x}{p_y}\right)^2$$`

there would be a new set of isovalue lines with a .hi-turquoise[different slope].

- Optimum in autarky would be different point tangent to highest isovalue line of new slope: .b[Point B]
]
]

]

---

# Indifference Curves

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# Indifference Curves

- Consider a bundle of goods x and y: A = (2,5)
]
.pull-right[

]

---

# Indifference Curves

- Consider a bundle of goods x and y: A = (2,5)

- Consider another bundle: B = (5,2)
  - More x but less y
]
.pull-right[

]

---

# Indifference Curves

- Consider a bundle of goods x and y: A = (2,5)

- Consider another bundle: B = (5,2)
  - More x but less y
  
- Consider a third bundle: C = (10,1)
  - Even more x but even less y
]
.pull-right[

]

---

# Indifference Curves

- Consider a bundle of goods x and y: A = (2,5)

- Consider another bundle: B = (5,2)
  - More x but less y
  
- Consider a third bundle: C = (10,1)
  - Even more x but even less y

- Suppose you are indifferent between `$A \sim B \sim C$`: these bundles are on the same .hi-blue[indifference curve]
]
.pull-right[

]

---

# Indifference Curves

]
]
.pull-right[

]

---

# Indifference Curves

.pull-left[
.smaller[
- Country is .hi[indifferent] between all bundles on the same indifference curve

- Bundles *above* curve are .hi-green[preferred over] bundles on curve
  - `$D \succ A \sim B \sim C$`
  - On a .hi-green[higher curve]

]

]
.pull-right[

]

---

# Indifference Curves

.pull-left[
.smaller[
- Country is .hi[indifferent] between all bundles on the same indifference curve

- Bundles *above* curve are .hi-green[preferred over] bundles on curve
  - `$D \succ A \sim B \sim C$`
  - On a .hi-green[higher curve]

- Bundles **below** curve are .hi-red[less preferred] than bundles on curve
  - `$E \prec A \sim B \sim C$`
  - On a .hi-red[lower curve]
]

]
.pull-right[

]

---

# Marginal Rate of Substitution

.pull-left[
- To aquire 1 more unit of x, how many units of y are you willing to give up to remain indifferent?

]

---

# Marginal Rate of Substitution I

.pull-left[
- To aquire 1 more unit of x, how many units of y are you willing to give up to remain indifferent?

- .hi[Marginal Rate of Substitution (MRS)]: rate at which you trade off one good for the other and remain *indifferent*

- Again: **opportunity cost**: # of units of `$y$` you need to give up to acquire 1 more `$x$`

]

---

# MRS vs. Other Slopes

- Isovalue lines (slope) & MRT (PPF slope) measured the **production** tradeoff between `$x$` and `$y$` based on market prices

- **MRS** measures **consumption** tradeoff between `$x$` vs. `$y$` based on preferences

]

---

# Marginal Rate of Substitution

- MRS is the slope of the indifference curve
`$$MRS_{x,y}=-\frac{\Delta y}{\Delta x} = \frac{rise}{run}$$`

- Amount of `$y$` given up for 1 more `$x$`

- Note: slope (MRS) changes along the curve!

]
.pull-right[

]

---

# Autarky Optimum

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# Home's Autarky Optimum

- .blue[Home] produces and consumes at highest indifference curve tangent to its PPF

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-32-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Home's Autarky Optimum

- .blue[Home] produces and consumes at highest indifference curve tangent to its PPF

- At .blue[Home]'s autarky optimum:

`$$\underbrace{MRT}_{\text{PPF Slope}}=\underbrace{MRS}_{\text{I.C. Slope}}=\underbrace{\big(\frac{p_x}{p_y}\big)}_{\text{price line}}$$`

- This is .blue[Home]'s .hi[relative price in autarky]: the relative price (of `$x$`) where nation is maximizing its welfare in autarky

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-33-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Foreign's Autarky Equilibrium

- .red[Foreign] (with different PPF) also produces and consumes at highest indifference curve tangent to its PPF

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-34-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Foreign's Autarky Equilibrium

- .red[Foreign] (with different PPF) also produces and consumes at highest indifference curve tangent to its PPF

- At .red[Foreign]'s autarky optimum:

`$$\underbrace{MRT'}_{\text{PPF Slope}}=\underbrace{MRS'}_{\text{I.C. Slope}}=\underbrace{\big(\frac{p_x}{p_y}\big)'}_{\text{price line}}$$`

- This is .red[Foreign]'s .hi[relative price in autarky]: the relative price (of `$x$`) where nation is maximizing its welfare in autarky

]

<img src="1.6-slides_files/figure-html/unnamed-chunk-35-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Relative Prices in Autarky Equilibrium

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-36-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-37-1.png" width="504" style="display: block; margin: auto;" />
]

]

---

# Relative Prices in Autarky Equilibrium

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-38-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-39-1.png" width="504" style="display: block; margin: auto;" />
]

- Relative price of x (slope of PPF) is lower (flatter) in .blue[Home] than .red[Foreign]

`$$\color{blue}{\left(\frac{p_x}{p_y}\right)} < \color{red}{\left(\frac{p_x}{p_y}\right)'}$$`

]

---

# Comparative Advantage

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-40-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-41-1.png" width="504" style="display: block; margin: auto;" />
]

.smallest[
- .blue[Home] has a comparative advantage in x; will export x
- .red[Foreign] has a comparative advantage in y; will export y
]

---

# Recall from Ricardian Model: Price Adjustments

- .blue[Home] exports x `$\implies$` *less* x sold in .blue[Home] `$\implies$` `$\uparrow p_x$` in .blue[Home]

- As x arrives in .red[Foreign] `$\implies$` *more* x sold in .red[Foreign] `$\implies$` `$\downarrow p_x$` in .red[Foreign]

- .red[Foreign] exports y `$\implies$` *less* y sold in .red[Foreign] `$\implies$` `$\uparrow p_y$` in .red[Foreign]

- As y arrives in .blue[Home] `$\implies$` *more* y sold in .blue[Home] `$\implies$` `$\downarrow p_y$` in .blue[Home]

---

# Global Market for x

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# Global Market for x: Home

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-42-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-blue[Home]'s Supply of x

]

]

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# Global Market for x: Home

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/home-exp-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-blue[Home]'s Export Supply of x

]

- As relative price of x (slope) `$\uparrow$` from `$\left(\frac{p_x}{p_y}\right)^{H} \rightarrow \left(\frac{p_x}{p_y}\right)^{2}$`, .blue[Home] exports more x
]

---

# Global Market for x: Home

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-44-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-blue[Home]'s Export Supply of x

]

- As relative price of x (slope) `$\uparrow$` from `$\left(\frac{p_x}{p_y}\right)^{H} \rightarrow \left(\frac{p_x}{p_y}\right)^{2}$`, .blue[Home] exports more x

- Trace .blue[Home]'s .hi-purple[export supply curve for x] upward as relative price of x increases
]

---

# Global Market for x: Foreign

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-46-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]'s Import Demand for x

]

]

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# Global Market for x: Foreign

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/foreign-imp-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]'s Import Demand for x

]

- As relative price of x (slope) `$\downarrow$` from `$\left(\frac{p_x}{p_y}\right)^{F} \rightarrow \left(\frac{p_x}{p_y}\right)^{2}$`, .red[Foreign] imports more x
]

---

# Global Market for x: Foreign

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-48-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]'s Import Demand for x

]

- As relative price of x (slope) `$\downarrow$` from `$\left(\frac{p_x}{p_y}\right)^{F} \rightarrow \left(\frac{p_x}{p_y}\right)^{2}$`, .red[Foreign] imports more x

- Trace .red[Foreign]'s .hi-purple[import demand curve for x] upward as relative price of x decreases
]

---

# The Global Market for x

- Put together .blue[Home's export supply] and .red[Foreign's import demand] for x

- .hi[World equilibrium relative price of x]: `$\left(\frac{p_x}{p_y} \right)^{2}$` balances .blue[Home's exports] and .red[Foreign's imports] of x
]

<img src="1.6-slides_files/figure-html/market-1.png" width="504" style="display: block; margin: auto;" />
]

---

# The Global Market for x

.pull-left[
.smallest[
- Both countries began in autarky (.blue[A], .red[A']) with very different relative prices of x
  - Cheaper in .blue[Home] (has comparative advantage)
  - More expensive in .red[Foreign] (comparative disadvantage)

]
]

<img src="1.6-slides_files/figure-html/unnamed-chunk-50-1.png" width="504" style="display: block; margin: auto;" />
]
---

# The Global Market for x

- As countries trade, changes relative price of x in each country until both reach equilibrium world relative price (B,B'), where both countries have same relative price:

`$$\left(\frac{p_x}{p_y} \right)^{H} < \left(\frac{p_x}{p_y} \right)^{2} < \left(\frac{p_x}{p_y} \right)^{F}$$`
]
]
.pull-right[

<img src="1.6-slides_files/figure-html/unnamed-chunk-51-1.png" width="504" style="display: block; margin: auto;" />
]

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# The Complete Picture

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# Autarky Equilibrium

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-52-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-53-1.png" width="504" style="display: block; margin: auto;" />
]

.smallest[
- Countries begin in .hi[autarky] optimum with different relative prices
  - A is optimum for .blue[Home]
  - A' is optimum for .red[Foreign]
]

---

# Specialization

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-54-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-55-1.png" width="504" style="display: block; margin: auto;" />
]

- International trade changes the relative price of x `$(\uparrow$` for .blue[Home], `$\downarrow$` for .red[Foreign])

- .hi-purple[With international trade, countries face same world relative prices] (slope of dark purple dashed line)

]

---

# Specialization

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-56-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-57-1.png" width="504" style="display: block; margin: auto;" />
]

.smallest[
- Countries .hi[specialize]: produce *more* of comparative advantaged good, *less* of disadvantaged good
  - .blue[Home]: A `$\rightarrow$` B: produces more x, less y 
  - .red[Foreign]: A' `$\rightarrow$` B': produces less x, more y

- Note this is .hi-purple[incomplete specialization]: countries still produce both goods!
]

---

# Trade Triangles

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-58-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-59-1.png" width="504" style="display: block; margin: auto;" />
]

- .blue[Home] `$\rightarrow$` x `$\rightarrow$` .red[Foreign]
]

---

# Trade Triangles

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-60-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-61-1.png" width="504" style="display: block; margin: auto;" />
]

- .blue[Home] `$\rightarrow$` x `$\rightarrow$` .red[Foreign]

- .blue[Home] `$\leftarrow$` y `$\leftarrow$` .red[Foreign]

]

---

# Gains from Trade

### .hi-blue[Home]

<img src="1.6-slides_files/figure-html/unnamed-chunk-62-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.6-slides_files/figure-html/unnamed-chunk-63-1.png" width="504" style="display: block; margin: auto;" />
]

- Both countries exchange their imports & exports and consume at C and C'

- Both reach a higher indifference curve with trade, well beyond their PPFs!

]