1.8 — The Specific Factors Model

# 1.8 — The Specific Factors Model

## ECON 324 • International Trade • Spring 2023

### Ryan Safner Associate Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/tradeS23">ryansafner/tradeS23</a> <a href="https://tradeS23.classes.ryansafner.com"> tradeS23.classes.ryansafner.com</a>

---

# Outline

### [Assumptions of the Specific Factors Model](#3)

### [Allocating the Mobile Factor (Labor)](#11)

### [Distribution Effects Using our Two Country Trade Example](#23)

### [Takeaways from the Specific Factors Model](#44)

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# Assumptions of the Specific Factors Model

---

# Assumptions of the Specific Factors Model

- But in truth, some factors are .hi[specific] or .hi[immobile]: can only be used for the production of a specific set of goods or industry
  - e.g. programmers can only work in software, not in pro-football
  - e.g. equipment used to make beer barrels cannot switch to producing computer chips

- .hi-purple[Opening up trade will affect the distribution of income between fixed and mobile factors]
]
]
.pull-right[
.center[
![](../images/workers.png)
]
]

---

# Assumptions of the Specific Factors Model

- Imagine 2 countries, .blue[Home] and .red[Foreign]

- Countries have three factors of production:
   - labor `$(L)$`
   - capital `$(K)$`
   - land `$(T)$`

]

---

# Assumptions of the Specific Factors Model

- Each country has two industries, .b[manufacturing (M)] and .b[agriculture (A)]

- .b[Manufacturing] is produced using *capital* `$(K)$` and *labor* `$(L)$`

- .b[Agriculture] is produced using *land* `$(T)$` and *labor* `$(L)$`

- Land `$(T)$` and capital `$(K)$` are .hi[specific factors], only used to produce one good

- Labor `$(L)$` is a .hi[mobile factor] that can be used in *either* (or both) sectors

]
.pull-right[
.center[
![:scale 75%](../images/factoryrobots.jpg)
![:scale 75%](../images/peasants1.jpg)

]
]

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# Setting up the Model: Production Function

- An economy's production can be described as a set of production functions for manufacturing `$(M)$` and agriculture `$(A)$`

`$$\begin{align*}
Q_M&=Q_M(K,L_M)\\
Q_A&=Q_A(T,L_A)\\
\end{align*}$$`

- Each country can only allocate its labor force between two industries

`$$L_M+L_A=\bar{L}$$`

]

.pull-right[
.center[
![:scale 75%](../images/factoryrobots.jpg)
![:scale 75%](../images/peasants1.jpg)

]
]

---

# Diminishing Marginal Product of Labor

- Each industry exhibits .hi[diminishing returns to labor]

- .hi[Marginal product of labor in manufacturing `\$(MPL_{M})\$`]: additional manufacturing output produced by adding one more unit of labor (holding `$K$` constant)

`$$MPL_{M} = \frac{\Delta Q_M}{\Delta L_M}$$`

- Declines as more `$L$` is added to manufacturing production

]

]

---

# Diminishing Marginal Product of Labor

- Each industry exhibits .hi[diminishing returns to labor]

- .hi[Marginal product of labor in agriculture `\$(MPL_{A})\$`]: additional agriculture output produced by adding one more unit of labor (holding `$T$` constant)

`$$MPL_{A} = \frac{\Delta Q_A}{\Delta L_A}$$`

- Declines as more `$L$` is added to agriculture production

]

]

---

# PPF

- We get a PPF with increasing costs again

- Let's examine more *why*

]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" />

]

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# Allocating the Mobile Factor (Labor)

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# A Note About Labor

.pull-left[
.smaller[
- A simple (and very Ricardian) assumption about labor: it is measured in hours, and can equally be applied to each industry

`$$\bar{L}=L_M+L_A$$`

- Every labor hour allocated to agriculture is a labor hour *not* allocated to manufacturing, and vice versa
  - .hi-purple[Opportunity cost of labor]

- Visualize a “labor budget constraint” to understand movements along the PPF
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Allocating Labor

.pull-left[
.smallest[
- Shows relationship of moving along PPF `$\iff$` reallocating labor across industries

- If all labor in `$A$` (point A), country only produces `$A$`, no `$M$`
- If all labor in `$M$` (point D), country only produces `$M$`, no `$A$`

- Remember, each industry has diminishing returns to labor, and will have a particular `$MPL$` depending on how much land or capital there are
  - Hence, a 1 unit `$\uparrow \downarrow$` in `$L$` in one industry *does not* imply a 1 unit increase in output

]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Allocating Labor

.pull-left[
.smallest[
- As we move to the right of the PPF, we are pulling labor out of agriculture and into manufacturing

- Each single unit of labor we take out of `$A$` and put into `$M$` will:
  - Lower `$\downarrow Q_A$` by `$MPL_{A}$`
  - Raise `$\uparrow Q_M$` by `$MPL_{M}$`

- Or to put it inversely, to produce 1 more unit of `$M$`:
  - Reallocate `$\downarrow L_A$` input by `$\frac{1}{MPL_{A}}$`
  - Reallocate `$\uparrow L_M$` input by `$\frac{1}{MPL_{M}}$`
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Production Possibilities Frontier

.pull-left[
.smaller[
- .hi[Marginal rate of transformation (MRT)] .hi-purple[*increases*] as we produce more of a good
  - Again: .b[“slope”], .b[“relative price of M”], .b[“opportunity cost of M”]
  - Amount of `$A$` given up for 1 more `$M$`

`$$\underbrace{MRT}_{slope}=-\frac{MPL_A}{MPL_M}$$`

- Note `$A \, (y)$` on top and `$M \, (x)$` on bottom!
    - if you think in our Ricardian terms, `$l_x=\frac{1}{MPL_x}$` and `$l_y=\frac{1}{MPL_y}$`, so `$\frac{l_x}{l_y} \implies \frac{MPL_y}{MPL_x}$`

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" />

]

---

# Allocating Labor

- Because of diminishing returns, as we move labor out of `$A$` and into `$M$`, we lower `$MPL_M$` and raise `$MPL_A$`

- This is why the PPF has increasing opportunity costs, and is bent inwards the way it is!

- For a given amount of `$T$`, `$K$`, and `$L$`, we can determine the economy's output bundle `$(Q_M, Q_A)$` by knowing how much labor is allocated across `$(L_M, L_A)$`

- Now let's find how labor is allocated across industries
]

]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" />
]

---

# The Demand for Labor in Competitive Industries

.pull-left[
.quitesmall[
- Profit-maximizing firms in competitive labor markets will hire labor (hours) up to the point where the marginal benefit of hiring labor equals the marginal cost
  - .red[Marginal cost per labor-hour]: wage `$w$`
  - .blue[Marginal benefit per labor-hour]: marginal revenue product (marginal product `$\times$` price of output)

- In manufacturing:

`$$w = MPL_M * P_M$$`

- In agriculture:

`$$w = MPL_A * P_A$$`

- Again, if you want to remember why, see my slides on [Factor Markets](https://ios23.classes.ryansafner.com/slides/1.8-slides.html#20)
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" style="display: block; margin: auto;" />

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# The Demand for Labor in Both Industries

- Because we have assumed .hi-purple[labor is mobile] (and homogenous “labor hours”), workers will always move out of a lower-paying industry and into a higher-paying industry

- Thus, in equilibrium, .hi-turquoise[wages `\$w\$` must equalize across both industries], with the implication:

`$$\begin{align*}
(w =) \quad MPL_M * P_M &= MPL_A * P_A \quad(= w)\\
-\frac{MPL_A}{MPL_M} &= - \frac{P_M}{P_A}\\
\end{align*}$$`

]

---

# Labor and the PPF

- Thus, we finally see how it is that the slope of the PPF is equivalent to the relative price of `$M:$`

`$$MRT = -\frac{p_M}{p_A}$$`
  - Opportunity cost of `$M$`, slope is amount of `$A$` given up for `$1 M$`
  - (Back to `$x$` on top, `$y$` on bottom!)
  
- At the optimum production, PPF is tangent to a value line with slope the relative price of `$M$`
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" />

]

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# Labor Allocation

- Recall both industries in equilibrium must charge the same wage `$w_M=w_A=w^{\star}$`

- Moving from left to right, labor allocated to manufacturing, `$L_M$`

- Moving from right to left, labor allocated to agriculture, `$L_A$`
]
]
.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" />

]

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# A Change in Relative Prices on Labor Allocation

.pull-left[
.smaller[
- An increase in the relative price of manufacturing `$\left(\frac{p_M}{p_A}\right)$` will increase the demand for labor in manufacturing

- Because both industries have to compete for labor, wages do increase, but not as much as the increase in the relative price of manufacturing

- More labor will be used in manufacturing than in agriculture, and thus, the economy will produce more manufacturing and less agriculture
]
]
.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" />

]

---

# A Change in Relative Prices on PPF

- We can equivalently see this on the PPF

- Increase in the relative price of manufacturing

`$$\bigg(\displaystyle\frac{p_M}{p_A}\bigg)^1 \rightarrow \bigg(\displaystyle\frac{p_M}{p_A}\bigg)^2$$`

- Moving from `$A \rightarrow B$`
  - Slope steepens
  - Country will produce less agriculture, more manufacturing

]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-15-1.png" width="504" style="display: block; margin: auto;" />

]

---

# Distribution Effects Using our Two Country Trade Example

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# Our Two Country Trade Example: Autarky

### .hi-blue[Home]

]

### .hi-red[Foreign]

]

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

---

# Our Two Country Trade Example: Specialization

### .hi-blue[Home]

]

### .hi-red[Foreign]

]

.smallest[
- .blue[Home] has comparative advantage in manufacturing
- .red[Foreign] has comparative advantage in agriculture
]

---

# Our Two Country Trade Example: Specialization

### .hi-blue[Home]

<img src="1.8-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.8-slides_files/figure-html/unnamed-chunk-22-1.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 M, less A 
  - .red[Foreign]: A' `$\rightarrow$` B': produces less M, more A

]

---

# Relative Price Changes in Home

- Let's look at three groups at .blue[Home]:
  - Laborers `$(L)$`
  - Capitalists (owners of `$K)$`
  - Landowners (owners of `$T)$`

- Increase in the relative price of manufacturing from trade
  - decrease in relative price of agriculture
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" style="display: block; margin: auto;" />
]

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# Effects of Trade on Home's Income Distribution: L

.pull-left[
.quitesmall[
- Workers find their wage has increased (but less than increase in relative price of M)
`$$\frac{\Delta w}{w_1} < \cfrac{\Delta \left(\frac{P_M}{P_A}\right)}{\left(\frac{P_M}{P_A}\right)_1}$$`

- Amount of manufactures `$Q_M$` that can be purchased with wages has *fallen*!
  - .hi-purple[Real wage in terms of manufacturing, `\$\downarrow \frac{w}{p_M}\$`]

- Amount of agriculture `$Q_A$` that can be purchased with wages has *risen*!
  - .hi-purple[Real wage in terms of agriculture, `\$\uparrow \frac{w}{p_A}\$`]

- .hi-purple[Effect on workers is ambiguous]
 - Depends on their consumption preferences between `$M$` and `$A$`
]
]
.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" style="display: block; margin: auto;" />
]

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# Effects of Trade on Home's Income Distribution: K

- Total income to capitalists `$= \underbrace{(P_M * Q_M)}_{\text{Revenues in M}} - \underbrace{(W * L_M)}_{\text{Labor costs}}$`

- As more labor used in manufacturing, `$\uparrow MP_K$`: Each machine has more workers to work it.

- .hi-purple[Capital owners gain]
  - We saw (1) `$\uparrow$` relative price of manufacturing and (2) `$\downarrow$` real wage in terms of manufacturing
  - Thus, income to capital will rise more than proportionately to the rise in relative price of manufacturing

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" style="display: block; margin: auto;" />
]

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# Advanced Explanation for Capital

- Total output `$Q_M$` using `$L_M$` is equal to the area under the `$MPL_M$` curve up to `$L_M$`

- Labor is paid `$w = MPL_M * p_M$`
  - Rewrite as real wage (in terms of `$M)$`: `$\frac{w}{P_M}$`
  - This times the total number of workers `$L_M$` equals the .blue[total wages paid]

- All .green[residual income] goes to capital owners
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-26-1.png" width="504" style="display: block; margin: auto;" />
]

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# Advanced Explanation for Capital

.pull-left[
.smallest[
- Because trade raises the relative price of manufacturing, `$\frac{p_M}{p_A}$`, we saw:
  - Increase in labor `$L_M$`, and increase in *nominal* wage `$w$`, but
  - Decrease in real wage in terms of `$m$`, `$\frac{w}{p_M}$`

- .hi-green[Capital owners gain]
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Effects of Trade on Home's Income Distribution: T

- Total income to landowners `$= \underbrace{(P_AM * Q_A)}_{\text{Revenues in A}} - \underbrace{(W * L_A)}_{\text{Labor costs}}$`

- As less labor used in agriculture, `$\downarrow MP_T$`: Each piece of land has fewer workers to work it.

- .hi-purple[Land owners lose]
  - We saw (1) `$\downarrow$` relative price of agriculture and (2) `$\uparrow$` real wage in terms of agriculture
  - Thus, income to landowners will fall more than proportionately to the fall in relative price of agriculture

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-28-1.png" width="504" style="display: block; margin: auto;" />
]

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# Advanced Explanation for Land

- Total output `$Q_A$` using `$L_A$` is equal to the area under the `$MPL_A$` curve up to `$L_A$`

- Labor is paid `$w = MPL_A * p_A$`
  - Rewrite as real wage (in terms of `$A)$`: `$\frac{w}{P_A}$`
  - This times the total number of workers `$L_A$` equals the total wages paid

- All .green[residual income] goes to land owners (as rent)
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" style="display: block; margin: auto;" />
]

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# Advanced Explanation for Land

.pull-left[
.smallest[
- Because trade lowers the relative price of agriculture, `$\frac{p_A}{p_M}$`, we saw:
  - Decrease in labor `$L_A$`, but increase in *nominal* wage `$w$`, so
  - Increase in real wage in terms of `$A$`, `$\frac{w}{p_A}$`

- .b[Land owners lose]
]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-30-1.png" width="504" style="display: block; margin: auto;" />
]

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# Effects of Trade on Home's Income Distribution

EFfects of trade on .blue[Home]'s:

- .b[Labor]: ambiguous
  - real wage rises in terms of `$M$`, falls in terms of `$A$`
  
- .b[Capital]: income rises more than proportionate to `$M$` relative price increase

- .b[Land]: income falls more than proportionate to `$A$` relative price fall

]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-31-1.png" width="504" style="display: block; margin: auto;" />
]

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# Effects of Trade on Home Income Distribution

- .hi-purple[Factor specific to the sector whose relative price rises is *better off* with trade]
  - Capital for manufacturing
  
- .hi-purple[Factor specific to the sector whose relative price falls is *worse off* with trade]
  - Land for agriculture

- .hi-purple[The mobile factor is *not clearly* better or worse off with trade.]
  - Labor

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-32-1.png" width="504" style="display: block; margin: auto;" />
]

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# Specialization (Again)

### .hi-blue[Home]

<img src="1.8-slides_files/figure-html/unnamed-chunk-33-1.png" width="504" style="display: block; margin: auto;" />
]

### .hi-red[Foreign]

<img src="1.8-slides_files/figure-html/unnamed-chunk-34-1.png" width="504" style="display: block; margin: auto;" />
]

]

---

# Relative Price Changes in Foreign

- Let's look at three groups at .red[Foreign]:
  - Laborers `$(L)$`
  - Capitalists (owners of `$K)$`
  - Landowners (owners of `$T)$`

- Decrease in the relative price of manufacturing from trade
  - increase in relative price of agriculture
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-35-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Effects of Trade on Foreign's Income Distribution: L

.pull-left[
.quitesmall[
- Workers find their wage has increased (but less than increase in relative price of A)
`$$\frac{\Delta w}{w_1} < \cfrac{\Delta \left(\frac{P_A}{P_M}\right)}{\left(\frac{P_A}{P_M}\right)_1}$$`

- Amount of manufactures `$Q_M$` that can be purchased with wages has *risen*!
  - .hi-purple[Real wage in terms of manufacturing, `\$\uparrow \frac{w}{p_M}\$`]

- Amount of agriculture `$Q_A$` that can be purchased with wages has *fallen*!
  - .hi-purple[Real wage in terms of agriculture, `\$\downarrow \frac{w}{p_A}\$`]

- .hi-purple[Effect on workers is ambiguous]
 - Depends on their consumption preferences between `$M$` and `$A$`
]
]
.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-36-1.png" width="504" style="display: block; margin: auto;" />

]

---

# Effects of Trade on Foreign's Income Distribution: K

- Total income to capitalists `$= \underbrace{(P_M * Q_M)}_{\text{Revenues in M}} - \underbrace{(W * L_M)}_{\text{Labor costs}}$`

- As less labor used in manufacturing, `$\downarrow MP_K$`: Each machine has fewer workers to work it.

- .hi-purple[Capital owners lose]
  - We saw (1) `$\downarrow$` relative price of manufacturing and (2) `$\uparrow$` real wage in terms of manufacturing
  - Thus, income to capital will fall more than proportionately to the fall in relative price of manufacturing

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-37-1.png" width="504" style="display: block; margin: auto;" />
]

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# Effects of Trade on Foreign's Income Distribution: T

- Total income to landowners `$= \underbrace{(P_A * Q_A)}_{\text{Revenues in A}} - \underbrace{(W * L_A)}_{\text{Labor costs}}$`

- As more labor used in agriculture, `$\uparrow MP_T$`: Each piece of land has more workers to work it.

- .hi-purple[Land owners gain]
  - We saw (1) `$\uparrow$` relative price of agriculture and (2) `$\downarrow$` real wage in terms of agriculture
  - Thus, income to landowners will rise more than proportionately to the rise in relative price of agriculture

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-38-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Effects of Trade on Foreign's Income Distribution

EFfects of trade on .red[Foreign]'s:

- .b[Labor]: ambiguous
  - real wage rises in terms of `$M$`, falls in terms of `$A$`
  
- .b[Capital]: income falls more than proportionate to `$M$` relative price fall

- .b[Land]: income rises more than proportionate to `$A$` relative price increase

]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-39-1.png" width="504" style="display: block; margin: auto;" />
]

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# Effects of Trade on Foreign's Income Distribution

- .hi-purple[Factor specific to the sector whose relative price rises is *better off* with trade.]
  - Land for agriculture
  
- .hi-purple[Factor specific to the sector whose relative price falls is *worse off* with trade.]
  - Capital for manufacturing

- .hi-purple[The mobile factor is *not clearly* better or worse off with trade.]
  - Labor

]
]

.pull-right[
<img src="1.8-slides_files/figure-html/unnamed-chunk-40-1.png" width="504" style="display: block; margin: auto;" />
]

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# Takeways from The Specific Factors Model

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# Takeways from The Specific Factors Model

- .hi-purple[Changes in trade fall mainly upon the fixed/specific factors of production]
  - Increase in relative prices (exports) benefit fixed factor producing exports
  - Decrease in relative prices (imports) harm fixed factor competing with imports

- .hi-purple[Mobile factors face ambiguous change]
  - Can move from low-income industries to high-income industries
  
]

]

---

# Takeways from The Specific Factors Model

- Of course, our simple model aggregates labor into a single mobile factor

- In reality, different types of labor, some may be .hi[mobile] and some may be .hi[immoble] and .hi[specific]

- .hu-purple[Changes in trade patterns and relative prices will affect specific and mobile factors differently]
]

]

---

# Example of Mobile vs. Specific Labor

.content-box-green[
.hi-green[Example]: Auto-workers in Detroit in the 1980s were a relatively specific and immobile factor

- Geographically concentrated

- Skills specific to car assembly-lines
]
]

.pull-right[
.center[
![:scale 50%](../images/gmworkers.jpg)
![:scale 50%](../images/gmbuilding.jpg)
]
]

---

# Example of Mobile vs. Specific Labor

- Relative price of cars falls in U.S., U.S. factories produce fewer cars, wages & jobs in U.S. auto manufacturing diminish

- More .hi-purple[mobile] and .hi-purple[nonspecific] workers left Detroit for other industries
  - e.g. maybe they went to Texas to work in booming oil industry

- More .hi-purple[immobile] and .hi-purple[specific] workers lost jobs
  - Maybe geographically stuck in Detroit
  - Skills were too specific to auto industry, not transferrable to other industries
]
]

---

# Some More Examples

---

# Some More Examples

---

# Some More Examples

---

# Takeways from The Specific Factors Model

.pull-left[
.smallest[
- Again, .hi-purple[changes in trade fall mainly upon the fixed/specific factors of production]
  - Increase in relative prices (exports) benefit fixed factor producing exports
  - Decrease in relative prices (imports) harm fixed factor competing with imports

- .hi-purple[Mobile factors face ambiguous change]
  - Can move from low-income industries to high-income industries

- .hi[Policy implication]: if governments wish to protect domestic groups from adverse trade shocks, increase mobility and non-specific skills/uses
  - make labor, capital, land markets *more flexible* to reduce shocks from trade on domestic workers, capital-, & land-owners

]
]
.pull-right[
.center[
![](../images/stuck.jpg)
]

]