<p>You can consider a single competitive firm model with quadratic cost function say $c(y) = \frac{y^2}{2}$. This firm is in Venezuela. Let $\hat{p}_c$ and $\hat{p}_v$ be price ceilings in Colombia and Venezuela respectively. Also, $\hat{p}_c &gt; \hat{p}_v$ indicating the opportunity to smuggle. Also, assume demands are sufficiently high so that the ceilings are binding. Consider competitive suppliers' problem without smuggling:</p>

<p>\begin{eqnarray*} \displaystyle\max_{y\geq 0} &amp;&amp; p_vy - \frac{y^2}{2}\end{eqnarray*}
Solving it, we get the supply curve as $y = p_v$.</p>

<p>Given that the price in Venezuela is $\hat{p}_v$, equilibrium quantity in this case will be $y^* = \hat{p}_v$.</p>

<p>With an option to smuggle with the additional cost say $c(z) = \frac{z^2}{2}$, the profit maximization problem is:
\begin{eqnarray*} \displaystyle\max_{y\geq 0, z \geq 0} &amp;&amp; p_vy + p_cz - \frac{(y + z)^2}{2} - \frac{z^2}{2}\end{eqnarray*}</p>

<p>This gives us the new domestic supply function as:
$y = 2p_v - p_c$</p>

<p>and the quantity smuggled as
$z = p_c - p_v$</p>

<p>So, equilibrium quantity sold in Venezuela has fallen to
$y^{**} = 2\hat{p}_v - \hat{p}_c$ and the quantity smuggled is equal to $z^{**} = \hat{p}_c - \hat{p}_v$.</p>

CC BY-SA 3.0
2017-01-25T19:46:37.567Z
Thank you! I will try to graph with supply/demand curves. Equilibrium quantity sold in Colombia would decrease? "So, equilibrium quantity sold in Colombia has fallen to.."
John B
CC BY-SA 3.0
2017-01-26T07:17:24.347Z
I just mixed up the notation. I have fixed it now. The equilibrium quantity consumed in Venezuela has fallen. The quantity smuggled to Colombia will only raise the quantity consumed in Colombia.
Amit
CC BY-SA 3.0
2017-01-25T17:14:03.780Z
2017-01-26T07:13:57.753Z
2017-01-26T07:13:57.753Z
Amit
<p>I am attempting a (very) basic analysis of smuggling from one economy to the other (Venezuela to Colombia) - arising from price ceilings <em>ceteris paribus</em>. e1 are the equilibria without intervention (price controls), e2 are the equilibria after price ceilings, and e3 show what I assume the markets would look like if Venezuelans supplied Colombia (by smuggling) where the Venezuelans' profit = </p>

<p>[Colombia p-hat - cost of smuggling - Venezuela p-hat]</p>

<p>The supply curve for Colombia becomes the sum of the Venezuelan and Colombian supply curves above the [Venezuelan price ceiling + cost of smuggling], while there is a movement <em>along</em> the Venezuelan supply curve to the point of [Colombian price ceiling - cost of smuggling].</p>

<p>Is this correct?</p>

<p>Thanks!
<a href="https://i.sstatic.net/93oyL.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/93oyL.png" alt="enter image description here"></a>
<a href="https://i.sstatic.net/g6kOP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/g6kOP.png" alt="S-D Graph"></a></p>

CC BY-SA 3.0
2017-01-23T22:37:32.987Z
After more consideration, I believe that my results require two assumptions. (1) That there are not any domestic 'black markets'. That is, all domestic trade is legitimate at the price ceiling, and (2) All smuggling from Venezuela is coordinated by Venezuelan suppliers. That is, no consumers are purchasing and reselling over the border - rather, suppliers hire people to deliver and sell in Columbia for a wage returning the revenue to the supplier. Like pizza delivery.
John B
CC BY-SA 3.0
2017-01-25T11:42:13.353Z
Suppose $\hat{p}_c$, $\hat{p}_v$ are price ceilings in Colombia and Venezuela respectively. Also, suppose per unit cost of smuggling is constant and equal to $c$. If  $\hat{p}_c - c > \hat{p}_v$, then no firm will supply in Venezuela because it is more profitable to smuggle to Colombia.
Amit
CC BY-SA 3.0
2017-01-25T15:48:32.143Z
Thank you! How could I model smuggling between these two economies? It must be relatively simple.
John B
CC BY-SA 3.0
2017-01-25T17:14:37.927Z
Check the answer. I have posted a simple model.
Amit
<supply-and-demand>
Supply/Demand of Smuggling
John B
# Supply/Demand of Smuggling

(Asked by: username_0 on 2017-01-21)

I am attempting a (very) basic analysis of smuggling from one economy to the other (Venezuela to Colombia) - arising from price ceilings *ceteris paribus*. e1 are the equilibria without intervention (price controls), e2 are the equilibria after price ceilings, and e3 show what I assume the markets would look like if Venezuelans supplied Colombia (by smuggling) where the Venezuelans' profit =

[Colombia p-hat - cost of smuggling - Venezuela p-hat]

The supply curve for Colombia becomes the sum of the Venezuelan and Colombian supply curves above the [Venezuelan price ceiling + cost of smuggling], while there is a movement *along* the Venezuelan supply curve to the point of [Colombian price ceiling - cost of smuggling].

Is this correct?

Thanks!
[![enter image description here](https://i.sstatic.net/93oyL.png)](https://i.sstatic.net/93oyL.png)
[![S-D Graph](https://i.sstatic.net/g6kOP.png)](https://i.sstatic.net/g6kOP.png)

**Tags:** supply-and-demand

> **Comments:**
> **username_0**: After more consideration, I believe that my results require two assumptions. (1) That there are not any domestic 'black markets'. That is, all domestic trade is legitimate at the price ceiling, and (2) All smuggling from Venezuela is coordinated by Venezuelan suppliers. That is, no consumers are purchasing and reselling over the border - rather, suppliers hire people to deliver and sell in Columbia for a wage returning the revenue to the supplier. Like pizza delivery.  
> **username_1**: Suppose $\hat{p}_c$, $\hat{p}_v$ are price ceilings in Colombia and Venezuela respectively. Also, suppose per unit cost of smuggling is constant and equal to $c$. If  $\hat{p}_c - c > \hat{p}_v$, then no firm will supply in Venezuela because it is more profitable to smuggle to Colombia.  
> **username_0**: Thank you! How could I model smuggling between these two economies? It must be relatively simple.  
> **username_1**: Check the answer. I have posted a simple model.  

---

## 1 answers

### Answer by username_1 on 2017-01-25. Score: 4 | Accepted answer

You can consider a single competitive firm model with quadratic cost function say $c(y) = \frac{y^2}{2}$. This firm is in Venezuela. Let $\hat{p}\_c$ and $\hat{p}\_v$ be price ceilings in Colombia and Venezuela respectively. Also, $\hat{p}\_c > \hat{p}\_v$ indicating the opportunity to smuggle. Also, assume demands are sufficiently high so that the ceilings are binding. Consider competitive suppliers' problem without smuggling:

\begin{eqnarray\*} \displaystyle\max\_{y\geq 0} && p\_vy - \frac{y^2}{2}\end{eqnarray\*}
Solving it, we get the supply curve as $y = p\_v$.

Given that the price in Venezuela is $\hat{p}\_v$, equilibrium quantity in this case will be $y^\* = \hat{p}\_v$.

With an option to smuggle with the additional cost say $c(z) = \frac{z^2}{2}$, the profit maximization problem is:
\begin{eqnarray\*} \displaystyle\max\_{y\geq 0, z \geq 0} && p\_vy + p\_cz - \frac{(y + z)^2}{2} - \frac{z^2}{2}\end{eqnarray\*}

This gives us the new domestic supply function as:
$y = 2p\_v - p\_c$

and the quantity smuggled as
$z = p\_c - p\_v$

So, equilibrium quantity sold in Venezuela has fallen to
$y^{\*\*} = 2\hat{p}\_v - \hat{p}\_c$ and the quantity smuggled is equal to $z^{\*\*} = \hat{p}\_c - \hat{p}\_v$.

> **Comments:**
> **username_0**: Thank you! I will try to graph with supply/demand curves. Equilibrium quantity sold in Colombia would decrease? "So, equilibrium quantity sold in Colombia has fallen to.."  
> **username_1**: I just mixed up the notation. I have fixed it now. The equilibrium quantity consumed in Venezuela has fallen. The quantity smuggled to Colombia will only raise the quantity consumed in Colombia.  

---

