Everyone Keeps Getting Becker's Crime Model Wrong

In a recent JPE article, Steven Levitt claims that Gary Becker's 1968 paper on crime makes predictions that are at odds with reality. Specifically, he claims that Becker's model predicts that the most efficient way to deter criminals is by combining a low probability of punishment (p) with an extremely severe penalty or fine (f) when a criminal is caught. Obviously, no developed country has a criminal justice system that functions this way. So, either the whole world is wrong or Becker was mistaken. Right?

I really respect Levitt, but this just isn't so. 

Becker's predictions depend on the risk preferences of criminals. If criminals are risk averse (or risk neutral), then Levitt is right that the optimal policy is to have a very low p and a very high f. To see why, suppose you start off with the opposite policy--a high p and low f. What happens if you decide to cut p in half and double f? The expected value of the penalty stays the same (pf = (p/2)*(2f)), but crime becomes more risky because you have increased the variance of the outcomes. On the one hand, this is bad news for risk averse criminals because they receive less utility from riskier crimes (see Figure 1). On the other hand, this is good news for taxpayers because imposing penalties is typically cheaper than trying to catch criminals. In other words, you are deterring criminals more and at a lower cost. If you keep increasing f and decreasing p, you will find that the cheapest way to deter risk averse criminals is to have a very low p and very high f. Just like Levitt said!

Unfortunately for Levitt, Becker did not think that criminals are risk averse. Instead, he spends a good chunk of his 1968 article arguing that criminals are risk lovers. In that case, making crime riskier actually INCREASES their incentive to commit crimes (see Figure 2). So, having a very low p and very high f is no longer the optimal policy. Becker goes on to argue that actual US policy seems consistent with the implications of his optimality analysis. In other words, Becker argues the exact opposite of what Levitt says.

Levitt is not the first person to mischaracterize Becker's paper in this way. In 2015, Alex Tabarrok wrote a blog post making a similar argument. Tabarrok's post was later boosted by Tim Worstall and Noah Smith. It is a shame that this keeps happening, especially in places like the JPE! It leaves the impression that Becker's paper is inherently flawed, possibly not worth reading. In reality, it is actually a good example of how to apply theory to understanding real-world problems.

Figure 1. Making Crime Riskier Deters Risk Averse Criminals

Following Becker (1968), these figure assumes the following. If a criminal is not caught, they get to keep all of the income they "earned" (Y). If they are caught, they have to pay some penalty or fine (f), leaving them with (Y-f). The probability the criminal is caught is p. In this figure, I illustrate the impact of increasing f from f to 2f and decreasing p from p to p/2 on a risk averse criminal.


Figure 2. Making Crime Riskier Encourages Risk Loving Criminals

Following Becker (1968), these figure assumes the following. If a criminal is not caught, they get to keep all of the income they "earned" (Y). If they are caught, they have to pay some penalty or fine (f), leaving them with (Y-f). The probability the criminal is caught is p. In this figure, I illustrate the impact of increasing f from f to 2f and decreasing p from p to p/2 on a risk loving criminal.

Resolving a Cocaine Paradox with Derived Demand

Earlier this year, Tom Wainwright appeared on Russ Robert's EconTalk to discuss his new book, Narco-nomics. This book is about the economics of the drug trade. During their conversation, Wainwright described how governments in countries like Colombia eradicate millions of acres of coca leaf crop every year as part of the "war on drugs." The idea behind this policy is that by making coca more expensive, we will also make cocaine more expensive since it is the drug's key ingredient. However, to the chagrin of policy makers, the price of cocaine has not risen much (if at all).

Wainwright's explanation for this seeming paradox is that 1) the price of coca represents a small portion of the price of cocaine (less than 1%), and 2) drug cartels have market power that allows them to negotiate lower prices with coca leaf growers. These both sound like good reasons to me, but I think Wainwright maybe forgetting one other reason that output prices might not be rising along with input prices. Specifically, Roberts and Wainwright carry on their conversation as if the price of all other inputs into cocaine production stayed the same in the face of coca eradication. But why should we expect that?

Coca seems to have no substitutes in the production of cocaine. So all other inputs should be complements in the production process. That means an increase in the price of coca will lead cocaine producers to use less coca and less of all other inputs. As cocaine producers purchase less of these other inputs, the price of these other inputs will fall to clear their respective markets. As a result, the price of coca goes up, the price of other inputs goes down, and the price cocaine will increase by less than the price of coca (possibly much less if the supply of other inputs is very price inelastic).

Since the "other inputs" used in cocaine products beside coca leaf include the violent aspects of the drug trade, I wonder if this analysis implies that those services would be in less demand? If so, maybe coca eradication at least makes trafficking less violent? I kind of doubt it, but it is something to think about.

Anyways, if you want to think about this some more, you can do so more formally using the derived demand model that I explored in my last post. Here's a quick visual representation of the analysis above using the derived demand model. For simplicity, I drew this assuming you need 1 unit of coca and 1 unit of "other services" to make 1 unit of cocaine. As you can see, the price of coca goes up, the price of other inputs goes down, and the price of cocaine goes up by less than the price of coca.

Primer on Deriving Demand for Inputs in a Fixed Proportion Production Process

The demand for inputs in the production of final goods is ultimately derived from the demand for the final products themselves, which is why input demands are sometimes called "derived demands." This relationship can sometimes be lost in all the math surrounding modern textbook treatments. I think this is why it is best to introduce students to the concept of derived demand using an example where an industry uses a fixed-proportion production process. Here the math is so simple that it doesn't get in the way of the economics of how output markets influence input markets (and visa versa).

However, few modern textbooks discuss this special case (exceptions include Friedman's Price Theory and Becker's Economic Theory). I think that is a shame. So, I thought I'd write a short primer on deriving an industry's demand for inputs into a fixed proportion production process. First, I provide an intuitive explanation for how to derive the inverse demand for an input using Alfred Marshal's famous knife manufacturing example. Second, I provide a formal discussion of how to derive the elasticity of input demand. Third, I show how this simple example illustrates Marshall's four laws of derived demand. Lastly, I provide some links to additional reading.

There is nothing original here. I am basically just summarizing some old notes that I wanted in one place. Hopefully someone besides me finds them useful.

1. Knives, Blades, and Handles
Suppose that knives are produced using a fixed proportions technology. Specifically, one handle and two blades are combined to create one knife. Figure 1 illustrates the demand curve for completed knives and the supply curves of each input (note that the Pb represents the price of two blades). 

Figure 1. Demand Curve for Final Product and Supply Curves of Inputs

Source: Friedman (1972)

So, how do we derive the demand for one input like handles? Well, let's think about what each curve is telling us. The demand curve for knives shows the most that consumers are willing to pay for a given quantity of knives. Similarly, the supply curve for each input shows the least that suppliers would have to be paid to provide a given quantity of that input. Thus, the most that knife producers would be willing to pay for a given quantity of handles is the difference between the demand for knives and the supply of blades (see Note XIV in Marshall's Mathematical Appendix). Put another way, the "demand price" for handles equals the "demand price for knives" minus the "supply price for two blades":

Figure 2 illustrates the derived demand for handles. Note that no handles are purchased when the price for knives equals the supply price for blades. This is because blades are so expensive at that level of output there is no money left over for handles.

Figure 2. Derived Demand for Handles

Source: Friedman (1972)

2. Deriving the Elasticity of Demand for Handles

We can use the inverse demand function for handles above to derive the elasticity of demand. I provide the details here, but the ultimate result is:

This formula can be useful in applied settings. For example, the EPA used this formula to calculate the elasticity of demand for small, stationary combustion engines (a key input in the product of irrigation equipment among other things) when considering adding regulations on that industry (see page 4-2).

3. Marshall's Laws of Derived Demand

This expression for the elasticity of demand illustrates several of Marshall's 4 laws of derived demand.

1. "The demand for anything is likely to be more elastic, the more elastic is the demand for any further thing which it contributes to produce." (Note that as the elasticity of demand for knives increases, the elasticity of demand for handles increases).
2. "The demand for anything is likely to be more elastic, the more readily substitutes for the thing can be obtained." (Not illustrated here because there is no substitutes for handles)
3. "The demand for anything is likely to be less elastic, the less important is the part played by the cost of that thing in the total cost of some other thing, in the production of which it is employed." (Note that as Ph/Pk decreases, the elasticity of demand also decreases)
   
4. "The demand for anything is likely to be more elastic, the more elastic is the supply of co-operant agents of production." (Note that as the elasticity of supply for blades increases, the elasticity of demand for handles also increases)

Side Note: Hicks later showed that Marshall's third law only holds if the elasticity of final demand is greater than the elasticity of substitution. An intuitive explanation by Saul Hoffman for why this is the case can be found here. However, we don't need to worry about that special case with fixed proportion technologies because the elasticity of substitution across inputs is zero. So the condition will always be satisfied.

4. Additional Reading

For more info on this topic, I'd recommend checking out these resources:

• Becker, Gary. Economic theory. Transaction Publishers, 2007.
• Diewert, W.E. "A Note on the Elasticity of Derived Demand in the N-Factor Case," Economica (May 1971): 192-198.
• Friedman, Milton. Price theory. 1972.
• Muth, R., "The Derived Demand Curve for a Productive Factor and the Industry Supply Curve," Oxford Economic Papers 16 (1964): 221-234
• Hoffman, Saul D. "Revisiting Marshall's Third Law: Why Does Labor's Share Interact with the Elasticity of Substitution to Decrease the Elasticity of Labor Demand?." Journal of Economic Education 40, no. 4 (2009): 437-445.
• Thurman, Walter N. "Applied general equilibrium welfare analysis." American Journal of Agricultural Economics 73, no. 5 (1991): 1508-1516.