Queens University at Kingston

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A. Introduction

  1. To motivate the simple linear regression model LRM, we will discuss a simple economic policy question:
    Would an increase in the minimum wage increase unemployment among low-wage workers?
  2. Let's put this question in the context of demand and supply of labor. The supply curve of labor is the number of people willing to work as a function of the market wage. From Econ 110, the location of the curve depends upon people's taste for work, prices of other goods, etc.

  3. The demand curve for labor is the number of workers firms are willing to hire as a function of the market wage. The location of the curve depends upon technology, prices of other factors of production, etc.

    The competitive equilibrium looks like the following:

  4. The result of the model is an equilibrium wage w* and an equilibrium employment level N*. People who aren't willing to work at w*, but are willing to work for some wage, would be unemployed or in school or just hanging out.

  5. A minimum wage $w_m$ acts as a price floor within this simple model. If the minimum wage is not binding $(w_m \ge w*)$ , then it should have no effect on employment or unemployment. A binding minimum wage has the following effect:

    (Note: the upward sloping line should be labeled Labour Supply.) The result is a lower level of employment (Nm), or higher rate of unemployment. Of course, not everyone works at a minimum wage. We need to interpret this as the market for low-skill (low-wage) workers, such as teenagers who have dropped out of high school.

  6. With a log-linear demand curve we would write:


    $$\ln N^{D} = \beta_1 + \beta_2 \times \ln w + u\eqno{D1}$$
    where u captures determinants of labour demand: technology, output prices, prices of other factors of production, etc. $N^D$ is the quantity demanded written as a function of the market wage w. (You should refer to your intermediate microeconomics textbook or notes to remind yourself that a Cobb-Douglas production function generates a demand function of the form (D1).)

  7. The supply curve is irrelevant to the market outcome, because the price floor means that employment is determined by the supply curve. In other words, when the price floor (mininum wage) is binding, then the observed employment rate N is equal to the quantity demanded: N = $N^D$ . If the price floor is not binding, then N is found at the intersection of labour demand and labour supply. Even if they are both simple log-linear curves, the market outcome N would demand upon four parameters. With a price floor, we can focus attention upon one equation and two unknown parameters.
  8. The slope of the demand curve (graphing w on the vertical axis) would be $1/\beta_2$ . Since the values of $\beta_1$ and $\beta_2$ depend upon unobservable things such as production functions, these values are unknown constants or perhaps population parameters. (Even if we assume that we know the form of the demand curve we aren't so bold as to assume the slope and intercept.)
  9. How much $N^D$ (and through it unemployment) responds to the minimum wage depends upon the slope of the demand curve:
    $$ {\partial ln N^D \over \partial \ln w_m} = -\beta_2.$$
    Recalling the definition of supply elastiticity, we see that $\beta_2$ is the elasticity of the demand curve. Economic theory predicts that $\beta_2 \le 0$ (downward sloping demand curve). So since our initial question concerns a change in the mininum wage we can focus on not two unknown parameters but only one unknonwn parameter. We therefore can re-form our initial question into a much easier question:
    How big is $\beta_2$ in absolute value?
  10. If we look across labor markets (states, provinces, over time), then we will get several pairs of minimum wages $w_i$ and employment rates $N_i$ , where $i$ is an index the different labor market. According the demand-curve model, the relationship between the two values depends upon the other factors operating in the market, which we might call $u_i$ .
  11. For example, differences in the local wages commanded by high skilled workers (prices of other factors of production) would shift the demand curve for low skilled workers. Data from labour markets that have different minimum wages might therefore look like the following:

    In other words, the slope of the supply curve causes the cloud of observations to be downward sloping as long as the influence of other factors is small relative to the slope. But if the variation in $u_i$ is relatively large (or the variation in mininum wages is relatively small) the cloud of points might appear flat even though $\beta_2$ is less than zero.

  12. To answer even our simplified question, we would need to answer another question first:
    Using sample information, can we isolate the slope $\beta_2$ from the other factors that disturb the relationship between N and w?
  13. The change in other factors of demand across observations in the sample implies that we can't precisely measure the parameters $\beta_1$ and $\beta_2$ from data. Instead, we must estimate the unknown population paramter $\beta_2$ from data on a sample of employment rates and minimum wages.
  14. Suppose we have somehow carried out this estimation. Let the estimate of $\beta_2$ be called $\hat\beta_2$ . Based on $\hat\beta_2$ , it is simple to form the prediction:
    $$ \hat{\%\Delta N^D} = \hat\beta_2 \times \%\Delta w_m Change$$
    If a politician told you how much the mininum wage was going to change, you could could plug into this formula and predict for him/her the percentage change in $N^D$ . So the easy part of the job is carrying out the prediction. The hard part is coming up with an estimate of $\hat\beta_2$ in the first place.
  15. Because your estimate $\hat\beta_2$ is a random variable (it depends on sample information which includes the effect of the disturbance factor u), your prediction $\hat{\%\Delta N^D}$ is also a random variable. Your prediction about the effect of changing the mininum wage is itself an estimate. You, and most definitely your client or audience, will ask the further question:
    How precise is a prediction based on the estimate $\hat\beta_2$ ?
    Wanting to know how precise your estimate is trying to infer something about the actual change based on your prediction. You might be asked to form a confidence interval for the actual employment response. Or, you might be asked to test the hypothesis that there is no employment response, which is the same as testing the hypothesis that $\beta_2$ is zero.
  16. Summary
    • We begin by asking a specific policy question about the effect of raising the mininum wage.
    • Using basic economic theory and some simplifying assumptions (like log-linear demand), we reduced the question to asking a specific numerical question about one theoretical construct, the slope of the demand curve for low wage workers.
    • We then realized that finding this value requires us to ask a specific statistical question about estimating a slope from a cloud of points.
    • If we are successful in carrying out this step, then we are lead to ask a specific inferential question about how precise our estimate is.
    • The goal of econometrics
      To use statistical techniques to estimate and test economic models using economic data.
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