Labour Supply Part 2 PDF

Labour Supply part 2 Constrained optimisation, Lagrangean method, the primal and dual problem and estimation  Constrained Optimisation This involves constructing a new function, a Lagrangean function, that incorporates the constraint. In order to do so we include an extra paramet...

Labour Supply part 2 Constrained optimisation, Lagrangean method, the primal and dual problem and estimation  Constrained Optimisation This involves constructing a new function, a Lagrangean function, that incorporates the constraint. In order to do so we include an extra parameter known as the Lagrange multiplier, λ. We then proceed as we would in an unconstrained problem, by taking the derivatives of the function and setting them equal to zero  Equivalence with substitution As long as the constraint is satisfied with equality, then both methods will give the same results. We have made assumptions regarding our utility function, for example non-satiation, which ensures that that is the case.  Lagrangean method The Lagrangean is simply a linear combination of the objective function and the constraint. It is given by To maximise we take derivatives with respect to C, L and λ.  First order conditions These 3 equations can be solved for the optimum values of C, L and λ  Interpretation We can solve the FOCs as follows On the LHS we have the MRS and on the RHS we have the slope of the BC. Note that (3) implies that all income is spent and referring back to the problem, this means that λ must be non-zero. There is a more general statement of this to check for slackness (remember that Jonathon discussed this with you). These are known as Kuhn Tucker conditions.  Interpretation Equation (4) above implicitly defines the optimal demand for leisure/supply of labour, as a function of the exogenous variables. If we knew the functional form of the utility function we could solve for this analytically and check the comparative static properties. (see if the effect of changing the wage etc is what we found in the diagrams from the first lecture)  Too much maths It is possible to sign comparative static effects, just using the results above, as long as we know the signs of the partials derivatives. We could then use the matrix techniques that you learned, for example Cramer’s rule. We will leave this for another time.  Worked Example Let Utility be given by The time constraint is given by 16-L=h, we’ll let the person sleep for 8 hours This gives the budget constraint as wh+N=w(16-L)+N=pC  The Lagrangian method Maximise subject to the time constraint w(16-L)+N=pC The Lagrangian First order conditions  Note that (2)÷(1) gives The LHS is the MRS, the RHS is the slope of the budget constraint. We can solve (4) for either C or L and use this in the BC (3). Or  Which gives Let’s put in some numbers and draw a diagram. Suppose that N=40, w=10, p=2, this gives  Diagrammatically C 50 10 16  Now let’s play Comparative statics is just a posh or confusing way of saying ‘what happens to this value if I change that value’. So let’s find out what happens to labour supply when we change the value of w, N and p. Since , p has no effect. What do you think about that, should it appear or not?  So let’s play with w and N. , keep w=10 and let N increase by 10  What do we notice? The most obvious thing is that each time non-labour income increases by 10, leisure demand increases by 1/2 hour and labour supply therefore falls by the same amount. If we continued to increase non-labour income it is fairly obvious that at some point leisure demand would exceed 16 and labour supply would become negative.  Let’s investigate this. We know that the slope of the IC is given by MRS=C/L. Let’s work out the corresponding values for C  Now we can compute the MRS The MRS will always equal the slope of the budget constraint, which is w/p=5. Note that if we hold L=10 but allow C to change optimally, the MRS at this point is greater than 5, eg. At N=50 optimal consumption is 52.5, if we hold L=10, then the MRS is 52.5/10. Let’s see this  N=40 to N=50 C The increase in non-labour income shifts the budget constraint out. Notice that if we constrain the agent’s choice at L=10, h=6, the indifference curve will not be 50 tangent to the budget constraint. At h=6, wh=60. The increase in NLI results in an increase in the demand for 20 both consumption and leisure; see the utility functions, the 10 16 goods are imperfect substitutes/complements  Changing the wage Notice that in this case the leisure demand function does not allow for a backward bending supply curve for labour. Let’s see that. It should be obvious that as the value of w increases, the value of L falls. You should be able to show this and you can do so either by using calculus or numbers.  Calculus: this will show the slope of the relationship As the wage is always positive, this result is unambiguous, an increase in the wage reduces the demand for leisure and increases labour supply.  Using some numbers We have that N=40 and w=10 initially. Hold N=40 and let w=8 and w=12. In this case we will have  Things to note You should have noted that the slope is not constant. In the above example we changed the wage by two units either side of 10. On one side the change was a 0.5 and on the other 0.33. This should not be a surprise, you have seen that type of relation before, it is an asymptote.  , as the value of w goes to infinity, the term approaches zero and leisure demand approaches the value 8. As w approaches zero, the term gets bigger and bigger and approaches infinity, which means that L also approaches infinity. Using excel, we can graph this  This is the full curve, with wage varying from 0.1 to 99. Leisure demand: Wage start 0.1 You can see the 250 shape of leisure 200 demand and the 150 100 asymptote 50 0 0 20 40 60 80 100 120  In this graph we have removed the very low values for w to show the asymptote at 8 more clearly. Leisure demand: Wage start 2.5 20 18 16 14 12 10 8 6 4 2 0 0 20 40 60 80 100 120  The next graph shows the labour supply function. Notice that this function is monotonic, it is non-decreasing. Labour supply: wage start 2.5 This is at variance 9 8 7 with the concept of 6 5 a backward bending 4 3 supply curve. 2 1 0 0 20 40 60 80 100 120  Income and substitution effect To analyse the income and substitution effect, we can make use of a Slutsky decomposition. This can be written as Recall that the substitution effect holds utility constant, this is the first term. The intuitive rationale for calculating the income effect in this way is that the effect of a wage change on income depends on the number of hours sold h and the effect of that on labour supply is given by  an increase in non-labour income reduces labour supply (increases leisure) this is positive, as the wage increases, labour supply increases and leisure demand falls  The income effect is given by = where we have used the solutions thus far The total effect we found as so the substitution effect can be calculated from the Slutsky equation  implies or from our results This is greater than zero as it should be (negative for leisure demand)  The backward bending supply curve The issue that we have is that if we start from a utility function and derive analytical solutions in the manner above, the resultant leisure demand functions will not exhibit the non-linearities necessary to result in backward bending supply. There are two possibilities for addressing this  Ad hoc specification In specifying the model explaining labour supply we can include a term that allows for backward bending suplly. The simplest way to accomplish this is to estimate a model of the following type This includes both a linear and quadratic term in wages  Effect of wages on hours A backward bending supply curve requires that this be positive over some range and then negative. This requires some sign restrictions.  We need and. The issue is that this does not arise from underlying economic theory of utility maximisation. This is a common issue in micro- economics and has been the subject of a great deal of research in demand models. The solution, which is too technical for our purposes, is outlined in the next slide.  Duality Microeconomic theory makes use of the duality between maximisation of one objective being the same as the minimisation of another; for example a consumer may maximise utility subject to a budget constraint but may also minimise expenditure subject to some utility constraint. This duality can prove very useful in econometric modelling.  The reason for this is that whilst it is not difficult to derive analytic solutions from maximisation problems, the range of utility functions for which this can be achieved is restrictive. Some results of duality (Roy’s identity, Shephard’s Lemma) mean that it is not necessary to start from utility maximisation; estimable functions can be derived using these results.  Example Roy’s identity allows us to derive the Marshallian demand for a good from an indirect utility function. An indirect utility function is the maximised value that can be achieved given some parameters, such as the wage, prices etc. For our example above it is the utility function that we derive when we substitute in the optimal values for C and L.  Indirect utility function For the utility function , when we maximised this subject to our constraint, we derived optimal demand for leisure and consumption as  Indirect utility function The indirect utility function is thus Which upon rearranging gives This is the maximised level of utility.  Roy’s identity Roy’s identity The Marshallian demand for a good () is given by , where M is money income and is the price of good. For consumption in our example we would need , where N is non labour income  Example This can be quite tedious but is worth doing at least once. It is however the intuition that we are after here. What does this mean? This looks pretty ugly and is. But let’s persevere  Get some common denominators This makes things a little easier. Now we can bring everything together inside the brackets  Cancel It helps if you know what 16 squared is, and factor the denominator  Cancel again This confirms Roy’s identity. Now why is this so important? Because as long as the indirect utility function has some special properties (we won’t go into them here, leave that to another time) the indirect utility function CAN BE very general.  Using Roy’s identity we can derive much more interesting labour supply functions than we could if we adopted the primal approach. This is incredibly important because our models should be derived from optimising behaviour. If our models are ad hoc, then how does that relate back to falsification?  Reading This is described in detail in Stern, N. (1986), “On the specification of labor supply functions” in R.W. Blundell and I. Walker (eds.), Unemployment, Search and Labour Supply, Cambridge University Press, 121-42. Which you can find on the canvas pages  Other Estimation issues Mass points: these occur in the distribution when we have a significant number of observations at a single point. In labour supply we typically solve for optimal number of hours. In practice, jobs are normally associated with a set number of hours. In survey data we may find that mass points exist at various parts of the distribution.  Examples Mass point at 0: we cannot observe optimal choices at this point as discussed above. Mass point at 35 hours per week: If the question asks the normal number of hours worked, many jobs require 35 hours per week.  As you should know from above diagrams, the optimal choice for these individuals would be negative hours of labour supply. It is important to treat these observations correctly. Why?  TOBIT ANALYSIS Y* 40 Y *  40  1.2 X  u 30 20 10 0 0 10 20 30 40 50 60 X -10 -20 -30 -40 For example, suppose that we have a labor supply model with Y hours of labor supplied per week as a function of X, the wage that is offered. It is not possible to supply a negative number of hours. 4  TOBIT ANALYSIS Y 40 Y *  40  1.2 X  u 30 20 10 0 0 10 20 30 40 50 60 X -10 -20 -30 -40 Those individuals with negative Y* will simply not work. For them, the actual Y is 0. 5  TOBIT ANALYSIS Y 40 Y *  40  1.2 X  u 30 20 10 0 0 10 20 30 40 50 60 X -10 -20 -30 -40 What would happen if we ran an OLS regression anyway? Obviously, in this case the slope coefficient would be biased downwards. 6  TOBIT ANALYSIS Y 40 Y *  40  1.2 X  u 30 20 10 0 0 10 20 30 40 50 60 X -10 -20 -30 -40 Here is the sample with the constrained observations dropped. 9  TOBIT ANALYSIS Y 40 Y *  40  1.2 X  u 30 20 10 0 0 10 20 30 40 50 60 X -10 -20 -30 -40 An OLS regression again yields a downwards-biased estimate of the slope coefficient and an upwards-biased estimate of the intercept. We will investigate the reason for this. 10  Other issues Mass points at kink points: The way that we have presented this so far, we have assumed zero percent income tax. If we include the tax system in our model, then this greatly complicates the analysis. When we reach a tax threshold we need to consider whether to work more hours, thus passing through that threshold, or to remain at the kink point.  Kink point Diagrammatically C At h* hours the wage is subject to a tax rate, the slope of the BC is then w(1-t) rather than w. it is likely that there will be clustering at this number of hours and this causes problems for estimation. Not least is the issue that we will need to control for the fact that these are not optimal choices h* h  Kink point Diagrammatically C At h* need to work out what the optimal h would have been without the kink by extending the upper net wage rate back to Non labour income. This greatly increases the difficulty in estimating these models h* h  Sample selection bias Violates the independence assumption of OLS. Sample selection issues: We assume that our sample is made up of independently, identically distributed random variables. People choose to work or not work If this choice is non-random, eg determined by some deterministic factors, then the sample of workers is a non- random drawing from the population  This lead to bias in our estimated effects E.g. classic paper is Mroz, The Sensitivity of an Empirical Model of Married Women's Hours of Work to Economic and Statistical Assumptions, 1987, Econometrica 55(4):765-99 http://unionstats.gsu.edu/9220/ Mroz_Econometrica_LaborSupply_1987.p df  Reading: Mroz, Thomas A. (1987) The Sensitivity of an Empirical Model of Married Women's Hours of Work to Economic and Statistical Assumptions Econometrica, 55(4), 765-799  Sample Selection Bias Problem: interested in estimating return to education say. Estimate simple Mincer equation 𝑙𝑤𝑎𝑔𝑒𝑖 = 𝛼 + 𝛽1𝑒𝑑𝑢𝑐𝑖 + 𝛽2𝑒𝑥𝑝𝑒𝑟𝑖+𝑢I Mincer, J. (1958) Investment in Human Capital and Personal Income Distribution, Journal of Political Economy, 66, 281-302. https://www.jstor.org/stable/1827422? seq=20#metadata_info_tab_contents  In this simple equation the returns to education are 10.9 %. The “returns to experience” are 1.5 %. Both estimations are statistically significant (see for instance the p-values associated with the t-statistic values)  Data come from the 1975 wave of the Michigan panel study of income dynamics (PSID). ◦ Describe the main variables in the workfile.  Employment status variables.  Hours of work.  Wages.  Other variables: education…….  Estimated effects are sensitive to the specification of the model. If we omit variables from our wage equation, (or use an incorrect functional form, linear rather than semi log) then the estimated coefficients from our model will not reflect the true, underlying relationship. Why?  The bias in a simple model is as follows So the effect of the included variable can be over-estimated or under-estimated  Sample selection When we have sample selection (the sample is not a random sample of our target population) then the bias occurs when the variable affects both the dependent variable and also the probability of being in the sample.  Example: Education As an example consider the education level of a married woman. More educated women are more likely to be in the work force, so more likely to be in our sample. More education is also likely to be related to wages; higher wages on average. If we estimate the model ignoring this fact, the estimated return is likely to be overstated  A note on estimation Estimating these models involves a technique known as maximum likelihood estimation. Least squares estimates are calculated by fitting a regression line to the points from a data set that has the minimal sum of the deviations squared On the other hand the likelihood function indicates how likely the observed sample is as a function of possible parameter values. Therefore, maximizing the likelihood function determines the parameters that are most likely to produce the observed data.  Gauss-Markov and maximum likelihood When the gauss-markov assumptions are satisfied, Least Squares and Maximum Likelihood result in the same estimators. In the case that we are considering here our model is no longer a simple linear model. We have to jointly estimate both the wage or hours equation and also the probability of being in the sample. This requires maximum likelihood