Monday, March 16, 2026

Misinformation From Economists

I have found another source of economists confidently spouting mistakes, Economics Stack Exchange. This has been around for more than a decade.

If I went back in time, I think I would have trouble convincing my 20 year old self that standard introductory textbooks are incoherent nonsense, never corrected.

I quickly found questions on the Cambridge Capital Controversy. What technology do we need to have reswitching to occur? Why is reswitching and reverse capital-deepening a problem exactly? Why did the Cambridge Capital Controversy have no impact on economic modelling? The participants do not seem to have much to say on the topic.

Ten years, ten moths ago, a question was posed: Can capital still be paid its marginal product in the absence of a homogeneous capital stock? This question was inspired by a Krugman answer to critics of Piketty. One answer was offered:

Different sorts of capital used as separate production technologies prevent clean aggregation to a representative form of capital but does not prevent capital from being paid its marginal product...

On the margin the two sorts of capital don't have the same product and so aggregation doesn't make sense here. But in this setting, it is likely that the rental rate on capital would be equated (r1 = r1) because why would you buy one sort of capital when the other sort paid more?

The answer is foolish. The variables are supposed to be "rental prices". They might be in units of numeraire units per year per services of ton iron and numeraire units per year per services of square meters of the services of steel sheets, where the latter are of a specified thickness. You could change their values by a change in units. For example, the latter could be in square yards, not square meters. So it makes no sense to equate these values.

I suspect I can find more confusion.

Friday, March 13, 2026

Eurocommunism and Communist Parties In Coalition Governments In Europe

Eurocommunism was a tendency in communist parties in Europe during the 1970s. The Soviet suppression of the Prague Spring cast communist parties in Western Europe in a bad light. How could they follow Moscow's ead after that? So they started articulating their own path and asserted their independence from the Soviet Union.

This tendency was a moderating tendency. Ernest Mandel, a follower of Trotsky and therefore a critic of Stalin, decried this tendency. He called Eurocommunism "the bitter fruits of socialism in one country."

Anyways, two instances of these "bitter fruits" stand out to me. One is the historic compromise, led by Enrico Berlinguer, the leader of the Italian Communist Party (PCI). This involved support for the Christian Democrats (DC). I guess that the PCI did not enter the government in the elections of 1976, but refused to vote against the DC on no-confidence votes in parliament. In some sense, the communists were to the right of the socialists, let alone the workerists outside of the parties.

Another case is Francois Mitterand, a socialist, who was elected president of France in 1981. He took the French Communist Party (PCF) into his governing coalition. The communists did not do well, being sort of domesticated.

That was a while ago. But take a look at Portugal. Antonio Costa, a socialist, was elected Prime Minister in 2015 and served to 2024. This was a coalition government, called the Left Bloc. The Portuguese Communist Party and the Greens were also coalition members. Costa is now President of the European Council. Since 2024, Portugal's Prime Minister is Luís Montenegro, head of a more right-leaning coalition. To confuse me, his party is the Social Democratic Party. I think the names of the parties suggests, to an American, that they are more left-wing than they are now. As of February, the Portuguese president is Jose Seguro, a socialist. I gather that his election was a matter of staving off the far right, in some sense.

So the history of socialism and communism includes typical parliamentary machinations, compromises, coalitions, and so on. To understand Portugal, I should know more about Salazar and the Carnation Revolution.

Monday, March 09, 2026

A Switch Point Along The Same Wage Curve With Multiple Agricultural Commodities

Figure 1: Wage Curves Around An Anomalous Switch Points

This post presents another anomalous switch point. A switch point is anomalous in that it has properties that cannot hold for a switch point in a model of single production, with inputs of labor and circulating capital alone.

This example is one with multiple agricultural commodities and intensive and extensive rent. The technology and the endowments of land are the same as in this example.

Required net output, that is, final demand, varies. I start by postulating that final demand consists of 28 bushels wheat and 28 bushels rye. Under this assumption, Alpha, Beta, Epsilon, and Lambda are feasible techniques.

The cost-minimizing technique at a given rate of profits must be:

  • Feasible.
  • Have non-negative prices for all commodities produced under the technique, have a non-negative wage, and have non-negative rents on all scarce lands.
  • Such that no process not operated under the technique obtains extra profits.

Epsilon is cost-minimizing up to a rate of profits of approximately 223.6 percent. A reswitching of the order of efficiency occurs over the range at which Epsilon is cost-minimizing. After the switch point, as illustrated in Figure 1, Alpha is cost-minimizing.

Figure 1 also illustrates a fake switch point at a rate of profits of approximately 219.0 percent. The wage curves for Alpha and Delta intersect at the fake switch point. The wage curve for Alpha is also the wage curves for Epsilon and Zeta. Likewise, the wage curve for Delta is also the wage curves for Eta and Theta. Epsilon is the unique cost-minimizing technique at and around this fake switch point. The prices of produced commodities (iron, wheat, and rye) differ, at the switch point, between the techniques for the two intersecting wage curves. In this sense, the fake switch point resembles the one in the example from Bidard and Klimovsky. The rent on type 2 land is positive under Epsilon, which would not be the case is this switch point was non-fake. Nor are the rents on type 1 land zero under Eta and Theta at this fake.

But consider again the switch point between Epsilon and Alpha. Under Alpha, only type 1 land is farmed, but only partially. Epsilon extends Alpha to produce wheat on type 2 land, to the extent of its endoment. The switch point lies along a single wage curve, which is anomalous.

Suppose that final demand was small enough that both Alpha and Gamma were feasible. For example, Alpha, Beta, Gamma, and Delta are the only feasible techniques when required net output consists of 10 bushels wheat and 10 bushels rye. Then Gamma is cost-minimizing from before the switch point, from approximately 176.8 percent. Alpha is cost-minimizing after the switch point. As with Epslion, under Gamma wheat is produced on type 2 land. But, unlike Epsilon, type 2 land is not farmed under Gamma to the extent of its endowment and the process in which wheat is produced on type 1 land is no longer operated. With this final demand, the example is one of reswitching between Gamma and Delta, at a lower rate of profits than shown.

Or suppose final demand consisted of 30 bushels wheat and 30 bushels rye. Then Beta, Epsilon, Iota, Kappa, and Lambda are feasible. Then this is a switch point between Epsilon and Iota. The same processes are operated under Epsilon and Iota, but which land is scarce varies. Iota is a technique in which landlords obtain intensive rent on type 1 land. The rent on type 1 land is negative under Iota before the switch point.

I have now found switch points:

The above switch point between Epsilon and Alpha combines these two phenomena, in some sense. I have also found fake switch points:

These results suggest that concept of a switch point is not tightly tied to intersections of wage curves in models of joint production.

Friday, March 06, 2026

Michael Overton Interviewing Margaret Wright On Operations Research

I do not know what order these should be in or even if this is the entire interview. I did not know of either Margaret Wright or Michael Overton before stumbling on this. Apparently she was once president of the Society of Industrial and Applied Mathematicians, worked with George Dantzig at Stanford, and so on.

I think I have not appreciated how much progress has been made in the last few decades:

"... but look back and see the progress we've made. So in optimization at various points people have said well, for nonlinear problems, you can solve problems with hundreds of variables. Of course, that's a very imprecise statement. They never say what kind of problems or whatever. Today we had a talk where the person was talking about hundreds of thousands or millions of variables. That's an amazing change. It's an amazing change in capability. And it's not all due to computer hardware. I think sometimes people think, oh, machines are faster; machines are bigger. But it's smarter and better algorithms, which is the area you and I work in, right? Not making faster machines. We try to take advantage of them, but we don't make them run faster. And to me, that's astonishing. And the great part of it, I think there's still interesting problems that don't have many variables in this non-derivative optimization area." -- Margaret Wright

I do not understand interior point methods or why the Simplex algorithm works so well on average.

Tuesday, March 03, 2026

Anomalous Switch Points

This post is to remind me that I have discovered some anomalous switch points. I am introducing the concept of an anomalous switch point here.

Consider a switch point in a model of single production, with inputs of labor and circulating capital alone. At a (generic) switch point, one process replaces another in an industry that produces a commodity that exists in both techniques. Other processes can be introduced or removed if some capital goods are used only in one of the two processes for the common industry. The switch point is on the wage frontier formed by the outer envelope of the wage curves for all techniques. Two wage curves intersect at the switch point.

An anomalous switch point differs in some property from a generic switch point in models of circulating capital alone.

The anomalous switch points under consideration here are not flukes. A fluke switch point is one in which any permutation of some parameters destroys the qualitative property under consideration for the switch point. A switch point in which two wage curves are tangent on the frontier is an example of a fluke. A switch point in which two processes are replaced, one in each of two industries that exist in the techniques with intersecting wage curves. In this example with both types of flukes, and more, four wage curves intersect at switch point for the second kind of fluke.

Anomalous switch points include:

  • A switch point in which the same processes are operated for both techniques: This example is one of extensive rent. The levels of operation of the processes and the land that is scarce vary around the switch point.
  • Another switch point in which the same processes are operated for both techniques: This example is one of intensie and extensive rent with multiple agricultural commodities.
  • A switch point along a single wage curve: This example combines fixed capital and extensive rent. Two techniques, Nu and Omicron, have the same 'solving system' and, hence, the same wage curve. The techniques differ in the economic life of machine when operated on scarce land that pays extensive rent.

Other switch points can be considered anomalous. D'Agata (1983) has examples with non-unique and non-exisitng cost-minimizing techniques in a model of intensive rent. Two techniques are cost-minimizing before a switch point. No cost-minimizing technique exists after the switch point. I think D'Agata's examples are more challenging to Sraffa's price theory than mine. Woods (1990) also has an example in joint production, without rent.

A non-fluke switch point in a model of joint production in whcih three wage curves intersect is another anomalous switch point. Bidard and Klimovsky have a genuine switch point like this in the paper in which they introduce the concept of a fake switch point. I think I also have examples in my models that combine intensive and extensive rent.

I do not consider anomalous a switch point on the wage frontier, in which the frontier is not the outer envelope of wage curves. The wage frontier must be the outer envelope in models of single production, but researchers in joint production established last century that this property need not hold in models of joint production, including in models of extensive rent.

Fake switch points can also be anomalous. With joint production, a question arises about which process should be replaced when a new process is introduced. A fake switch point, in a model of joint production, is an intersection of (non-tangent) wage curves in which the cost-minimizing technique does not change. The intersection's location and existence depend on the numeraire. The price of a commodity produced under both techniques whose wage curves intersect varies among techniques.

A fake switch point can be anomalous in that it deviates from properties of Biard and Klimovsky's example:

  • Example fixed capital and extensive rent: Prices of commodities produced under both techniques do not vary among techniques at switch points. Price and rent vary between techniques only for non-commodities (that are free) under the non-cost-minimizing technique.

Technical terms: Switch point, Anomalous switch point, Fake switch point, Anomalous fake switch point, Fluke switch point.

References
  • D’Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica 35: 147-158.
  • Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.
  • Vienneau, Robert L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy 36(2): 827-843.
  • J. E. Woods (1990) The Production of Commodities: An Introduction to Sraffa, Humanities Press International

Saturday, February 28, 2026

A Reswitching Example With Extensive And Intensive Rent And Multiple Agricultural Commodities

Figure 1: Wage Curves For Feasible Techniques
1.0 Introduction

This post demonstrates a novel aspect of the reswitching of techniques. The cost-minimizing techniques in the example do not differ in which processes are operated. They differ in which lands are fully cultivated and thus obtain rent. In one technique, two commodities are produced, by distinct processes on the type of land that is fully farmed. In the other, one process, producing one commodity, is operated on the land that pays rent. In other words, the techniques that reswitch pay extensive and intensive rent, respectively.

The reswitching example depends on more than one agricultural commodity being produced. When the technique with extensive rent is cost-minimizing, two processes are operated on type 2 land. Type 2 land is not fully farmed. Two processes producing the same commodity cannot be operated on non-scarce land, away from a switch point, when prices of production prevail.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. Iron is an industrial commodity, produced with no inputs from land. Two types of land are available for producing the agricultural commodities, wheat and rye. Wheat is produced by two processes, each operating on a different type of land. The same is true for rye. Inputs and outputs are specified in physical terms. For example, the inputs for process II, per bushel wheat produced, are 5/2 person-year, the services of one acre of type 1 land, 1/200 ton iron, 1/4 bushels wheat, and 1/300 bushels rye. Each process exhibits constant returns to scale (CRS), up to the limits imposed by the endowments of the lands.

Table 1: Processes Comprising the Technology
InputsIndustries
IronWheatRye
IIIIIIIVV
Labora0,1 = 1/3a0,2 = 5/2a0,3 = 7/20a0,4 = 1a0,5 = 3/2
Type 1 Land0c1,2 = 10c1,4 = 20
Type 2 Land00c2,3 = 50c2,5 = 1
Irona1,1 = 1/6a1,2 = 1/200a1,3 = 1/100a1,4 = 1a1,5 = 0
Wheata2,1 = 1/200a2,2 = 1/4a2,3 = 3/10a2,4 = 0a2,5 = 1/4
Ryea3,1 = 1/300a3,2 = 1/300a3,3 = 0a3,4 = 0a3,5 = 0
OUPUTS1 ton iron1 bushel wheat1 bushel wheat1 bushel rye1 bushel rye

The specification of the problem is completed by defining the available endowments of land and the level and composition of final demand. Accordingly, assume 100 acres of each type of land are available. Suppose the required net output, also known as final demand, consists of 8 bushels wheat and 60 bushels rye.

Table 2 shows the available techniques for these parameters. Land is free for techniques Alpha, Beta, Gamma, and Delta. Techniques Epsilon, Zeta, Eta, and Theta pay extensive rent. Intensive rent is obtained by landlords for Iota, Kappa, Lambda, Mu, and Nu. Under Nu, intensive rent is obtained on both types of land.

Table 2: Technique
NameProcessesType 1 LandType 2 Land
AlphaI, II, IVPartially FarmedFallow
BetaI, II, VPartially FarmedPartially Farmed
GammaI, III, IVPartially FarmedPartially Farmed
DeltaI, III, VFallowPartially Farmed
EpsilonI, II, III, IVPartially FarmedFully Farmed
ZetaI, II, IV, VPartially FarmedFully Farmed
EtaI, II, III, VFully FarmedPartially Farmed
ThetaI, III, IV, VFully FarmedPartially Farmed
IotaI, II, III, IVFully FarmedPartially Farmed
KappaI, II, IV, VFully FarmedPartially Farmed
LambdaI, II, III, VPartially FarmedFully Farmed
MuI, III, IV, VPartially FarmedFully Farmed
NuI, II, III, IV, VFully FarmedFully Farmed

Consider the Delta technique, for an example of relationships between these techniques. Under Delta, both wheat and rye are grown on type 2 land, but generally not to the limit imposed by the endowments of land. These same wheat and rye-producing processes are operated in Eta and Theta. In both of these techniques with extensive rent, type 2 land is still not farmed to the extent of its endowment. In Eta, wheat is produced on type 1 land to the extent of the endowment of type 1 land. In Theta, rye is produced on type 1 land to the extent of its endowment. But compare Theta to Mu. In Mu, the processes in Delta are also supplemented by growing rye on type 1 land, as in Theta. Type 1 land is not totally farmed, however, under Mu. Type 2 land is totally farmed to the extent of its endowment, with both wheat and rye contining to be produced in parallel on that type of land.

3.0 Quantity Flows and Feasible Techniques

The space of the final demand vector can be partitioned into regions by where each technique is feasible. Assume the final demand for iron is zero. Figure 2 shows a partition of the two-dimensional space for net output of wheat and rye. The point of final demand is indicated for the reswitching example in this post, with a final demand of 8 bushels wheat and 60 bushels rye.

Figure 2: Final Demands for Feasible Techniques

At the lowest level of final demand, Alpha, Beta, Gamma, and Delta are feasible. Land is in excess supply, and the model reduces to a model of circulating capital. As output expands, towards the specified final demand for the example, Delta becomes infeasible. At the limit of Delta's fesibility, net outputs of wheat and rye can be traded off. Lambda and Mu become feasible, with type 2 land obtaining intensive rent.

As final demand continues to expand, Alpha becomes infeasible. Here, too, at the limit for Alpha, the net outputs of wheat and rye can be traded off. Iota and Kappa are the other techniques that pay intensive rent, and they relate to Alpha in the same way that Lambda and Mu relate to Delta. The feasible techniques are now Beta, Gamma, Iota, Kappa, Lambda, and Mu.

As output continues to expand towards the specified point of final demand, Gamma becomes infeasible, with a constraint imposed by the endowment of type 1 land. Iota becomes infeasible, as well. Epsilon and Theta, which pay extensive rent, become feasible. For the reswitching example, Beta, Theta, Kappa, Lambda, and Mu are feasible.

4.0 Price Equations

A system of equations is defined for prices of production for each technique. I take Mu as an example.. Processes I, III, IV, and V are operated under Mu. Processes III and V combine to bring the entire endowment of type 2 land under cultivation. The prices of production for Mu, in obvious notation, satisfy the following equations

(p1 a1,1 + p2 a2,1 + p3 a3,1)(1 + r) + w a0,1 = p1

(p1 a1,3 + p2 a2,3 + p3 a3,3)(1 + r) + rho2 c2,3 + w a0,3 = p2

(p1 a1,4 + p2 a2,4 + p3 a3,4)(1 + r) + w a0,4 = p3

(p1 a1,5 + p2 a2,5 + p3 a3,5)(1 + r) + rho2 c2,5 + w a0,5 = p3

Each equation applies to a process operated under Mu. It is assumed that wages and rents are paid out of the surplus product, at the end of the period of production. The prices of advanced capital goods incur a charge for the rate of prices. A final equation equates the price of the numeraire to unity.

p1 d1 + p2 d2 + p3 d3 = 1

The above equations specify prices of production for Mu. A similar price system characterizes prices of production for each of the other techniques in the example.

The price system for Mu consists of five equations in six variables, r, w, p1, p1, p1, and rho2. The solution has one degree of freedom. If the rate of profits is taken as given, each of the other five variables can be expressed as a function of the rate of profits. The wage curve is a function of the rate of profits for a given technique. The rent curve is the corresponding function for rent.

The wage curves for the feasible techniques are plotted in Figure 3, at the top of this post. Wage curves are only shown for a technique for the ranges of the rate of profits in which rents are positive. Figure 3 shows graphs of the rent curves for the feasible technique. No rent is paid under Beta, and Beta is never cost-minimizing. Theta is cost-minimizing at high and low rates of profits. Mu is cost-minimizing for intermediate rates of profits.

Figure 3: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques

Assertions above about the ranges of the rates of profits in which Theta and Mu are cost-minimizing have yet to be demonstrated. The cost-minimizing technique at a given rate of profits is:

  • Feasible
  • Such that no price of a produced commodity, wage, rate of profits, or rent is negative
  • Such that extra profits cannot be obtained, at prices associated with the technique, by operating processes outside the technique

Extra profits exist when the difference between revenues and costs for a process is positive. Costs include a charge on the prices of the advanced capital goods for the given rate of profits, and the difference is taken for a unit level of operation of the process. Prices of production for a technique are such that extra profits are zero for the processes operated by that technique.

Figure 4 illustrates extra profits for Beta. Processes III and IV are not operated by Beta. Extra profits can always be obtained, under Beta, by operating process IV, whatever the rate of profits for which prices are found. Beta is never cost-minimizing.

Figure 4: Beta is Never Cost-Minimizing

Figure 5 shows that Theta is cost-minimizing whenever rent is positive for prices of production for the technique. The graph also makes obvious that the conditions, that rent be non-negative and that extra profits are not available, are independent of one another.

Figure 5: Theta is Cost-Minimizing at High and Low Rates of Profits

Figures 6 and 7 demonstrate that Kappa and Lambda are never cost-minimizing. Figure 8 demonstrates that Mu is cost-minimizing for the range of the rate of profits in which rent is positive for prices of production associated with the technique.

Figure 6: Kappa is Never Cost-Minimizing

Figure 7: Lambda is Never Cost-Minimizing

Figure 8: Mu is Cost-Minimizing at Intermediate Rates of Profits

6.0 The Demand for Labor and for Capital

The analysis of the choice of technique need make no mention of supply and demand functions. But the results allow us to plot the real wage against employment, for the given final demand (Figure 9). The relation can be interpreted as an economy-wide demand curve for labor. The switch points appear as horizontal segments in this graph. The 'perverse' switch point is a step function approximation to an upward-sloping demand curve for labor. The widespread tendency to draw downward-sloping labor demand functions, given ideal assumptions such as competitive markets, no search costs, and so on, lacks a coherent justification.

Figure 9: The Demand for Labor

The rate of profits can also be plotted against the value of advanced capital goods (Figure 10). Here, too, switch points are horizontal segments. The value of capital is the iron, wheat, and rye advanced at the start of the production period and evaluate at prices of production. The variation of the value of capital between switch points is known as a price Wicksell effect. The variation at a switch point across techniques is the real Wicksell effect. What happens at the 'perverse' switch point has also been called reverse capital deepening. At any rate, justification for drawing downward-sloping demand curves for capital, by default, is also lacking.

Figure 10: The Demand for Capital

7.0 Conclusion

This post extends the well-established critique of economic theory to which Sraffa (1960) is a prelude. In this example, the techniques that reswitch do not differ in which processes are operated. They differ in the scale at which these processes are operated, thereby resulting in a variation in which lands are scarce. The usual consequences of 'perverse' switch points appear in which, for example, a higher wage is associated with firms wanting to employ more workers to produce a given final demand.

Thursday, February 26, 2026

Balderdash In Teaching Of Labor Economics

Borjas (2016) seems to be a standard textbook in labor economics. It has the usual obsolete idea that labor markets would clear, as shown by supply and demand, under ideal conditions. Unemployment is to be explained by deviations from these ideal conditions. Why would a professor in such a backwoods place like Harvard be expected to know about results in capital theory from two-thirds of a century ago?

The last chapter, chapter 12, is on unemployment. It concludes with this summary:

  • Although the unemployment rate in the United States drifted upward between 1960 and 1990, the economic expansion of the 1990s reduced the unemployment rates substantially.
  • Even a well-functioning competitive economy experiences frictional unemployment because some workers will unavoidably be 'in between' jobs. Structural unemployment arises when there is an imbalance between the supply of workers and the demand for workers.
  • The steady-state rate of unemployment depends on the transition probabilities among employment, unemployment, and the nonmarket sector.
  • Although most spells of unemployment do not last very long, most weeks of unemployment can be attributed to workers who are in very long spells.
  • The asking wage makes the worker indifferent between continuing his search activities and accepting the job offer at hand. An increase in the benefits from search raises the asking wage and lengthens the duration of the unemployment spell; an increase in search costs reduces the asking wage and shortens the duration of the unemployment spell.
  • Unemployment insurance lengthens the duration of unemployment spells and increases the probability that workers are laid off temporarily.
  • The intertemporal substitution hypothesis argues that the huge shifts in labor supply observed over the business cycle may be the result of workers reallocating their time so as to purchase leisure when it is cheap (that is, during recessions).
  • The sectoral shifts hypothesis argues that structural unemployment arises because the skills of workers cannot be easily transferred across sectors. The skills of workers laid off from declining industries have to be retooled before they can find jobs in growing industries.
  • Efficiency wages arise when it is difficult to monitor workers' output. The above-market efficiency wage generates involuntary unemployment.
  • Implicit contract theory argues that workers prefer employment contracts under which incomes are relatively stable over the business cycle, even if such contracts imply reductions in hours of work during recessions.
  • A downward-sloping Phillips curve can exist only in the short run. In the long run, there is no trade-off between inflation and unemployment.
  • The combination of high unemployment insurance benefits, employment protection restrictions, and wage rigidity probably accounts for the high levels of unemployment observed in Europe in the 1980s and 1990s.

Borjas does note that the claim about European unemployment seems to be empirically untrue in that European institutions did not change much, and yet unemployment became lower. I suppose Borjas feels obligated to present theories of imperfections and rigidities that keep wages above the supposed market-clearing wage. Note some of these blame workers, who prefer leisure, would otherwise take advantage of information asymmetries by shirking, or obtain unemployment benefits. The science regerettably shows why we need to beat up on workers.

But, of course, this is pure ideology. Borjas teaches the obsolete theory that wages and employment would be successfully modeled in the long run by the intersection of well-behaved supply and demand curves, absent imperfections. He probably does not know better.

Reference

George J. Borjas. 2016. Labor Economics. Seventh edition. McGraw Hill.