Graphical explanation for empirical studies of immigrant's impact on wages (with some unavoidable maths)
Note to the heterodox readers, if any: This post is unabashedly neoclassical.
Note to everyone else; this post is a bit of a mess. I could go through and clean up notation, label all the graphs, add in relevant links and edit it for readability and um, make it more inflammatory, but after drawing all the graphs and typing in all the formulas in LaTex I'm just feeling too lazy right now.
Here's some relevant links:
George Borjas
Brad DeLong
Mark Thoma
Ok.
Solow Model:
On the x-axis is capital per worker, K/L. The bright blue line represents the addition to capital stock per worker, the saving rate times per capita output, s*Y/L. The dark blue line is output per worker as a function of capital per worker. The red line represents the subtraction from capital per worker due to capital depreciation and maybe (native) population growth, (n+d)*K/L. Where subtraction from capital per worker equals addition to capital per worker, capital per worker is constant (for simplicity I'm assuming there's no technological growth which would drive additional capital accumulation). This is the long run steady state of this economy. On the y-axis we can read of the level of output per capita (as well as things like total savings and consumption per capita).
Now we want to relate wages to per capita output. If we take the commonly used Cobb Douglas aggregate production function to represent the relationship between capital, labor and output;
so that per capita output is
^{\alpha}=k^{\alpha}$)
where K is capital, L is labor (later on, an index of labor input) and Y is aggregate output.
and assume labor gets paid its marginal product then we'll have an equation for wages
*(K/L)^{\alpha}=(1-\alpha)y$)
Inverting that (so we can line up our graphs nicely) we have
$)
This is shown below:
The reason why the slope of the green line is greater than 1 is because some of the per capita output goes to capital owners rather than workers. Parameter a generally measures the extent of diminishing returns to capital but here it also turns out to represent the share of total output going to capital owners. The higher the share a, the steeper the green line, the lower the wages. Additionally we can find the return on capital, r, by looking at the purple line tangent to the production function (dark blue curve - in the graph I tangented the wrong curve and I don't feel like redrawing it). Since all the debate has been about immigration's impact on wages, I'm not gonna worry too much about r for the rest of the post.
So how does immigration come in? Well, the simplest way to introduce it is to assume it represents and increase in the number of workers. As the number of workers increases due to immigration capital per worker (K/L) falls and so does output per worker (Y/L). We find the new Y/L (in graphs denoted by lower case y, just like K/L is k) by moving down the dark blue production function, come over to the graph on the left and find the new, lower wage level.
This is essentially what some really old, "naive", studies of the effects of immigration on wages did. But it isn't the end of the story. Because now K/L is below it's steady state value, return to capital is higher and as a result addition to capital per worker is greater than subtraction per worker. So the process of capital accumulation starts again and K/L returns over time to its old level, as does Y/L and as do wages.
(note for the mathematically savvy; you can actually linearize the system of differential equations around the steady state and compute the rate of convergence from which one could estimate how fast wages recover to their pre-immigration levels)
This is equivalent to estimating the elasticity of labor demand. But here I can just tell you what it is. It's alpha, or if you believe that capital's share in output is 1/3, it's 1/3. So this simple analysis would say that for every 10% increase in the number of workers due to immigration, wages would fall, on impact by 3.3% and then slowly go back to where they were. The whole process is illustrated below. This is where you're gonna hafta start clicking on the images to enlarge'em.
But like I said, this is a fairly naive approach. That's because it implicitly assumes that all workers, foreign and domestic, skilled and unskilled are perfect substitutes for each other. This doesn't mean that workers are of the same quality, it's just that quality is related linearly. One engineer equals three doctors equals five lawyers equals a hundred construction workers equals six hundred and sixty six economists. For the purposes of production, if you're a firm, it doesn't matter whether you hire the five lawyers or the six hundred sixty six economists.
Obviously that's not very good. So let's start differentiating workers. First let's just differentiate them by skill level. Suppose that now L is not the number of workers but an index of a labor input, composed of skilled and unskilled workers. Specifically let's assume that the labor input index is constant elasticity of substitution and is given by:
^{1/\rho}$)
where U is number of unskilled workers, S is number of skilled workers and rho measures the elasticity of substitution in production between'em.
Then, again assuming that each kind of worker gets paid her marginal product we get wages:
y(\frac {dL} {dU})=(1-\alpha)y(\frac {L} {U})^{1-\rho}$)
y(\frac {dL} {dS})=(1-\alpha)y(\frac {L} {S})^{1-\rho}$)
and inverting again we have
}(\frac {U} {L})^{1-\rho}$)
}(\frac {S} {L})^{1-\rho}$)
where the last part follows after some simple algebra. This means we have two green lines - one that relates skilled workers' wages to per capita ... we actually got to be careful now, it's no longer per capita output, it's output per unit of labor input, but the two are related in a straight forward manner so I might slip in what follows. Likewise now it's capital per unit of labor input rather than capital per worker ... anyway, two green lines, one for the unskilled and one for the skilled workers.
So what happens now when there's immigration, which here I'll take to be an increase in the number of unskilled workers? Well, as before, each worker (or unit of labor input) has less capita to work with so Y/L goes down as before. But note that now the slopes of the green lines depend on the the number of workers, skilled and unskilled. I'll let you take the derivatives (it's simple for the skilled since there only L changes, but for the unskilled both L and U change). Anyway if you do that, you'll get that how the lines shift depends on rho - which measures the elasticity of substitution between skilled and unskilled workers.
Well, most estimates of rho suggest that skilled and unskilled workers are complements. To build a house you need some relatively unskilled construction workers, but you also need skilled architects, surveyors, etc. You can't build the house if you don't have one group or the other. This means that rho is negative (and it's a pain to always worry about a negative parameter, but there you go). If you took your derivatives correctly you'll see that this means that the slope of the green line for the skilled workers shifts downward, while the green line for the unskilled workers shifts upward. This means that the effect on unskilled workers' wages is acerbated but skilled workers gain from immigration. If you were to do an empirical study based on the simple model based above, rather than this one, then the gains to skilled workers would instead show up, incorrectly, as gains to capital (if it wasn't just a complete mess).
Here's the graph, and to avoid clutter I'm omitting labels - think of it as one of those puzzles where you have to fill in the missing parts.
Of course, once the shock occurs, the process of capital accumulation starts up again, which the graph illustrates. Now the green lines are permanently shifted, but output per ... indexed labor, starts going back to its old level. This means that now both wages of unskilled and skilled start rising. So the wages of the unskilled recover somewhat (math can tell you that it's not all the way) while the wages of the skilled keep on rising. The purple arrows in the upper graphs indicate how wages move, while the red lines in the lower graphs show the time path of wages for the two worker groups.
Here's the same thing with good old supply and demand, but it's less pretty and you don't see everything that's going on:
The increase in the number of unskilled workers is a shift in labor supply in the upper graph represented by a shift in the blue line. This lowers the wage rate for unskilled workers. But through complementarity effects it also increases the demand for skilled labor, increasing their wages as seen in the lower graph. Then capital accumulates and the demands for both kinds of labor shift up. (Again, I omitted all the labels out of laziness. Wages are on the y-axis, amount of each type of labor is on the x-axis).
Of course all this begs the question; who are the skilled workers? Well, if you look at Ottaviano and Peri, then for the US, the answer seems to be "anyone's who graduated from high school", which means most workers see some gains in the short run and even larger gains in the long run. But it's true, the very bottom would be hurt. Or would it?
Well, this kind of study is still fairly basic. I guess you could call these kinds of studies "second generation" - they estimate the effects separately for various skill levels of workers and allow for capital accumulation that might result. But they still differentiate workers only very coarsely.
The above mentioned Ottaviano and Peri goes much further and differentiates workers by skill level, occupation and whether they're immigrants or not. It's not implausible that foreign born and native born workers, even when they have the same education and are employed in the same industry would not be perfectly substitutable for each other. It'd be too much work here to go through all the differentiations possible. Let's just differentiate between three types of workers here: skilled-native, unskilled-native, and unskilled-immigrant. So we're assuming a labor index like this:
^{\rho/\epsilon}]^{1/\rho}$)
Where now S is number of skilled native workers, U_n is number of unskilled native workers and I is number of immigrants. Now rho measures elasticity of substitution between skilled and unskilled workers and epsilon measures elasticity of substitution between foreign and native workers.
Let's define the unskilled labor index as:
Since we've got three groups now, we'll have three green lines. The equations for the wages are now:
y(\frac {dL} {dS})=(1-\alpha)y(\frac {L} {S})^{1-\rho}$)
y(\frac {dL} {dU_n})=(1-\alpha)y(\frac {L} {L_u})^{1-\rho}(\frac {L_u} {U_n})^{1-\epsilon}$)
y(\frac {dL} {dI})=(1-\alpha)y(\frac {L} {L_u})^{1-\rho}(\frac {L_u} {I})^{1-\epsilon}$)
And if we invert these we'll get the three green lines.
Now we increase the number of immigrants. The effect on native-skilled workers is pretty straight forward because it is essentially the same as the effect on skilled workers in the model right above. The green line shifts down and skilled workers gain today and even more in the future. The effect on PREVIOUS immigrants is also pretty straightforward - above we were increasing the number of unskilled workers, here we're increasing the number of immigrants, so the effect on previous immigrant's wages when the new ones come is to decrease their wages on impact, which then gradually recover somewhat over time.
The difference is in what happens to native unskilled workers. Again, if you do the math, you'll see that it depends on elasticities of substitution (now rho and epsilon) and which one is bigger. So theoretically, now, the answer is "it depends". The green line for native-unskilled workers could shift up or it could shift down. It could even shift down enough so that even unskilled workers gain, today and in the future. For this to be true however, there would have to be a lot of complementarity between native and foreign unskilled workers (epsilon would have to be negative and large in absolute value). Here again is a stripped down graph:
So like I said, for the native unskilled workers it could go either way. They could loose some and then recover or they could gain and then gain some more. What it depends on is the relative magnitudes of elasticity of substitution between native-unskilled and foreign-unskilled and the elasticity between the unskilled as a group and skilled as a group. As it turns out most estimates suggest that the complementarity between skill groups is larger than between foreign and native (note that actually you don't need complementarity. You need enough less-than-perfect substitutability) which means that the loose-then-recover-some scenario is more plausible for the unskilled (again here roughly meaning "high school drop outs"). But it's not that much larger. In fact what the estimates suggest is that the green line for unskilled workers shift down though not quite enough to offset the fall in wages due to decreased capital per worker. What this means is that the likely effect on unskilled workers' wages is "small loss in short run, slight gain in the long run", or, more or less 0.
0.
And in fact, that's roughly speaking (in terms of orders of magnitude) what Ottaviano and Peri find although that study being more detailed is not directly comparable to all them graphs above. As often is the case The World doesn't really feel like providing unambiguous, strong, one sided evidence that would make taking an ideological stance easy (until you think about the gains to the migrants themselves).
In a way the bottom line here, if you've read this far, is that the main people that immigrants are hurting through their effect on wages are those most like them. In other words, previous immigrants.
HAPPY FOURTH OF JULY!
Note to everyone else; this post is a bit of a mess. I could go through and clean up notation, label all the graphs, add in relevant links and edit it for readability and um, make it more inflammatory, but after drawing all the graphs and typing in all the formulas in LaTex I'm just feeling too lazy right now.
Here's some relevant links:
George Borjas
Brad DeLong
Mark Thoma
Ok.
Solow Model:
On the x-axis is capital per worker, K/L. The bright blue line represents the addition to capital stock per worker, the saving rate times per capita output, s*Y/L. The dark blue line is output per worker as a function of capital per worker. The red line represents the subtraction from capital per worker due to capital depreciation and maybe (native) population growth, (n+d)*K/L. Where subtraction from capital per worker equals addition to capital per worker, capital per worker is constant (for simplicity I'm assuming there's no technological growth which would drive additional capital accumulation). This is the long run steady state of this economy. On the y-axis we can read of the level of output per capita (as well as things like total savings and consumption per capita).
Now we want to relate wages to per capita output. If we take the commonly used Cobb Douglas aggregate production function to represent the relationship between capital, labor and output;
so that per capita output is
where K is capital, L is labor (later on, an index of labor input) and Y is aggregate output.
and assume labor gets paid its marginal product then we'll have an equation for wages
Inverting that (so we can line up our graphs nicely) we have
This is shown below:
The reason why the slope of the green line is greater than 1 is because some of the per capita output goes to capital owners rather than workers. Parameter a generally measures the extent of diminishing returns to capital but here it also turns out to represent the share of total output going to capital owners. The higher the share a, the steeper the green line, the lower the wages. Additionally we can find the return on capital, r, by looking at the purple line tangent to the production function (dark blue curve - in the graph I tangented the wrong curve and I don't feel like redrawing it). Since all the debate has been about immigration's impact on wages, I'm not gonna worry too much about r for the rest of the post.
So how does immigration come in? Well, the simplest way to introduce it is to assume it represents and increase in the number of workers. As the number of workers increases due to immigration capital per worker (K/L) falls and so does output per worker (Y/L). We find the new Y/L (in graphs denoted by lower case y, just like K/L is k) by moving down the dark blue production function, come over to the graph on the left and find the new, lower wage level.
This is essentially what some really old, "naive", studies of the effects of immigration on wages did. But it isn't the end of the story. Because now K/L is below it's steady state value, return to capital is higher and as a result addition to capital per worker is greater than subtraction per worker. So the process of capital accumulation starts again and K/L returns over time to its old level, as does Y/L and as do wages.
(note for the mathematically savvy; you can actually linearize the system of differential equations around the steady state and compute the rate of convergence from which one could estimate how fast wages recover to their pre-immigration levels)
This is equivalent to estimating the elasticity of labor demand. But here I can just tell you what it is. It's alpha, or if you believe that capital's share in output is 1/3, it's 1/3. So this simple analysis would say that for every 10% increase in the number of workers due to immigration, wages would fall, on impact by 3.3% and then slowly go back to where they were. The whole process is illustrated below. This is where you're gonna hafta start clicking on the images to enlarge'em.
But like I said, this is a fairly naive approach. That's because it implicitly assumes that all workers, foreign and domestic, skilled and unskilled are perfect substitutes for each other. This doesn't mean that workers are of the same quality, it's just that quality is related linearly. One engineer equals three doctors equals five lawyers equals a hundred construction workers equals six hundred and sixty six economists. For the purposes of production, if you're a firm, it doesn't matter whether you hire the five lawyers or the six hundred sixty six economists.
Obviously that's not very good. So let's start differentiating workers. First let's just differentiate them by skill level. Suppose that now L is not the number of workers but an index of a labor input, composed of skilled and unskilled workers. Specifically let's assume that the labor input index is constant elasticity of substitution and is given by:
where U is number of unskilled workers, S is number of skilled workers and rho measures the elasticity of substitution in production between'em.
Then, again assuming that each kind of worker gets paid her marginal product we get wages:
and inverting again we have
where the last part follows after some simple algebra. This means we have two green lines - one that relates skilled workers' wages to per capita ... we actually got to be careful now, it's no longer per capita output, it's output per unit of labor input, but the two are related in a straight forward manner so I might slip in what follows. Likewise now it's capital per unit of labor input rather than capital per worker ... anyway, two green lines, one for the unskilled and one for the skilled workers.
So what happens now when there's immigration, which here I'll take to be an increase in the number of unskilled workers? Well, as before, each worker (or unit of labor input) has less capita to work with so Y/L goes down as before. But note that now the slopes of the green lines depend on the the number of workers, skilled and unskilled. I'll let you take the derivatives (it's simple for the skilled since there only L changes, but for the unskilled both L and U change). Anyway if you do that, you'll get that how the lines shift depends on rho - which measures the elasticity of substitution between skilled and unskilled workers.
Well, most estimates of rho suggest that skilled and unskilled workers are complements. To build a house you need some relatively unskilled construction workers, but you also need skilled architects, surveyors, etc. You can't build the house if you don't have one group or the other. This means that rho is negative (and it's a pain to always worry about a negative parameter, but there you go). If you took your derivatives correctly you'll see that this means that the slope of the green line for the skilled workers shifts downward, while the green line for the unskilled workers shifts upward. This means that the effect on unskilled workers' wages is acerbated but skilled workers gain from immigration. If you were to do an empirical study based on the simple model based above, rather than this one, then the gains to skilled workers would instead show up, incorrectly, as gains to capital (if it wasn't just a complete mess).
Here's the graph, and to avoid clutter I'm omitting labels - think of it as one of those puzzles where you have to fill in the missing parts.
Of course, once the shock occurs, the process of capital accumulation starts up again, which the graph illustrates. Now the green lines are permanently shifted, but output per ... indexed labor, starts going back to its old level. This means that now both wages of unskilled and skilled start rising. So the wages of the unskilled recover somewhat (math can tell you that it's not all the way) while the wages of the skilled keep on rising. The purple arrows in the upper graphs indicate how wages move, while the red lines in the lower graphs show the time path of wages for the two worker groups.
Here's the same thing with good old supply and demand, but it's less pretty and you don't see everything that's going on:
The increase in the number of unskilled workers is a shift in labor supply in the upper graph represented by a shift in the blue line. This lowers the wage rate for unskilled workers. But through complementarity effects it also increases the demand for skilled labor, increasing their wages as seen in the lower graph. Then capital accumulates and the demands for both kinds of labor shift up. (Again, I omitted all the labels out of laziness. Wages are on the y-axis, amount of each type of labor is on the x-axis).
Of course all this begs the question; who are the skilled workers? Well, if you look at Ottaviano and Peri, then for the US, the answer seems to be "anyone's who graduated from high school", which means most workers see some gains in the short run and even larger gains in the long run. But it's true, the very bottom would be hurt. Or would it?
Well, this kind of study is still fairly basic. I guess you could call these kinds of studies "second generation" - they estimate the effects separately for various skill levels of workers and allow for capital accumulation that might result. But they still differentiate workers only very coarsely.
The above mentioned Ottaviano and Peri goes much further and differentiates workers by skill level, occupation and whether they're immigrants or not. It's not implausible that foreign born and native born workers, even when they have the same education and are employed in the same industry would not be perfectly substitutable for each other. It'd be too much work here to go through all the differentiations possible. Let's just differentiate between three types of workers here: skilled-native, unskilled-native, and unskilled-immigrant. So we're assuming a labor index like this:
Where now S is number of skilled native workers, U_n is number of unskilled native workers and I is number of immigrants. Now rho measures elasticity of substitution between skilled and unskilled workers and epsilon measures elasticity of substitution between foreign and native workers.
Let's define the unskilled labor index as:
Since we've got three groups now, we'll have three green lines. The equations for the wages are now:
And if we invert these we'll get the three green lines.
Now we increase the number of immigrants. The effect on native-skilled workers is pretty straight forward because it is essentially the same as the effect on skilled workers in the model right above. The green line shifts down and skilled workers gain today and even more in the future. The effect on PREVIOUS immigrants is also pretty straightforward - above we were increasing the number of unskilled workers, here we're increasing the number of immigrants, so the effect on previous immigrant's wages when the new ones come is to decrease their wages on impact, which then gradually recover somewhat over time.
The difference is in what happens to native unskilled workers. Again, if you do the math, you'll see that it depends on elasticities of substitution (now rho and epsilon) and which one is bigger. So theoretically, now, the answer is "it depends". The green line for native-unskilled workers could shift up or it could shift down. It could even shift down enough so that even unskilled workers gain, today and in the future. For this to be true however, there would have to be a lot of complementarity between native and foreign unskilled workers (epsilon would have to be negative and large in absolute value). Here again is a stripped down graph:
So like I said, for the native unskilled workers it could go either way. They could loose some and then recover or they could gain and then gain some more. What it depends on is the relative magnitudes of elasticity of substitution between native-unskilled and foreign-unskilled and the elasticity between the unskilled as a group and skilled as a group. As it turns out most estimates suggest that the complementarity between skill groups is larger than between foreign and native (note that actually you don't need complementarity. You need enough less-than-perfect substitutability) which means that the loose-then-recover-some scenario is more plausible for the unskilled (again here roughly meaning "high school drop outs"). But it's not that much larger. In fact what the estimates suggest is that the green line for unskilled workers shift down though not quite enough to offset the fall in wages due to decreased capital per worker. What this means is that the likely effect on unskilled workers' wages is "small loss in short run, slight gain in the long run", or, more or less 0.
0.
And in fact, that's roughly speaking (in terms of orders of magnitude) what Ottaviano and Peri find although that study being more detailed is not directly comparable to all them graphs above. As often is the case The World doesn't really feel like providing unambiguous, strong, one sided evidence that would make taking an ideological stance easy (until you think about the gains to the migrants themselves).
In a way the bottom line here, if you've read this far, is that the main people that immigrants are hurting through their effect on wages are those most like them. In other words, previous immigrants.
HAPPY FOURTH OF JULY!
posted by YouNotSneaky! at 12:09 PM
23 Comments:
OMG! This has totally blown my mind! I'm sure it would have made Pigou's nipples hard. In any case, it make my week a lot better!
A few comments...
When you quote the O&P study for the first time you say that "unskilled natives" = "high-school graduates" when you mean "drop-outs".
In Modigliani's world, the short-run adjustment would not be in wages only but mostly in unemployment. Depending on unemployment insurance, I'm not sure if this is better or worse from a welfare point of view.
And last but not least, being a "unskilled immigrant" is not a life sentence. After a few years I imagine that immigrants become natives. From a dynastic household perspective, they might even become skilled natives, in the long run! (For example, Arnold Schwarzenegger, although he's not representative.)
When you quote the O&P study for the first time you say that "unskilled natives" = "high-school graduates" when you mean "drop-outs".
Thanks, I fixed it. There's probably a lot more typos.
the short-run adjustment would not be in wages only but mostly in unemployment.
Well, this is Solow growth model/micro labor so the assumption is wages clear markets or the Fed's doing it job of keeping the economy at full employment. Like I said, unabashedly neoclassical.
And last but not least, being a "unskilled immigrant" is not a life sentence. After a few years I imagine that immigrants become natives. From a dynastic household perspective, they might even become skilled natives, in the long run!
That's actually part of the point that Brad DeLong was making in response to George Borjas.
Anyway - you can put that stuff in, just now you'll have take a total derivative for a given wage since immigrants becoming natives means dI<0 and dU or dS>0. So that throws more ambiguity into the shifts.
Hey! Since when is "neoclassical" a reason for disclaimers? (Just noticed the disclaimer for your heterodox readership/reader :-).)
I love Solow! (His model, not him... well, OK, maybe him too, a little.) It's one of the 3 models I know! :-)
The Solow model is actually a straightforward model to illustrate some of the problems with the way economists think about capital. Suppose the economy is in a steady state, with capital stock per worker of k*. Now suppose that a third of the population dies off suddenly. Does this increase the capital stock to 1.5 times K* right after the loss? Well, if one thinks about the *composition* (i.e. the kinds of machines and buildings) of the capital stock, it's pretty clear that it will not be the same as the capital stock that would have been accumulated if the saving rate had been high enough to obtain this capital stock in steady-state for the original population. So we're back to Robinson's commentary on leets, where capital is like a mechano set that can be costlessly and instanteously transformed. The notion of an aggregate production may have some kind of validity along a equilibrium path. It's not clear that it means much in situations where the economy is "shocked" off this path.
I know this is only residually related to the point you're making, but given some of our previous discussions, I couldn't resist...
You're right and I'm aware of it. Hence the disclaimer at the beginning of the post.
But... suppose I reject Solow and APFs and all that. How can I think about the impact of immigration on wages? If I explicitly worry about Wicksel effects then anything can happen and that doesn't get me anywhere does it?
But... suppose I reject Solow and APFs and all that. How can I think about the impact of immigration on wages?
You're taking a model that was designed to account for a specific set of "facts", and trying to use it to make predictions about another phenomena (immigration). Your additional structure (CES aggregator across "types" of labor) is ad hoc and you're dealing with a disequilibrium situation (which is my major irritation with RBC theory). The problem, at the end of the day, is how can we know if the predictions are reliable, if we are facing a novel situation? And how can we preclude further alternations to the model (e.g addition of capital goods that are specific to skill-cateogries, etc.), that may well change the predicted outcomes?
I don't object to the use of such models, since novel situations do apply and it seems somewhat nihilist to say that nothing should be done to try to predict what might happen. But economists should be upfront about the limitations of their models. Just because certain modelling elements appeared to be useful in one context doesn't mean they will be of any use in other.
You're taking a model that was designed to account for a specific set of "facts", and trying to use it to make predictions about another phenomena (immigration).
Actually I think that the fact that the model is flexible enough so that it allows you to study different phenomenon is a point in its favor.
Your additional structure (CES aggregator across "types" of labor) is ad hoc and you're dealing with a disequilibrium situation
I'm not sure if I'd call the additional structure "ad hoc" - it's more general than the standard stuff.
Here's what we wanna know:
Impact of immigration on wages.
Here's issues we want to include in the analysis (which early studies neglected):
1. Possibility that workers of different skill levels are not perfect substitutes.
2. Possibility that foreign and domestic workers are not perfect substitutes.
3. Possibility that capital responds to changes in labor force.
Now.
Data can tell us - within the framework - whether 1 and 2 are valid concerns. If they're not then the estimated coefficients will be 1 or whatever. It would probably also be possible here to go with some input-output matrix or linear activity model (which forces the different groups of workers to be either perfect substitutes or perfect complements) - which would avoid the CCC problems (although introducing others - see the previous parentheses) but I don't think the answer would be that different.
On 3 - the way it's done empirically (honestly I'm not 100% sure here) it just looks at response of capital to an increase in labor. It doesn't force it to go back to the steady state ("equilibrium") or anything. So that's a bit more frame-work free.
And how can we preclude further alternations to the model (e.g addition of capital goods that are specific to skill-cateogries, etc.), that may well change the predicted outcomes?
That would be a potentially interesting extension. Difficult, but interesting. And if it changes the outcome, I'd change my mind. It's what you're suppose to do.
But economists should be upfront about the limitations of their models.
Yes, and that's why there's the disclaimer.
Actually I think that the fact that the model is flexible enough so that it allows you to study different phenomenon is a point in its favor.
My initial comment is fairly esoteric. In microeconomics, a production function is taught as a purely technical relationship (i.e. if I had this set of inputs, what's the maximum level of output that I could obtain). Once you start aggregating up, the interpretation changes somewhat (if a firm optimally hired $K worth of capital goods and L units of labour, what is the maximum output that the firm could produce.) Going from a long-run optimal allocation to a short-run situation where output plans change is not as simple as a movement across isoquants in (K-L) space, if one interprets K as a set of optimal chosen capital goods. Now I suppose the argument one could make is that you won't be "far off" using this approach, but there is no way of knowing the error introduced by this treatment.
Aggregate production functions are even more problematic (outside of the one good model), as they aren't technical relationships, or even relationships between some optimally chosen inputs and the resulting level of output. This implies that their use may only be justified in a certain context (where the structure of the capital stock is continuously in equilibrium).
Really, this just brings me back to my earlier point about mathematics not neccessarily yielding greater clarity. What is the correct interpretation of a APF? And once we agree upon that interpretation, what uses does this interpretation permit?
Re first 2 paragraphs: Yes, yes, I know. But I WANNA KNOW THE EFFECT OF IMMIGRATION ON WAGES!!! Is there another framework I can use??? Gimme gimme gimme!!! There's practical questions that need answers and gardens that need to be cultivated.
Really, this just brings me back to my earlier point about mathematics not necessarily yielding greater clarity. What is the correct interpretation of a APF? And once we agree upon that interpretation, what uses does this interpretation permit?
But if I spell out the word C-A-P-I-T-A-L does that clear up any misunderstandings that arise if I just write "K"? My point was and is that the ambiguity here is intrinsic to the concept, not the presentation/formulation.
There's practical questions that need answers and gardens that need to be cultivated.
I already conceded this point. The question I'm asking, I suppose, is how reliable are the predictions that are made from such a model, given that the model elements have only demonstrated "success" in explaining a very specific set of facts (the Kaldor facts)? This to some degree relates to McCloskey's point...economists can trace through the logical implications of any number of assumptions. At what point do such exercises have anything useful to say about the world?
For the record, I'm not a heterodox economist. Borjas makes a comment on his web-page about the role of modeling in understanding the economic effects of immigration. As he notes, labour economists tend to be agnostic about theoretical predictions, instead preferring to let the data speak. (This is of course the transgression that Card and Krueger made in their minimum wage studies.)
Well, I feel like I've been handing out demerits left and right for the past two days so I really need to shut up about prominent public figures, but
Borjas makes a comment on his web-page about the role of modeling in understanding the economic effects of immigration. As he notes, labour economists tend to be agnostic about theoretical predictions, instead preferring to let the data speak.
when he says that he's just being disingenuous (to put it politely). What does it mean to "let the data speak"? You got some data and you run some regressions. You can just slap a linear regression model, find the correlations, be silent about omitted variables or model specification and claim that immigration depresses wages because you "let the data speak?". But what you've done is estimated an elasticity (of labor demand). And you might even say "I've estimated an elasticity and the data speak for itself". But how the hot air balloons in the shape of Snoopy do you know that you've estimated the right elasticity? You need theory.
No. Wait. Not exactly. You can estimate a more general version of the regression model and see if the estimated coefficients suggest that the simple one you did before - the one where "data spoke for itself" - is valid.
Which is where the uh, "disingenousiness" (sp? whatever) comes in. Somehow, a restricted, special case, wave hands about other stuff model lets "data speak for itself", but a more general empirical model which embeds the simple one is "theoretical predictions".
What does it mean to "let the data speak"?
I'm not going to speak for Borjas. To me, a difference between say, trade economists, and labor economists, is what Borjas mentions. The trade economists to a large degree accept the validity of the theories as a matter of faith (this is changing, hopefully for the better, with the emergence of empirical trade in the 15 years). In contrast, the labor economists often view theory as a set of conjectures that have yet to be empirically established. Outside the realm of the "natural experiment" (if such a beast truly exists), it's not clear even with data that one can do so, and regression equations are models in their own right. So yes, I agree that data can rarely speak for itself.
It seems to me, however, that many economists place undo faith in their theories, not because there is sound empirical evidence to support them, but because they find the logic appealing. And my question is whether the predictions that come from such a process are reliable.
But if I spell out the word C-A-P-I-T-A-L does that clear up any misunderstandings that arise if I just write "K"? My point was and is that the ambiguity here is intrinsic to the concept, not the presentation/formulation.
The point of the CCC was not that the concept of capital was ambiguous. The point was that using an aggregate measure in a supply and demand model was incoherent (Franklin Fisher and others have reached similar conclusions within the aggregation literature). The "ambiguity" that you mention is simply a neoclassical faith that there must be some way of measuring capital that justifies the intuition of the single good model, or that the goodness of fit in certain contexts must exist for the "right" reasons.
This, by the way, likely characterizes the "pigheadedness" of the Post-Keynesians. No amount of arguments about "fitting the data" will ever convince them that the neoclassical model dentifies a deep causal determinantion of the distribution of income, when the theory is based on a theoretically incohorent variable.
Actually, it's not incoherent. Hicks offers two limiting cases where a composite (capital) good acts perfectly as the underlying composition. Fixed proportions or constant price elasticities, if I remember correctly, most likely I don't.
As long as we restrict study to these two extreme cases, we're fine! ;-)
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