Where I'm Stuck

# Where I'm Stuck ## Part 1: Working Backwards from Extended Labor Market Game I'm starting from the Extended Labor Market Game, from page 33, and trying to work backwards. The Extended Labor Market Game as it is on that page looks like: | | | Worker | | | - | - | - | - | | | | $\textsf{Work}$ | $\textsf{Home}$ | | **Employer** | $W^C$ | 10, **8** | 0, 2 | | | $W^M$ | 15, **3** | 0, 2 | | | 0 | **20**, 0 | 0, **2** | Which threw me off a bit because there's no Nash in this game (strictly dominant is bolded, weakly dominant underlined) But, I think the logic of the text around it works out if the form changes to the following (and I'm guessing there's something I'm missing about credible threats or... something like that, because the worker would never accept 0, so maybe we cross that off in the above table bc of iterated elimination of something or other?) | | | Worker | | | - | - | - | - | | | | $\textsf{Factory}$ ($F$) | $\textsf{Home}$ ($H$) | | **Employer** | $W^C$ | 10, **8** | 0, 2 | | | $W^M$ | \***15**, **3**\* | 0, 2 | Because yeah, in this game we get a Nash, $\mathbf{s}^* = (W^M, \textsf{Work})$ (I changed $\textsf{Work}$ to $\textsf{Factory}$ in my notebook since $W$ is already used for the wages), at which $u^*_E = 15, u^*_W = 3$. But now there's no "punishment option" for the employer here...? The only deviation would be to pay more, which could make sense in a complete-contract thing where more $ means more output, but not here? (Since here the total surplus, 18, is the same regardless of the chosen wage). But, if we do calculate the loss measures for this game, we get that $E$ can unilaterally deviate to move the outcome from $\mathbf{s}^* = (W^M, F)$ to $\widetilde{\mathbf{s}}_E = (W^C, F)$, at which $u_E(\widetilde{\mathbf{s}}_E) = 10, u_W(\widetilde{\mathbf{s}}_E) = 8$, producing $$ \mathcal{I}_{E \rightarrow W} = \frac{3-8}{3} = -\frac{5}{3}, \mathcal{A}_{E \rightarrow W} = \frac{15-10}{15} = \frac{1}{3} \implies \mathcal{C}_{E \rightarrow W} = \frac{-5/3}{1/3} = -5 $$ and then $W$ can unilaterally deviate to move the outcome from $\mathbf{s}^*$ to $\widetilde{\mathbf{s}}_W = (W^M, H)$, at which $u_E(\widetilde{\mathbf{s}}_W) = 0, u_W(\widetilde{\mathbf{s}}_E) = 2$, producing $$ \mathcal{I}_{W \rightarrow E} = \frac{15-0}{15} = 1, \mathcal{A}_{W \rightarrow E} = \frac{3-2}{3} = \frac{1}{3} \implies \mathcal{C}_{W \rightarrow E} = \frac{1}{1/3} = 3. $$ So this says that having the monopsony wage option leads to negative capacity for the employer and positive capacity for the worker, which seems like the opposite of the truth? I'm also slightly confused about a separate thing because, in the text around the model on page 33, the deviation is **not** unilateral, since the worker gets to re-adjust their choice of strategy **after** the unilateral deviation of the employer, hence it re-equilibrates into the (0, 2) outcome in your description, right? But, in my head there were a bunch of reasons for only modeling unilateral deviations (without subsequent re-adjustment and re-equilibration), e.g. that we want to capture the *worst* thing that an agent could do to the other, where the constraints on this worst thing come from the boundedly-rational costs-to-self of choosing unilateral punishment deviation, rather than from the unboundedly-rational new outcome that would eventually result after (potentially many "rounds" of) re-adjustment? That might be an independent/orthogonal issue though, so I just wanted to mention it before moving to the model I added on page 18. ## Part 2: Eliciting Higher Output via Higher Wage So then, for the model I ended up putting up closer to the top on page 18, in place of the Invisible Hand Game, I moved instead to the efficiency wage model, but where bc of complete contracts, the higher wage just directly produces a higher joint surplus? Because (and I guess this is where we keep accidentally tug-of-war-ing, in hindsight :shocked_face_with_exploding_head: ), even though I know **in theory** that *both* monopsony and efficiency wage effects give rise to asymmetries of power, when I actually sit down and try to work out the math, it feels like the efficiency wage setup is the only one of the two where I'm actually able to derive a result that makes sense to me? I say all that because, yeah, after wrestling for a while with the monopsony model written out up above here in Part 1, the model that's in the .tex right now is one where the employer pays a **higher** wage in order to elicit a greater total output (hence, an efficiency wage model, from my understanding of the term). Idk why but, that's the only one of the two that is clicking in my head, because now the employer has an actual punishment option -- let me write out the results for this efficiency-wage model real quick, so you can see what I mean. The model there is: | | | Worker | | | - | - | - | - | | | | $\textsf{Factory}$ ($F$) | $\textsf{Home}$ ($H$) | | **Employer** | $w = 3$ | \***9**, **3**\* | 0, 1 | | | $w = 2$ | 8, **2** | 0, 1 | Which encodes a case where an additional unit of wage (from 2 to 3) produces an additional 2 units of total output (from 10 to 12). And then, with this model, we get Nash $\mathbf{s}^* = (3, F)$ generating $u^*_E = 9, u^*_W = 3$. Employer can punish by moving to $w = 2$, so the outcome becomes $\widetilde{\mathbf{s}}_E = (2, F)$, generating $u_E(\widetilde{\mathbf{s}}_E) = 8, u_W(\widetilde{\mathbf{s}}_E) = 2$, so that $$ \mathcal{I}_{E \rightarrow W} = \frac{3-2}{3} = \frac{1}{3}, \mathcal{A}_{E \rightarrow W} = \frac{9-8}{9} = \frac{1}{9} \implies \mathcal{C}_{E \rightarrow W} = \frac{1/3}{1/9} = 3, $$ and Worker can punish by moving to $H$, shifting the outcome to $\widetilde{\mathbf{s}}_W = (3, H)$, generating $u_E(\widetilde{\mathbf{s}}_W) = 0, u_W(\widetilde{\mathbf{s}}_W) = 1$, so that $$ \mathcal{I}_{W \rightarrow E} = \frac{9-0}{9} = 1, \mathcal{A}_{W \rightarrow E} = \frac{3-1}{3} = \frac{2}{3} \implies \mathcal{C}_{W \rightarrow E} = \frac{1}{2/3} = \frac{3}{2}, $$ so that, to me at least, this gives the "intuitive" result that the employer's capacity to dominate is exactly twice the worker's capacity to dominate. ## Part 3: Making the Numbers in Part 2 Less Arbitrary So then as a third step, where I got stuck/lost in the sauce I guess, I tried to play devil's advocate for myself in terms of questioning whether these are just "convenient" numbers, and whether the intuitiveness (to me) of the Part 2 result would vanish if they changed. So then it's pretty easy to explain where I got lost: * I tried fully modeling the total output $y(e)$ as a function of worker's chosen effort $e(w)$ (itself a function of the wage, so I chose it to be $e(w) = w$ as... the most parsimonious choice in my head, since diminishing returns come in later via the output function), so that if the worker agrees to work they get $u_W = w$ while the employer gets the rest of the output minus the wage, $u_E = y(e(w)) - w$. * So, to turn this into actual numbers I tried simplifying so that the worker's outside option was $z = 0$, meaning that the offered wage could start at $w = 0 + \varepsilon$, then increase up to (say) $w = 1$ to elicit maximal effort. * Then (I guess this is probably where I went off the rails a bit), as the last step to try and get numbers for a normal-form representation, I used the simplest strictly-concave output function I could think of, $y(e) = 10\sqrt{e}$, trying to represent the diminishing returns from additional effort (since otherwise the "optimal" wage is just $w = \infty$, right?) And so this led to the following model: | | | Worker | | | - | - | - | - | | | | $\textsf{Factory}$ ($F$) | $\textsf{Home}$ ($H$) | | **Employer** | $w = 1$ | \***9**, **1**\* | 0, 0 | | | $w = 0 + \varepsilon$ | $\sqrt{\varepsilon}(10 - \sqrt{\varepsilon})$, $\mathbf{\varepsilon}$ | 0, 0 | And so, even though I'll spare you the calculations with that $\sqrt{\varepsilon}(10-\sqrt{\varepsilon})$ term, it still seems like this "works", in the sense that even once I fully model-out the output function, effort function, etc., it comes out as expected, that the efficiency wage model leads to a situation where employer has more power the more they increase the offered wage above the worker's outside option (0 in this case)? It's just clicking with me in a way that the monopsony model from Part 1 above isn't, so, that's where I'm stuck and I wanted to write it out here before adding/changing more stuff in the .tex!