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When preferences are strict, the strong core is unique and the top-trading cycle procedure (TTCP) is the unique mechanism yielding the strong core allocation. Here are the highlights. Lemma 1: Let $x$ be the allocation generated by the Top-Trading Cycle Procedure. Then $x$ is in the strong core. Lemma 2: Let $P$ be the set of players, $\Omega = \{ \omega^{... 3 Well, you cannot take the derivative of$E(n)$with respect to$n$, because$n$is an integer variable. More generally, you want to prove a property with respect to$n$. The problem you have is that the corresponding domain is not a convex set: say, for$n$and$n+1$,$0<\lambda <1$, the value$\lambda n + (1-\lambda) (n+1) = n+1-\lambda$does not ... 3 Gale and Shapley barely make any assumptions about preferences. They don't need a functional form, simply an ordinal ranking of the options. Moreover, there are no transfers in this setting, only the discrete matching options. Even Kelso and Crawford do not use quasi-linear utility. They assume some utility function that is increasing and continuous in ... 3 I am aware of classic models such as Kreps & Scheinkman (1983). I am also aware of Burdett, Shi & Wright (2001), but note that their buyers are ex-post symmetric as both of them have value$v_i=v=1$. I am also aware of the literature on competing auctioneers, e.g., Peters & Severinov (1997) and Virag (2010). The buyers' equilibrium selection ... 3 Let$m$cobb-douglas. This means$m(u,v) = A u^\alpha v^{1-\alpha}$for$\alpha < 1$and$q(\theta) = A \theta^{-\alpha}$. Then $$\frac{\theta q'(\theta)}{q(\theta)} = -\alpha > -1$$ To what extent this holds for other functional forms I'm not sure, but I haven't seen anything but Cobb-Douglas being used as the matching function. 2$\beta$appears to be a decision variable, at firm level. Then its value should be determined optimally, under some criterion. So what is the criterion, what is the objective function which should be optimized over the value of$\beta$? It appears that we should maximize the value of the option$V(\beta)$$$V(\beta) = \frac{-c + q(\theta)(1-2\beta)J}{\rho -... 2 The total number of matched applications is$$\sum_{x=1}^\infty g(x)xv = E[x]v$$The number of applications matched with vacancies with x-many applications is$$g(x)xv$$Hence$$Prob(\text{matching with x} | \text{matched} = \frac{g(x)x}{E[x]}$$Using this in the equation in the question gives$$ \tilde f = a \sum_{x=1}^\infty \frac{g(x)x}{E[x]}\... 2 The Neo-Schumpeterian literature models firm behavior with "an search and select" approach. Nelson and Winter (2009) where the pioneers in the subject, it is a pleasant and interesting reading. Nelson, Richard R.; Winter, Sidney G. An evolutionary theory of economic change. Harvard University Press, 2009 2 Think about it this way. The function is measuring the number of new matches being created. If we increase either the vacancies or the amount of unemployed, the function will generate more matches. Therefore, the partial derivatives are expected to be positive. For example consider the matching function with something other than employment. If we consider a ... 2 In the version of the McCall search model I'm familiar with, once you accept an offer of wage$w$, you receive that wage$w$in every period starting from the period of acceptance. Hence, from the perspective of the current period, your utility from accepting that offer is $$w + \beta w + \beta^2 w + \cdots =\frac{w}{1-\beta}$$ In case this equality ... 2 Chapters 6-8 (especially 8) in Modern Labor Economics: Theory and Public Policy, a textbook by Ehrenberg and Smith, cover the topics concerning labor supply. I guess the parts in Ch. 8 that discuss hedonic wage theory would be most relevant to your inquiry. 2 When you take the integral of a$\max_{\{\cdot\}}$operator, I think you have to split the integral into two separate integrals with different supports on them. Even if your value function is complicated and there is no differentiability, you only need continuity for an existence of a solution to solve the optimization problem. 2 I think I get the answer but I am not 100% sure. Any comment would be welcome. I realized that this is related to the search-matching nature of the model (comparing to competitive labor market), so that if capital is not perfectly reversible the worker can bargain with the firm. Note that even with perfect reversibility, given the match-specific surplus, the ... 2 I am not sure if this is exactly what you are looking for, but this paper has model that sounds like what you want: Shimer, Robert, 2006. “On-the-job search and strategic bargaining, ”European EconomicReview50, 811—830. Have a look at this one too: Shimer, R. (2007). Mismatch. American Economic Review, 97(4), 1074-1101. 2 The authors apply in the second step the exact same rules they used in the first step, regarding the calculation of the total differential of an integral, taking also into account that the integral is taken over a function of the integrating variable and not just the integrating variable itself. The first step is $$d[u(x|F)]=d\left[\int_{b_0}^{x}\frac{m}... 1 q(\theta) is defined as the job-filling rate. Note that market tightness \theta is not necessarily constant over time (Pissarides makes a dynamic analysis at some point). It may help to denote it \theta_t. As an approximation, q(\theta_t)\delta t is a probability for a firm to meet a worker between t and t+\delta t for \delta t small enough. ... 1 Note that function p actually depends upon a single variable u/v and I prefer avoiding the abuse of notation (source of confusion) and write p(u/v) = P(u,v). This implies: $$p'(u/v) = P_u(u,v)v = P_v(u,v) \cdot (-v^2/u).$$ So if P(u,v)v = s total differentiation along ds=0 yields P_u(u,v)vdu + (P+... 1 Search and matching models are fairly standard in macroeconomics. I guess this is fair to say since Peter Diamond, Dale Mortensen, and Christopher Pissarides won the Nobel prize in 2010. These are the names behind the canonical DMP model that most graduate students have to study in their first year. Although I am not a macroeconomist, I am confident to say ... 1 It depends on what you define as wage curve. I did not check in the textbooks, but you suggest that Pissarides and Cahuc et al. do not use the same mathemaical expression for the "wage curve". 1) In Cahuc et al., the expression combines the surplus sharing equation with the Bellman equations without using the free entry condition of firms. The wage w is ... 1 There's two implementations that should somewhat cover your needs, matchingR which does reduced-form algorithmic matching and matchingMarkets, which estimates estructural matching models with some baysean tools to correct for endogeneity. The problem is you're looking for some very specific models which may require some tweaks on these implementations. 1 A vacant job is an asset if we consider that the firm can match with a worker and get a positive expected profit. Firms post vacancies up to a point where V=0. If V were positive, more firms would enter, would post additional vacancies, and would get a `rent' (by matching with worker). 1 The deviance from a "normal" market boils down to how that market clears. Matching markets are markets in which prices alone don't clear the market, as thought in mainstream economic theory, and so the market is clear using the help of other institutions. Take it from Alvin Roth's informative EconTalk interview: "...labor markets are like these--they are ... 1 Let's refresh the definition of a partial derivative. partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Example: Let f(t, x) = 3t + x^2. The (total derivative) is \frac{d}{... 1 I suppose one way to solve this - but I'm still looking for a proper way without - is to fall back to matching. If there is matching between the intermediate and the final stage, at zero search cost and symmetric arrival rates, the surplus will be split given the Nashbargaining weights - which then determines the price. 1 From$$ 1 - e^{-b} = a_{F}(b) = \frac{k}{y (1-\beta)}$$we get$$b = -\ln\left(1-\frac{k}{y (1-\beta)}\right) \tag{1}$$For this to exist (be a real number) it must be the case that$$1-\frac{k}{y (1-\beta)} > 0 \implies \frac k y < (1-\beta) \tag{2}$$which is already assumed. Then, uniqueness of the solution of$(1)$is guaranteed because ... 1 This is a pretty general algorithm, you can probably tailor a better one to your specific problem. If you stick to this discreet strategy space it seems to me you would have to find the equilibria via brute force. Basically you would look at all bid profiles$(x_1,x_2,x_3)\$ and check whether it is an equilibrium. To do this, you would have to check if ...