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Econ-189-A1 · revision r1 · on Econ-189

GPT 6 Astra Ultra attempt on Econ-189

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. Measuring employer wage-setting power Sharp competition-wage bounds in a declared labor-supply model
  2. Short recap and the counterfactual being identified
  3. Preferences, technology, and the intervention
  4. Sharp identified sets from baseline records
  5. Which exogenous responses remove the ambiguity?
  6. Economic restrictions and the unresolved target
  7. References
  8. Completion Estimate

Measuring employer wage-setting power Sharp competition-wage bounds in a declared labor-supply model

Problem ID: J42-185. Status: Unresolved; rigorous restricted results.

Short recap and the counterfactual being identified

The target is the log change in the same baseline worker's selected wage when employer wage-setting power is removed, with production, preferences, and resources held fixed. An observed productivity--wage wedge is not in general that counterfactual change. We solve the identified-set calculation in an explicit one-employer family with isoelastic supply and decreasing marginal revenue product, and show exactly which additional responses identify its primitives. We do not solve the general sorting, amenities, contract, entry, or equilibrium-selection problem.

Kline [kline], Section 3.1.2, shows that productivity passthrough alone need not identify wage-setting power; Section 3.1.3 discusses using employment responses as well. Those passages were inspected in the full chapter. Our calculation uses a more restrictive model to turn this measurement issue into sharp bounds for a specified competition intervention. The familiar monopsony first-order condition is not presented as a new result.

Preferences, technology, and the intervention

Fix observed baseline wage $w_0>0$ and employment $L_0>0$. Parameters lie in a known rectangle \[ e\in[\underline e,\overline e]\subset(0,\infty),\qquad h\in[\underline h,\overline h]\subset[0,1). \] Here $e$ is a labor-supply elasticity and $h$ is the elasticity of the marginal product with respect to employment, taken with a minus sign. Workers have unit labor endowments, identical productive skill, no job amenities, no incentive pay, and quasilinear utility equal to wage minus a reservation cost $v$ when employed. There is one employer and an outside option normalized to utility zero. On the relevant wage range the measure of workers with $v\leq w$ is \[ L(w)=L_0(w/w_0)^e,\qquad w(L)=w_0(L/L_0)^{1/e}. \tag{1} \] One can implement this with a finite population measure on reservation costs: choose a common sufficiently large $\overline L$ and use quantile function $v(u)=w_0(u/L_0)^{1/e}$ for $u\in[0,\overline L]$. Choose $\overline L$ larger than all baseline and competitive employment levels computed below. Compactness of the parameter rectangle makes this possible. Extend supply by saturation above its largest reservation cost.

The firm's differentiable revenue on $L>0$ has marginal product \[ p(L)=p_0(L/L_0)^{-h},\qquad R(L)=K+\frac{p_0L_0}{1-h}(L/L_0)^{1-h},\qquad p_0=w_0(1+1/e). \tag{2} \] The constant $K$ is output from a fixed input; it has no effect on employment choices. The limit at $L=0$ exists since $h<1$. Revenue is output in a numeraire good with exogenous unit price; this is a technology, not an endogenous product-market price schedule. Fixed inputs and $K$ are held fixed under the competition intervention. The baseline employer posts a uniform wage and hires every worker willing to accept it. Profit is $R(L(w))-wL(w)$, with no entry or wage discrimination.

Define intervention $C$ to replace wage posting by price-taking labor demand, with the same firm technology and worker supply. A competitive wage $w^C$ clears $L(w^C)$ against a profit-maximizing choice of $L$ at that wage. If $h=0$, demand at wage $p_0$ is a correspondence; the declared selector chooses the market-clearing supply quantity. The wage remains unique. Firm profits accrue to an owner; wage payments and profits are feasible transfers out of output, and profits are nonnegative in the construction with $K\geq0$ below.

Let $B$ be the fixed set of baseline employed workers, those with reservation cost at most $w_0$. The target is their average log wage change. All workers in $B$ remain employed under $C$, as proved below; their selected job stays at the same employer. Thus there is no undefined log wage or composition change on $B$. New entrants are reported separately.

Theorem (Equilibrium and the counterfactual wage): For every parameter pair in the rectangle the unique baseline employment optimum is $L_0$ and its wage is $w_0$. Competitive employment and wage satisfy \[ \frac{L^C}{L_0}=(1+1/e)^{e/(1+eh)},\qquad \frac{w^C}{w_0}=(1+1/e)^{1/(1+eh)}. \tag{3} \] Every baseline worker is retained and the target is \[ m(e,h)=\frac{\log(1+1/e)}{1+eh}. \tag{4} \] The increase in employment is $L_0[(1+1/e)^{e/(1+eh)}-1]$.
Proof: Below the supply cap, choosing a wage is equivalent to choosing employment. The marginal wage bill is \[ \frac{d}{dL}[w(L)L]=(1+1/e)w_0(L/L_0)^{1/e}. \] The ratio of marginal revenue (2) to this marginal expenditure is $(L/L_0)^{-(h+1/e)}$. Hence profit is strictly increasing for $L<L_0$ and strictly decreasing for $L>L_0$, up to the cap. Wages above the cap's clearing wage only increase the wage bill with no extra output, so they cannot improve profit. This proves the unique baseline choice.

A competitive equilibrium obeys $p(L^C)=w(L^C)$, giving $(L^C/L_0)^{h+1/e}=1+1/e$. Both sides of the intersection are monotone in opposite directions, and supply is strictly increasing, so the intersection and wage are unique. When $h>0$, concavity of $R$ makes this firm's unique demand optimum at $w^C$. When $h=0$, $w^C=p_0$ makes the firm indifferent across feasible employment and the specified quantity is profit maximizing. The cap was chosen beyond this intersection. Equation (1) then gives the second formula in (3). Since $1+1/e>1$, both employment and wage increase. Workers with reservation cost at most $w_0$ still prefer employment, proving retention of the fixed baseline group. Everyone receives the common new wage, so its log ratio is (4). The employment formula follows by subtraction. ∎

The baseline technological wedge is $\log(p_0/w_0)=\log(1+1/e)$, whereas the wage gain divides that wedge by $1+eh$. Endogenous expansion reduces marginal productivity when $h>0$. Holding the baseline marginal product fixed when computing the competitive wage would therefore answer a different intervention.

Sharp identified sets from baseline records

Specify the observable law to contain the common baseline wage, employment, and total revenue $R_0$, but neither reservation costs nor a direct observation of marginal product. Worker labels among the employed can be assigned identically across models. Suppose \[ R_0\geq\max_{e,h}\frac{w_0(1+1/e)L_0}{1-h}. \] For each pair choose $K=R_0-p_0L_0/(1-h)\geq0$. Thus every pair has the same observed baseline records. The unobserved technology and reservation-cost distribution may differ across compatible models; within each model they remain fixed under $C$.
Theorem (Attainable sharp wage-gain bounds): Under exactly this observation scheme and primitive rectangle, the sharp identified set is the interval \[ \left[ \frac{\log(1+1/\overline e)}{1+\overline e\,\overline h}, \frac{\log(1+1/\underline e)}{1+\underline e\,\underline h} \right]. \tag{5} \] If marginal product $p_0>w_0$ is also exactly observed, then $e=(p_0/w_0-1)^{-1}$ is identified when it lies in the rectangle, and the sharp interval is \[ \left[\frac{\log(p_0/w_0)}{1+e\overline h}, \frac{\log(p_0/w_0)}{1+e\underline h}\right]. \tag{6} \] If that implied elasticity is outside the rectangle, no model is compatible.
Proof: Equation (4) is continuous. Its numerator is positive and strictly decreasing in $e$, while its positive denominator is nondecreasing in $e$ and in $h$. It is strictly decreasing in $h$ because $e>0$. These facts place its minimum and maximum at the opposite corners shown in (5). Every parameter pair is observationally compatible by the explicit $K$ construction, and the equilibrium proof establishes a complete admissible model for each pair. The rectangle is connected; along a line segment joining the two corners, continuity and the intermediate value theorem attain every value between the endpoints. Thus (5) is a sharp set, not merely coordinate bounds on parameters. With observed $p_0$, equation (2) uniquely determines $e$. Varying $h$ continuously over its interval proves (6) by the same construction. ∎

The production-information qualification matters. Observing the entire function $R(L)$ over the relevant range would identify its derivative and curvature in this family; such data cannot be treated as equivalent to observing the single revenue level $R_0$. Likewise, if the fixed-input component $K$ were separately known, not every rectangle point would remain compatible with the revenue observation.

Which exogenous responses remove the ambiguity?

Let a scalar productivity shifter $z$ multiply the variable part of revenue in (2) by $e^z$, with supply, preferences, and $K$ fixed. Consider a neighborhood of $z=0$ where no capacity constraint binds. Assume the structural response to $z$ is observed, for example because $z$ is exogenously assigned and has no direct effect on supply or amenities. Define local elasticities \[ \rho=\left.\frac{d\log w(z)}{dz}\right|_{z=0},\qquad \nu=\left.\frac{d\log L(z)}{dz}\right|_{z=0}. \]
Proposition (Wage and employment responses): In this family \[ \rho=\frac1{1+eh},\qquad \nu=\frac e{1+eh}. \] Thus joint observation of $(\rho,\nu)$, with $\rho,\nu>0$, identifies \[ e=\nu/\rho,\qquad h=(1-\rho)/\nu,\qquad m=\rho\log(1+\rho/\nu). \tag{7} \] Observation of $\rho$ alone leaves the feasible elasticity set \[ E_\rho=\{e\in[\underline e,\overline e]: (\rho^{-1}-1)/e\in[\underline h,\overline h]\}. \] If this set is a nonempty interval $[e_-,e_+]$, the sharp gain interval is $[\rho\log(1+1/e_+),\rho\log(1+1/e_-)]$.
Proof: The baseline first-order equation under the shifter is $e^z(L(z)/L_0)^{-h}=(L(z)/L_0)^{1/e}$. Hence $\log(L(z)/L_0)=ez/(1+eh)$ and, using supply, $\log(w(z)/w_0)=z/(1+eh)$. Differentiation gives $\rho,\nu$, and algebra gives (7). If only $\rho$ is observed, its equation restricts the product $eh$ and gives precisely $E_\rho$. For $\rho=1$, this requires $h=0$; otherwise $\rho\in(0,1)$ and interval intersection with the inequalities on $(\rho^{-1}-1)/e$ gives an interval or the empty set. Every compatible pair has the identical wage response $w(z)=w_0e^{\rho z}$ on a common sufficiently small neighborhood. Baseline $K$ is adjusted as before. Equation (4) equals $\rho\log(1+1/e)$ along this feasible set and is strictly decreasing and continuous in $e$, proving the sharp interval assertion. ∎

The wage-only statement deliberately does not suppose employment responses are also hidden in the observable law: once $\nu$ is observed the joint formula applies. A reduced-form wage response is useful only under the exclusion that the shock does not also change supply or amenities. There is no theorem here identifying $e$ from an arbitrary wage shock.

Economic restrictions and the unresolved target

The bounds are informative because skill, amenity, incentive, and productivity-measurement confounding have been ruled out by explicit primitive restrictions; they have not been removed by a statistical identity. No baseline worker becomes nonemployed in this family, so that mass is exactly zero. The intervention adds workers from the same fixed population and keeps the same production technology and numeraire resources. It is a partial-equilibrium labor market with a declared price-taking benchmark, not an economy-wide entry or reallocation model.

A full solution to the catalogue question must allow model-consistent amenity and incentive responses, multiple employers and selected jobs, measurement error in marginal products, and equilibrium changes in outside options. Bounds (5)--(7) cannot be transported to those settings without rebuilding the observable equivalence and sharpness arguments. The present result locates a concrete remaining ambiguity even in the smaller family: a perfectly measured baseline wedge need not determine the competitive wage when the employment response changes marginal productivity.

References

Completion Estimate

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