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In order to be eligible for benefits, a claimant must be unemployed "through no fault of their own." When an employee is terminated they will generally satisfy this requirement unless they are terminated for reasons amounting to "misconduct." Misconduct is conduct evidencing an intention or reckless disregard for the interests of the employer. Things like stealing or showing up to work drunk are examples of misconduct. Being let go because of a different kind of management style definitely would not be "misconduct."
There is a second issue presented by our facts, though. This is the offer fo a lower paying position. Employees are only eligible for benefits if they are unemployed "through no fault of their own." When an employee turns down an offer of suitable work, they will be regarded as unemployed "through fault" since they have made the voluntary decision to remain unemeployed. This holds true regardless of whether the employee was released from their last job for reasons not amounting to misconduct.
Determining whether a job offer is an offer of suitable work will take into account the type of work being performed and the rate at which it is being paid. A job that pays $45,000 less and is not doing the same kind of work almost certainly would not be considered an offer of "suitable work." So, it most cases rejecting such an offer would not be a bar to unemployment benefits.
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