Assume a variant of the static GE model in which the representative household has preferences , and is a standard quasi-concave utility function. Let the capital input is fixed at, and let be the representative rm's production function, and it is a standard concave production function. Imagine if the government subsidizes employment, It does this by paying firms a subsidy for each unit of labor employed, and the government finances this subsidy by taxing households using a lump-sum tax . The government also balances its budget. The rate of subsidy , the subsidy rate is an exogenous variable and the tax is an endogenous variable.

1) Please write down the household's optimization problem. Depict the solution to the household's problem in a diagram.

2) Please write down the firm's profit maximization problem. Depict the solution to the firm's problem in a diagram.

Please solve it using a diagrammatic approach.

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