Question

The representative firm has a production function Y=30N. The representative household utility function is u(c,ℓ∣g)=ln(c+0.5g)+2.5ln(ℓ) Based on the above utility function, 50% of government spending is a direct substitute for private consumption. Suppose that g=1. The government collects a flat-rate labor income tax with a tax rate of τ=10%. At the equilibrium, what is the optimal amount of consumption for the representative household?

Answer 1

To find the optimal consumption for the representative household at equilibrium, we need to maximize the given utility function with respect to consumption. By substituting the given values and taking the derivative, we can solve for the optimal consumption. The optimal consumption can then be plugged into the production function to find the corresponding level of output.

Step-by-step explanation:

In order to determine the optimal amount of consumption for the representative household at equilibrium, we need to consider the given utility function and the tax rate. The utility function given is u(c,ℓ∣g)=ln(c+0.5g)+2.5ln(ℓ), where c represents consumption and g represents government spending. In this case, g is set to 1. Since 50% of government spending is a direct substitute for private consumption, we can substitute g with 0.5c. Thus, the utility function becomes u(c,ℓ∣c)=ln(c+0.5c)+2.5ln(ℓ).

To find the optimal consumption at equilibrium, we need to maximize the utility function with respect to c. Taking the derivative of the utility function with respect to c and setting it equal to zero, we can solve for c. The optimal consumption is the value of c that maximizes the utility function. Once we have c, we can plug it back into the production function Y=30N to find the corresponding level of output.

Note: To solve the utility maximization problem, we may need additional information such as the representative household's budget constraint.