Question

question 1. 44 points a one-period macroeconomic model
consider an economy in which the representative consumer preferences are described by
\\u(c, l) = 0.9 \log c + 0.1 \log l.\\
the total number of hours available to the representative consumer is \\(h = 1\\), and the real market wage is \\(w\\). the representative firm produces the final consumption good using the technology function
\\y = z n^d,\\
where \\(n^d\\) is the labor demand, and \\(z = 2\\). assume the government sets the level of its spending to \\(g = 0.75\\), which has to be financed through a proportional tax \\(t\\).

1. 6 points write down the consumers optimization problem and find the optimal consumption bundles.

Explanation:

Formulate the consumer's budget constraint

The representative consumer maximizes utility subject to their budget constraint.
The consumer has a total time endowment of \(h = 1\), which is split between leisure \(l\) and labor supply \(N^s\):

$$l + N^s = 1 \implies N^s = 1 - l$$

The consumer earns a real wage \(w\) for each unit of labor supplied.
The government levies a proportional tax \(t\) on labor income.
Therefore, the consumer's budget constraint is:

$$C = (1 - t) w N^s = (1 - t) w (1 - l)$$

Rearranging this gives the standard budget constraint:

$$C + (1 - t) w l = (1 - t) w$$

Set up the utility maximization problem

Using the Utility Maximization concept, we write the consumer's optimization problem:

$$\max_{C, l} \quad U(C, l) = 0.9 \log C + 0.1 \log l$$

subject to:

$$C + (1 - t) w l = (1 - t) w$$

and the boundary conditions:

$$C \ge 0, \quad 0 \le l \le 1$$

Solve for the optimal consumption bundle

For a Cobb-Douglas utility function of the form \(U(C, l) = a \log C + b \log l\), the optimal expenditure shares on consumption and leisure are proportional to their weights in the utility function.
The total potential income (wealth) of the consumer is:

$$I = (1 - t) w$$

The optimal expenditure on consumption is:

$$C = \frac{a}{a + b} I = \frac{0.9}{0.9 + 0.1} (1 - t) w = 0.9 (1 - t) w$$

The optimal expenditure on leisure is:

$$(1 - t) w l = \frac{b}{a + b} I = \frac{0.1}{0.9 + 0.1} (1 - t) w = 0.1 (1 - t) w$$

Solving for \(l\):

$$l = 0.1$$

State the final optimal bundle

The optimal choices of consumption and leisure as functions of the wage rate \(w\) and the tax rate \(t\) are:

$$C^* = 0.9 (1 - t) w$$

$$l^* = 0.1$$

Answer:

The consumer's optimization problem is:

$$\max_{C, l} \quad U(C, l) = 0.9 \log C + 0.1 \log l$$

subject to:

$$C = (1 - t) w (1 - l)$$

with \(C \ge 0\) and \(0 \le l \le 1\).

The optimal consumption bundle is:

$$C^* = 0.9 (1 - t) w$$

$$l^* = 0.1$$