To draw the Marshallian demand curve for $x_1$ given the utility function $u(x_1, x_2) = 2x_1 + x_2$, we proceed as follows:

1. Understand the Marshallian Demand Function

Marshallian demand shows the quantity of a good ($x_1$) a consumer will demand at different prices ($p_1$), given their income ($m$) and the prices of other goods ($p_2$).

Given:

• Utility: $u(x_1, x_2) = 2x_1 + x_2$
• Income: $m = 30$
• Price of $x_2$: $p_2 = 2$

Optimization Problem:

Maximize $u(x_1, x_2) = 2x_1 + x_2$, subject to the budget constraint: $p_1x_1 + p_2x_2 = m$ Substitute $p_2 = 2$ into the budget constraint: $p_1x_1 + 2x_2 = 30$

Step 1: Solve for $x_2$ in terms of $x_1$ using the budget constraint:

$x_2 = \frac{30 - p_1x_1}{2}$

Step 2: Substitute $x_2$ into the utility function:

$u(x_1, x_2) = 2x_1 + \frac{30 - p_1x_1}{2}$

Simplify: $u(x_1) = 2x_1 + 15 - \frac{p_1x_1}{2}$

Step 3: Maximize $u(x_1)$ with respect to $x_1$:

Take the derivative of $u(x_1)$ with respect to $x_1$ and set it to zero: $\frac{d}{dx_1}\left(2x_1 + 15 - \frac{p_1x_1}{2}\right) = 2 - \frac{p_1}{2}$

Setting $2 - \frac{p_1}{2} = 0$, solve for $p_1$: $p_1 = 4$

Thus, $x_1$ depends on $p_1$:

1. When $p_1 \leq 4$, $x_1$ is fully affordable.
2. When $p_1 > 4$, ( x_1 = tighten-->import matplotlib.pyplot as plt import numpy as np

Parameters

income = 30 # m = 30 p2 = 2 # p2 = 2

Marshallian demand for x1 as a function of p1

def marshallian_x1(p1, income, p2): return income / p1 if p1 >= 4 else income / 4

Prices of x1 (p1)

p1_values = np.linspace(0.1, 10, 100) x1_values = [marshallian_x1(p1, income, p2) for p1 in p1_values]

Plot the demand curve

plt.figure(figsize=(8, 6)) plt.plot(p1_values, x1_values, label="Marshallian Demand for $x_1$", linewidth=2) plt.axvline(4, color="red", linestyle="--", label="Critical Price ($p_1 = 4$)") plt.title("Marshallian Demand Curve for $x_1$") plt.xlabel("Price of $x_1$ ($p_1$)") plt.ylabel("Quantity of $x_1$") plt.grid(alpha=0.4) plt.legend() plt.show() <<ImageDisplayed>>successHere is the Marshallian demand curve for $x_1$. The curve shows the relationship between the price of $x_1$ ($p_1$) and the quantity demanded.

Explanation of the Curve:

1. Flat Region ($p_1 \leq 4$):

   • When $p_1 \leq 4$, the consumer allocates their income optimally to buy $x_1$, keeping $x_1 = m / p_1$.

2. Decreasing Region ($p_1 > 4$):

   • Beyond the critical price ($p_1 > 4$), $x_1$ demand reduces rapidly due to diminishing utility returns relative to the budget constraint.

Would you like more details or help with further calculations? Here are five related questions to deepen understanding:

1. How does the budget constraint affect the shape of the demand curve?
2. Can we derive the demand for $x_2$ from this setup?
3. What happens if the income or the price of $x_2$ changes?
4. How do we interpret the slope of the curve in economic terms?
5. What if the utility function were different, e.g., Cobb-Douglas?

Tip: In economic analysis, always check how the marginal utility compares to the price ratio for optimal consumption decisions.