Question One Let $$ x_{1} $$ be the food consumption of a household, and $$ x_{2} $$ be the consumption of clothes. The utility preferences of this household can be represented as $$ U\left(x_{1}, x_{2}\right)=3 \ln x_{1}+5 \ln x_{2} $$. Additionally, this household faces the unit price for $$ x_{1} $$ which is 10 Tsh and unit price for $$ x_{2} $$ which is 4 Tsh. Determine the Marshallian demands of each good considering a budget of Tsh 100 . Question Two The Government of the United Republic of Tanzania (URT) is now implementing Second, third and fourth Phases of its Bus Rapid Transport (BRT) Project in Dar es Salaam. The second phase of Bus Rapid Transit in Tanzania commute from Gerezani-Kariakoo to Mbagala-Rangitatu. The third phase run from Kariakoo to Gongo la Mboto while the fourth phase runs from Morocco to Tegeta. It was found that, at current prices the Price Elasticity of Demand for Commuter Services at peak hours in 0.6 and the Price Elasticity of demand during off-peak services is 4.0 . Assuming the company that operates Buss Rapid Transport (BRT) wishes to increase its revenue. As an expert of economics, what price changes and why would you recommend to this company for: a) Peak-time commuter services b) Off-peak services

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### Question One

To determine the Marshallian demands for the goods $$ x_1 $$ (food) and $$ x_2 $$ (clothes), we start with the utility function:

$$
U(x_1, x_2) = 3 \ln x_1 + 5 \ln x_2
$$

The prices for the goods are given as:
- Price of $$ x_1 $$ (food) = 10 Tsh
- Price of $$ x_2 $$ (clothes) = 4 Tsh

The budget constraint is:

$$
10x_1 + 4x_2 = 100
$$

To find the Marshallian demands, we can use the method of Lagrange multipliers. We want to maximize the utility function subject to the budget constraint.

The Lagrangian is given by:

$$
\mathcal{L} = 3 \ln x_1 + 5 \ln x_2 + \lambda (100 - 10x_1 - 4x_2)
$$

Taking the first-order conditions, we differentiate with respect to $$ x_1 $$, $$ x_2 $$, and $$ \lambda $$:

1. $$ \frac{\partial \mathcal{L}}{\partial x_1} = \frac{3}{x_1} - 10\lambda = 0 $$
2. $$ \frac{\partial \mathcal{L}}{\partial x_2} = \frac{5}{x_2} - 4\lambda = 0 $$
3. $$ \frac{\partial \mathcal{L}}{\partial \lambda} = 100 - 10x_1 - 4x_2 = 0 $$

From the first two equations, we can express $$ \lambda $$:

From (1):

$$
\lambda = \frac{3}{10x_1}
$$

From (2):

$$
\lambda = \frac{5}{4x_2}
$$

Setting these equal to each other:

$$
\frac{3}{10x_1} = \frac{5}{4x_2}
$$

Cross-multiplying gives:

$$
3 \cdot 4x_2 = 5 \cdot 10x_1
$$

$$
12x_2 = 50x_1
$$

$$
x_2 = \frac{50}{12} x_1 = \frac{25}{6} x_1
$$

Now, substitute $$ x_2 $$ back into the budget constraint:

$$
10x_1 + 4\left(\frac{25}{6} x_1\right) = 100
$$

$$
10x_1 + \frac{100}{6} x_1 = 100
$$

To combine the terms, convert $$ 10x_1 $$ to sixths:

$$
\frac{60}{6} x_1 + \frac{100}{6} x_1 = 100
$$

$$
\frac{160}{6} x_1 = 100
$$

$$
x_1 = \frac{100 \cdot 6}{160} = \frac{600}{160} = \frac{15}{4} = 3.75
$$

Now, substitute $$ x_1 $$ back to find $$ x_2 $$:

$$
x_2 = \frac{25}{6} \cdot \frac{15}{4} = \frac{375}{24} = 15.625
$$

Thus, the Marshallian demands are:

$$
x_1^* = 3.75, \quad x_2^* = 15.625
$$

### Question Two

To analyze the price changes for the Bus Rapid Transport (BRT) services, we need to consider the price elasticity of demand.

**a) Peak-time commuter services:**

The price elasticity of demand during peak hours is 0.6, which indicates that demand is inelastic. This means that a price increase will lead to a less than proportional decrease in quantity demanded, thus increasing total revenue.

**Recommendation:** The company should consider increasing prices for peak-time commuter services. Since demand is inelastic, the increase in price will lead to an increase in total revenue.

**b) Off-peak services:**

The price elasticity of demand during off-peak hours is 4.0, indicating that demand is elastic. This means that a price increase will lead to a more than proportional decrease in quantity demanded, thus decreasing total revenue.

**Recommendation:** The company should consider decreasing prices for off-peak services. Since demand is elastic, a decrease in price will lead to a more than proportional increase in quantity demanded, thus increasing total revenue.

In summary:
- **Peak-time services:** Increase prices.
- **Off-peak services:** Decrease prices.
For Question One, the Marshallian demands are $$ x_1^* = 3.75 $$ and $$ x_2^* = 15.625 $$. For Question Two, increase prices for peak-time services and decrease prices for off-peak services.

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