Give step-by-step solution with explanation and final answer:

solve all part a, b, c, d and for mulitple choice give explanantion why other options are not correctr on CRRRE RE [la
Suppose Juan's

Question

Give step-by-step solution with explanation and final answer:

solve all part a, b, c, d and for mulitple choice give explanantion why other options are not correctr on CRRRE RE [la
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VIPSolutions · Accepted Answer

Let's solve each part step by step, with explanations for each answer and why other options are incorrect.

---

### **a. What is the name for preferences like Juan's and Charlie's?**

#### **Juan's utility function:**
$$ U = 2X + Y $$

This is a **linear utility function**, where goods can be substituted at a constant rate. This is called **perfect substitutes**.

- **Negative exponential:** Not applicable; that's for risk preferences, not for linear goods.
- **Perfect complements:** Would be of the form $ U = \min(aX, bY) $.
- **Perfect substitutes:** **Correct!** Linear form, constant rate of substitution.
- **Cobb-Douglas:** Would be of the form $ U = X^a Y^b $.

**Answer for Juan:**  
**Perfect substitutes**

#### **Charlie's utility function:**
$$ U = \min\{3X, 2Y\} $$

This is the classic form for **perfect complements** (goods consumed in fixed proportions).

- **Cobb-Douglas:** Not of the min form.
- **Negative exponential:** Not applicable.
- **Perfect complements:** **Correct!** Min function means fixed proportion consumption.
- **Perfect substitutes:** Would be a linear function.

**Answer for Charlie:**  
**Perfect complements**

---

### **b. What is the marginal utility of X for Juan given his current holdings?**

Juan's utility: $ U = 2X + Y $

**Marginal utility of X** is the partial derivative with respect to X:
$$
\frac{\partial U}{\partial X} = 2
$$

So, no matter his holdings, the marginal utility of X is **2**.

- **1:** Incorrect, not the coefficient of X.
- **2:** **Correct!**
- **0:** Incorrect, not the coefficient.
- **3:** Incorrect.

**Answer:**  
**2**

---

### **c. What is the marginal utility of X for Charlie given his current holdings?**

Charlie's utility: $ U = \min\{3X, 2Y\} $

For perfect complements, **marginal utility is determined by which argument is smaller, i.e., which is the limiting factor**:
- At current holdings: $ X = 3, Y = 4 $
- $ 3X = 9, 2Y = 8 $
- So, utility is limited by $ 2Y $ (since 8 < 9).

**Marginal utility of X:** If you increase X by 1, $ 3X $ goes from 9 to 12, but since $ 2Y $ is still 8, the minimum is still 8, so **utility does not increase**.

- **3:** Incorrect, not the coefficient.
- **2:** Incorrect.
- **0:** **Correct!** Utility does not increase.
- **1:** Incorrect.

**Answer:**  
**0**

---

### **d. Suppose Charlie gives 1/6 unit of X to Juan, and Juan gives Charlie 1/4 unit of Y.**

Let’s calculate if each person is better off after the trade.

#### **Current holdings:**
- Juan: $ X = 2, Y = 2 $
- Charlie: $ X = 3, Y = 4 $

#### **After trade:**
- Juan gets $ +1/6 $ X, gives $ -1/4 $ Y
  - New: $ X = 2 + 1/6 = 13/6 $, $ Y = 2 - 1/4 = 7/4 $
- Charlie gives $ -1/6 $ X, gets $ +1/4 $ Y
  - New: $ X = 3 - 1/6 = 17/6 $, $ Y = 4 + 1/4 = 17/4 $

---

#### **Charlie's utility before and after:**

- **Before:** $ U = \min\{3 \times 3, 2 \times 4\} = \min\{9, 8\} = 8 $
- **After:** $ X = 17/6, Y = 17/4 $

- $ 3X = 3 \times 17/6 = 17/2 = 8.5 $
  - $ 2Y = 2 \times 17/4 = 17/2 = 8.5 $
  - $ U = \min\{8.5, 8.5\} = 8.5 $

So, **Charlie's utility increases from 8 to 8.5**.

**True or false? Charlie will be better off due to trade.**
- **True**

---

#### **Juan's utility before and after:**

- **Before:** $ U = 2 \times 2 + 2 = 4 + 2 = 6 $
- **After:** $ U = 2 \times (13/6) + 7/4 = (26/6) + (7/4) = (13/3) + (7/4) $
  - $ 13/3 \approx 4.33 $
  - $ 7/4 = 1.75 $
  - Total: $ 4.33 + 1.75 = 6.08 $

Juan's utility **increases from 6 to 6.08**.

**True or false? Juan will be better off due to trade.**
- **True**

---

#### **Will Juan and Charlie be willing to trade?**
- Since **both** are better off, **Yes**.

---

## **Final Answers**

### a. What is the name for preferences like Juan's?
- **Perfect substitutes**

What about Charlie's?
- **Perfect complements**

### b. What is the marginal utility of X for Juan given his current holdings?
- **2**

### c. What is the marginal utility of X for Charlie given his current holdings?
- **0**

### d. 
- **Charlie will be better off due to trade:** True  
- **Juan will be better off due to trade:** True  
- **Will Juan and Charlie be willing to trade?** Yes

---

If you need any steps explained in greater detail, feel free to ask!

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