Question 1202381
**1. Find the Marginal Rate of Substitution (MRS)**

* The MRS represents the rate at which a consumer is willing to trade one good for another while maintaining the same level of utility. 
* For the utility function U = XY, the MRS is given by:
   * MRS = - (dU/dX) / (dU/dY) = - Y/X 

**2. Set up the Tangency Condition**

* At the utility-maximizing point, the MRS must equal the price ratio of the two goods:
   * MRS = - (dY/dX) = - (px/py) 
   * Y/X = px/py 
   * Y = (px/py) * X

**3. Substitute Y in the Budget Constraint**

* Budget Constraint: 936 = pxX + pyY
* Substitute Y = (px/py) * X: 
   * 936 = pxX + py * ((px/py) * X) 
   * 936 = pxX + pxX 
   * 936 = 2 * pxX

**4. Solve for X as a function of px**

* X = 936 / (2 * px) 
* **X = 468 / px** 

**5. Find X0 and U0 when px = 9 and py = 7**

* **X0 = 468 / 9 = 52**

* To find Y0, substitute X0 and px, py into the budget constraint:
   * 936 = 9 * 52 + 7 * Y0 
   * 936 = 468 + 7 * Y0 
   * Y0 = (936 - 468) / 7 = 66

* **U0 = X0 * Y0 = 52 * 66 = 3432**

**Therefore:**

* **X = 468 / px**
* **X0 = 52**
* **U0 = 3432**