SOLUTION: Suppose that you have two consumption choices: good X, and good Y. An indifference curve is the set of consumption choices with a CONSTANT utility. For example if consuming 10X and

Question 1202381: Suppose that you have two consumption choices: good X, and good Y. An indifference curve is the set of consumption choices with a CONSTANT utility. For example if consuming 10X and 6Y gives me the same utility as consuming 11X and 5Y, then these are both points on the same indifference curve. An indifference map is the set of all indifference curves with EVERY given utility.
Consider the indifference map given by:
U=XY , where U is a measure of utility.
A budget curve gives the set of possible consumption choices with a given income. If you have an income of $936 and the price of good X is given by px, and the price of good Y given by py. The equation for the budget line is given by: 936=pxX+pyY.
A utility maximizing combination of goods X and Y occurs when the budget line is tangent to a indifference curve.
Find X as a function of its price. (If Y represents all other goods, than this function is just a demand curve for X).
X=
(Use px for px)
Let X0 and U0 be the values for X and U when px=9 and py=7.
X0=
U0=
Answer by GingerAle(43) (Show Source): You can put this solution on YOUR website!
**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**

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