Can someone help me with budget constraint slope and MRS ?

i searched many websites, but all gave very conflicting opinions.
this is my conclusion, please correct me if im wrong.

1) MRS = slope of indifference curve <- correct? if wrong, what is MRS, and how is the slope of indifference curve calculated?

2) MRS = Px / Py and MRS = dX / dY (d means delta, change in) and MRS = MUx/MUy , this always is true whether or not its at a optimum point. correct?

3) slope of budget constraint = Px/Py and = dX / dY (d means delta, change in) regardless of at optimum point or not. true?

something is wrong with my conclusion for sure, because if my premiss is correct then slope of budget constraint is always =
MRS, which is not true.

Answer:
I assume you know a bit of calculus.

1) correct. Obtain it by totally differentiating the utility function U = f(X,Y), viz: dU = f'xdX + f'ydY = 0 and solve for dY/dX which is the MRS. In words: at the margin, the units of Y to be lost per unit of X gained, without any loss or gain of utility.

2) MRS = dY/dX (use this form if you are using Px/Py) - see above.

3) Slope of the budget line = Px/Py (in absolute value). Note that dY/dX is used for any curve. For example:

Y = B/Px - (Px/Py)X; dY/dX = - Px/Py = slope of budget line.

So if you are using dY/dX for MRS, then no it isn't always true that MRS = Px/Py. As you suspect, this is only true at the optimum. (technically: this must be true at the optimum but need not be true elsewhere.)

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