Question

can someone explain this step by step? especially don't understand how they got m/px? why we can't use the lagrange. and not sure how they drew the graph for this question. SOMEONE PLEASE HELP MIDTERM SOON AND WILL GIVE BIG THUMBS UP!!!! confused where 4x20+5x0 is coming from

Problem 4 Eric's preferences for goods x and y are represented by the following utility function: U(X,Y) = 4X +5Y. The price of good X is px = 2 and the price of good Y is Py = 3. Finally, Eric's income is I = 10. a) Find the marginal rate of substitution. Comment if it is increasing, constant or decreasing in X. b) Write the constrained optimization problem that Eric faces. c) Find the optimal bundle (X*,Y*). d) Find Eric's utility at the optimal point, and find another bundle that gives him the same level of satisfaction. It the alternative bundle feasible? Why? e) Draw a graph that includes the following curves: ( Eric's budget constraint, (ii) Eric's optimal bundle and (iii) the indifference curve associated with Eric's optimal bundle. * MRS = =-MOX - -ų . MRs is constant. X MUY constrained Optimisation : Max U=Ux+ 5y: Perfect substifutes: Lagrange not I sit 2x + 3y =10 usable © Corner solution. Abs.value = 1 MRT - -2 :Absolute value MUY - MUY MRS MRS > MRT ++ only consume x.. / >ų @ Otility 1 x - 2 = 2 4x+5y = 4X20 +500 = 80: Alternate bundle not жос 7 sahn at at

Answer 1

a)

U(X, Y) = 4X + 5Y

X and Y are perfect substitutes of each other. There exists a corner solution in this case. Eric either spends his entire income on good X or on good Y.

Marginal Rate of Substitution = MUx / MUy

MUx = Derivative of U wrt X = 4

MUy = Derivative of U wrt Y = 5

Hence,  Marginal Rate of Substitution = MUx / MUy = 4/5

Since, it is a constant numbere, so MRS in constant in X

b.

Constrained Optimization problem for Eric.

Maximize U(X, Y) subject to Budget Constraint.

Let Budget Constraint: M = Px*X + Py*Y

Px = 2, Py = 3 , M = 10

Budget Constraint: 10 = 2X + 3Y

Hence, the Constrained Optimization problem is following:

Maximize U(X, Y) st 10 >= 2X + 3Y

c.

Now, the optimal bundle will have 0 units of either of the two goods are both are perfect substitute of each other. A slight fall in the price of good X would induce Eric to demand all X and no Y.

Similarly, A slight fall in the price of good Y would induce Eric to demand only Y and 0 units of X

So, the optimal bundle is either (X* , 0) or (0, Y*)

This depends on the MRS and Price Ratio whichever is greater.

Let say MRS > Price Ratio

MUx / MUy > Px / Py

MUx / Px > MUy / Py

Eric derives more utility from good X and therefore will consume only X

Similarly,

Let say MRS < Price Ratio

MUx / MUy < Px / Py

MUx / Px < MUy / Py

Eric derives more utility from good Y and therefore will consume only Y

In our case, Price Ratio = 2/3 = .67

And MRS = 4/5 = 0.8

Since, MRS > Price Ratio. Eric will consume only X and 0 units of Y

So, Y = 0. Put this Y in Budget Constraint: 10 = 2X + 3Y

10 = 2X + 3*0

X* = 5

So, OPTIMAL BUNDLE: (X*, Y*) = (5 , 0)

d)

Eric's utility at this optimal bundle:

U = 4X + 5Y

U = 4*5 + 5*0

U = 20

Now, Eric's budget income = 20

Since, all the Eric's income has been exhausted in purchasing X and Y, there will no other bundle that would be feasible for Eric.

e)

Now, Budget Line: 10 = 2X + 3Y

3Y = 10 - 2X

Y = 10/3 - 2/3X

Hence, the slope of budget line = 2/3

Here, the Y intercept = 10/3 ( X = 0)

and X intercept = 5 (Put Y =0)

Now, U = 4X + 5Y

5Y = U - 4X

Y = 1/5*U - 4/5X

Slope of IC = 4/5 = 0.8

Slope of IC > Slope of budget line

Therefore, the ICs will be steeper than the budget line

Now, at optimal bundle Y =0

and X* = 5

10/3 T → IC (ii) Budget Line (i) + ii) Optimal "Bundle

If you are unclear on something, then please ask in comments.

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