Question 5 Please show all calculations to get credit (8 points) a. Penny purcha

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Question 5 Please show all calculations to get credit (8 points) a. Penny purchases two goods, food (F) and clothing (C). The utility that Ann receives by consuming food (F) and clothing Cis given by UlF, Q) - F?C. She has income/-240 and faces prices P, $8 and Py $2The Marginal Rate of Substitution(RSis and the equation of the budget line is given by I P,F +PcC. The tangency condition requires that MRS is equal to the slope of the budget line Write the Statement of the problem, Lagrangian, First Order Conditions as done in class and find out the equilibrium quantity of purchase of F* and C*. This is the baseline case. Now, If the price of clothing increases to $8 (everything else remaining unchanged) rewrite the Statement of the problem, Lagrangian, First Order Conditions as done in class and find out the new equilibrium quantity of purchase of F and C; call them Fnew and Cnew. This is called Case 1 points +2.5 points 5 points) What must the price of x fall to in order for her to be exactly as well off as before the change in P, i.e. We need to calculate p. such that, with the new prices of product y, Penny reaches exactly the same indifference curve as in the baseline case? (3 points) MUF MUC ?? b. Formula Sheet ables au(x) au(xy) MU00, for one variable; MU My or two Tangency Condition: MUP MU Q(L.K) aL aQ(u,k) ?? Tangency Condition: MPL MPK r Steps to Cost Minimization & minimized cost function Steps to Utility Maximization Step 1: Find the production function and set the quantity target- this is your constraint (lsoquant Identify the utility function Step 2: Find the (unminimized) cost function-this is your Identify the budget constraint objective function (lsocost) Step &: Write the statement of the problem: what do you want to minimize? Subject to what constraint? What variables are Write the choice function . Write the lagrangian Find out the 3 first order conditions . Use the first two first order conditions to find out solate the tangency condition and the constraint s endogenous? What variables are exogenous? Step 4: Find out the first order conditions & use them to Step S: Compute equilibrium employment of labor, and the tangency condition acta e union of the esogenous parameters Le, the demand functions Use the tangency condition and the third first order condition to find the optimal consumption Step6:Stick in the factor demand functions into the cost of x and y and U function to find out the minimized cost function

Explanation / Answer

There are two products, foods and clothing. Foods are assumed to be x and clothing are asumed to be y.

The utility function can be written as:

U(F,C)= F2C

or, U(x,y)= x2y

income (I)= 240, Px= $8, Py= $2

where Px and Py are the prices of food and clothing respectively.

The budget line also can be written as:

I= Pxx+Pyy

or, 8x+2y= 240

then the Lagrangian function can be written as:

L= x2y+?(240-8x-2y)

by doing first order condition the following values can be found out:

x= 20, y= 40, U= 16000

Also the tangancy condition has satisfied. i.e.,

MRS=-(MUx/MUy)= -(Px/Py)

here, -(Px/Py)= -(8/2)= -4

MUx=2xy and MUy= x2

then, -(MUx/MUy)= -(2y/x)=-4 (by putting the values of x and y respectively)

This is the baseline case.

b) Now, it is supposed that the price of clothing, i.e., Py has been increased from $2 to $8. All other things remain unchanged.

then, the budget line will be:

I= Pxx+Pyy

or, 8x+8y= 240

or, x+y= 30

then, the Lagrangian function can be written as,

L= x2y+?(30-x-y)

by doing first order condition the following values can be found out:

or, ?L/?x= 2xy-?=0 ........(1)

or, ?L/?y= x2-?=0 ..........(2)

or, ?L/??= 30-x-y=0 .......(3)

from equations (1) and (2) it can be found out that,

x=2y

putting this value of x in equation (3)

30-2y-y=0

or, y=10

then, x= 2.10= 20

So, the budget lin will be,

8.20+8.10= 240- it will also remain unchanged.

Only the amount clothing will be changed because of chsnge in the price of clothing. As price increases, so as law ofdemand, the amount of clothing falls.

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