Question

Hello, someone posted answer to this question but i am sure part C & D were wrong. Can someone kindly answer it correctly? Thanks in advance.

Linda receives utility from buying clothes and grocery items (At the JBS Grocery shop). Her budget is $304. The JBS shop is offering a $16 coupon if the amount spent on groceries is equal or greater than $100. Let Y be the quantity of units of clothes consumed with a price of $1 per unit and X the quantity of grocery items consumed with a price of $2 per unit/item. Linda's utility function is: U (X,Y) = 3/4LnY + 1/4 LnX

1) Find the optimal allocation of groceries (X) and clothes (Y) for Linda if her budget is $304.

2) Find the optimal allocation of groceries (X) and clothes (Y) if Linda's budget is $320. (NOTE***Here i do understand how the answer is x= 40 units of groceries and y=240 units of clothing BUT i do not understand how it will look graphically because at 40units of groceries means Linda spent 40units*2$/unit= 80$ but then how did she get the 16$ coupon if she spent 80$ and not a 100$. It doesn't make sense to me?

3) Based on your answers of part a and b, Could you tell what would Linda's optimal allocation be if she receives the offer from ABC shop? (Note*** Here I do not understand why the answer is x=40 and y=240 is it because linda is in theory richer?)

4) Could Linda's utility be higher if she chooses to buy 50 grocery items ? Could you explain your answer drawing an indifference curve? Would you consider this result a corner solution? (Note*** I think her utility stays the same but then again i do not know why. Would really appreciate some explanation. Please also let me know if this is a corner solution!)

Answer 1

Answer 1 :-

As per the utility function given in the question Linda receives highest utility when she spends 1/4th of income on groceries and 3/4th of her income on clothes . Thus when her income is 304 .

Amount of clothes she buys = 304*3/4 = 228 units .

Amount of groceries she buys = ((304/4)/2)= 38 units

Answer 2 :-

When her income rises to $320 ,

The amount of clothes she purchases = (320*3)/4 = 240 units

The amount of groceries she purchases = ((320/4)/2) = 40 units

Answer 3 :-

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In the given figure it is evident that Linda can only avail the offer when she chooses to spend atleast 100$ on groceries .

Thus technically her allocation of quantity becomes (50+8) = 58 units

At a price of 100$ only .

However the offer of $16 is treated as an additional income that enables the consumer to purchase more .

Thus in the above figure the line starts from $116(100+16) . After the consumer spends $100 that is at $16(116-100) he has purchases 50 units already and now this addition will enable her to purchase 8 more units totalling up-to 58 units .

This effect is represented by the area's between A to B .

Answer 4 :-

In the above figure the budget line 1 represents the allocation when her income is $304 . The maximum point of utility is point F on IC 2 .

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Now BL2 represents budget line when her income is $320 . Where she receives highest satisfaction at point E at IC1 .

To understand the effect of the coupon we must first understand that she has to change her allocation in order to get the reward .

If we consider that her income is $320 .

She receives of a utility of ( 240+40) utils .= 280 utils

Now she is ready to change her budget allocation and purchase atleast 50 units of groceries in order to a coupon of $16 which will fetch her 8 more units without extra cost .

Thus the new budget line starts from A to C and then slopes to D representing coupon quantity of 8 units from A to C and quantity purchases from C to D .

When she buys 50 units of X automatically her clothes consumption reduces to 220 units . Thus her new expenditure falls on point G at IC3 .

Now she is getting a satisfaction of (220+58) = 278 utils which is lower than the previous allocation.

Thus even if she receives a coupon and moves to a higher indifference curve IC3 her utility is lower than before.

In the figure we can see that point E is above point G .