Steve’s preferences for hamburgers (X) and chicken sandwiches (Y) can be represe

Question

Steve’s preferences for hamburgers (X) and chicken sandwiches (Y) can be represented by u(X,Y)=X 0.5Y 0.5 . a) If the price of a can of a hamburger is pX, the price of a chicken sandwich is pY, and Steve’s monthly budget for fast food is w, derive Steve’s monthly demand functions for hamburgers X(pX,pY,w) and chicken sandwiches Y(pX,pY,w). Start by stating Steve’s maximization problem. b) What would Steve consume if pX=$10, pY=$7.50 and w=$300? c) An outbreak of mad cow disease reduces the local supply of beef. Imported beef is more expensive, driving up the price of hamburgers to $15. How many hamburgers and chicken sandwiches does Steve now consume? d) Graphically decompose the total effect of the price change into an income effect and substitution effect. Plot hamburger consumption on the horizontal axis. Label clearly. e) Are hamburgers a normal or inferior good? Explain. Just a sample study question for my test tomorrow. Really need to understand so if you can be as detailed as possible that would be greatly appreciated. Thanks!!

Explanation / Answer

price of a can of a hamburger is pX : pX=$10,

the price of a chicken sandwich is pY : pY=$7.50

monthly budget for fast food : w=$300

Budget Line: $300 = $10x + $7.5y

Two conditions must be satisfied: (I) MRS = Px/ Py and (ii) I = Px + Py with both prices and income taken as givens.  

(I) MRS = Px/ Py = 10/7.5 =1.33   I = Px + Py = $300 = $10x + $7.5y

So our two equations are:

MRS= MUx/MUy

MUx=0.5x^-0.5y^0.5

MUy=0.5x^0.5y^-0.5

(i) y/x = $10/$7.5 = 1.33 which can be rearranged as y = 1.33x,   and (ii) $300 = $10x + $7.5y

Next, substitute (i) into (ii) and solve for x as follows-- $300 = $10x + $7.5(1.33x)

so x = 15. We can now find y by plugging x= 15 into (I) : y=20 =1.33*15

The demand curve is an equation which shows quantity demanded as a function of price. Using above two equations again , we can find the demand curve as follows:

y/x = Px/$2 which solving for y gives us   y = (Pxx)/4. If we substitute this equation into the budget constraint and solve for x we get the demand curve--

$300 = Pxx + 2(Pxx/4) = Pxx + Pxx/2 = 1.5(Pxx)

$200 = (P­xx) which gives us    200/Px = x. This is the Demand Curve

First, find the equation for the MRS. We know MRSxy = MUx/MUy where MUx = ¶U/¶x = axa-1yb and   MUy = ¶U/¶ y = bxa yb-1 . Therefore, MRS =(axa-1yb)/(bxa yb-1) = (a/b)(y/x).

Second, set MRS equal to Px/Py and solve for y:

(a/b)(y/x)= Px/Py or, y = (b/a)(Pxx/Py).

Third, plug the above result into budget line and solve for x:

I = Pxx + Py [(b/a)(Pxx/Py)] = [(a+b)/a][ Pxx]. Solving for x we get: x = [a/(a+b)](I/Px ). This is the demand curve for Cobb-Douglas utility functions.

For y, following the same steps, you would get   y = [b/(a+b)](I/Py).

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