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Homework answers / question archive / University of California, San Diego - ECON 100A Homework 3  Question1)Matt gets utility from candy bars (C) and movies (M)

University of California, San Diego - ECON 100A Homework 3  Question1)Matt gets utility from candy bars (C) and movies (M)

Economics

University of California, San Diego - ECON 100A

Homework 3

 Question1)Matt gets utility from candy bars (C) and movies (M).  Matt ALWAYS eats 2 candy bars at each movie he attends.  Extra candy without seeing another movie is of no value to Matt, just as going to the movies is of no value without the candy bars.  Matt maximizes his utility from movies and candy bars subject to a budget constraint. 

 

1A. What type of utility function represents Matt’s preferences?  Write down an expression for Matt’s utility.     

 

1B.  Draw a picture of Matt’s utility maximizing problem. (See 1F below)

 

1C.  Will you be able to use the Lagrange multiplier technique to solve for Matt’s utility maximizing combinations of movies and candy bars?  Why or why not?

 

 

1D. What is Matt’s utility maximizing combination of movies, M*(PM, PC, I), and candy bars,          C*(PM, PC, I)?

 

1E.  What is the Matt’s indirect utility function, i.e., V(PM, PC, I)=U(M*(PM, PC, I),C*(PM, PC,I))?

 

 

1F.  What are the numerical values of C*, M* and U*?  Put these values on your graph,  

 

 

 

 

Question 2:  Now, let’s see if you can do the general perfect complements problem.  Matt gets utility from X and Y.  Matt always consumes αX with βY.  

 

2A.  What type of utility function represents Matt’s preferences?  Write down an expression for Matt’s utility.

 

 

2B. What is Matt’s utility maximizing combination of X*(Px, Py, I) and Y*(Px, Py, I)?

Use the facts that 

  1. that there will be no excess X or Y (so 1/αX*=1/βY*) and 
  2. the utility maximizing bundle must be on the budget constraint (PxX*+ PyY* =I).

 

 

2C.  What is the Matt’s indirect utility function, i.e., V(Px, Py, I)=U(X*(Px, Py, I), Y*(Px, Py, I))? 

 

 

2D. What share of his budget does Matt spend on X*?  What share of his budget does Matt spend on Y*?  Prove that the budget shares sum to one.

 

 

 

 

 

 

2E.  Prove that the demand function for X is scale invariant (homogenous of degree zero).

 

Question 3:  Toni gets utility from X and Y.  Toni thinks that one unit of X is exactly as good as four units as Y.

 

3A. What type of utility function represents Toni’s preferences?  Write down an expression for Toni’s utility.

 

 

 

3B.  Draw a picture of Toni’s utility maximizing problem.  (see 3F below)

 

3C.  Will you be able to use the Lagrange multiplier technique to solve for Toni’s utility maximizing combinations of X and Y?  Why or why not?

 

 

3D. What are Toni’s ordinary demand functions, X* (Px, Py, I), and Y* (Px, Py, I)?

 

    Use two conditions:

  1. The fact that either X*=0 or Y*=0.
  2. The budget constraint

 

 

 

3E.  What is the Toni’s indirect utility function, i.e., V(Px, Py, I)=U(X*(Px, Py, I),Y*(Px, Py, I))?

 

 

3F.  What are the numerical values of X*, Y* and U*?  Put these values on your graph, 

 

 

 

Question 4:  Now, let’s see if you can do the general perfect substitutes problem.  Toni gets utility from X and Y.  Toni thinks that α units of X are exactly as good as β units of Y.  

 

4A.  What type of utility function represents Toni’s preferences?  Write down an expression for Toni’s utility.

 

     

 

4B. What are Toni’s ordinary demand functions, X*(Px, Py, I) and Y*(Px, Py, I)?

 

  

 

4C. What is the Toni’s indirect utility function, i.e., V(Px, Py, I)=U(X*(Px, Py, I), Y*(Px, Py, I))? 

 

4D. What share of her budget does Toni spend on X*?  What share of her budget does Toni spend on Y*? 

 

 

 

Question 5:  Consider the utility maximization problem subject to a budget constraint with the following utility function:  maxU(x, y) =x+ 2 y

x,y

5A.  Are the commodities x and y goods?  Prove your answer.  What does this imply about the indifference curves for U?

 

5B. Do the indifference curves for this utility function exhibit diminishing MRS?  Prove your answer.

 

 

5C.  Write down the first order conditions for this constrained optimization problem.

 

 

5D.  Solve for the optimal consumption bundle, x* and y*, as a function of px , py, and I.

 

 

5E.  A key property of ordinary demand functions is that they are scale invariant (homogeneous of degree zero in all prices and income).  Prove that these demand functions are homogenous of degree zero.

 

 

5F.  What share of her budget does this consumer devote to the consumption of x*? to y*?  What do these two budget shares sum to?      

 

     

 

 

Question 6:  Consider the consumer’s problem to maximize her utility subject to a budget constraint

x0.5 y0.5

with the following utility function:  maxU(x, y) = +           

 

x,y        .5         .5

 

  1. Do the indifference curves for this utility function exhibit diminishing MRS?  Prove your answer.

 

 

  1. Write down the first order conditions for this constrained optimization problem.

x0.5           y0.5

 

  1. Solve for the optimal consumption bundle, x* and y*, as a function of px , py , and I.

 

 

 

 (After you have learn comparative statics, try))

  1. Prove that x*(px, py, I) and y*(px, py, I) are normal goods.

 

x*(px, py,I)      y*(px, py,I)

y   px         x         py

 

  1. Is there an inverse relationship between own price changes and the consumption of x* and y* (i.e., does the ”Law of Demand” hold for x* and y*?

 

Question 7:  Consider the consumer’s problem to maximize her utility subject to a budget constraint with the following utility function:  max!,!? ?, ? =??!?!

 

  1. What are the first order conditions for this constrained optimization problem?

 

  1. Solve for the ordinary demand functions, x*( px , py , I) and y*( px , py , I).

 

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