Econ 100A Problem Set #4: Demand Spring 2022
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Econ 100A
Problem Set #4: Demand
Spring 2022
1. Matt gets utility from candy bars (C) and movies (M). Matt always eats exactly 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. (On PS3, #7(d) you solved for Matt’s demand functions, C*(PC, PM, Y) and M*(PC, PM, Y).) Verify that his demand functions satisfy Walras’s Law.
2. Consider the general perfect complements problem. Matt gets utility from x1 and x2 . He always consumes α units ofx1 with β units ofx2 . The price ofx1 isp1, the price ofx2 isp2 and his income is Y.
a. Write down an expression for Matt’s utility function.
b. Find Matt’s utility maximizing bundle, x1*(p1, p2, Y) and x2*(p1, p2, Y).
c. What share of his budget does Matt spend on x1? On x2? Show that the budget shares sum to one.
d. Prove that the demand function for x1 is scale invariant (homogeneous of degree zero).
3. Shirley enjoys drinking beer. Her local pub serves 16 oz (S) and 24 oz beers (L). She does not care about the size of each beer; she only cares about how much beer she drinks. For
the remaining parts of this problem put 16 oz beers on the x-axis and 24 oz beers on the y-axis. (On PS3, #8(g) you solved for Shirley’s demand functions, S*(PS, PL, Y) and L*(PS, PL, Y).)
a. What share of her budget does Shirley spend on 16 oz beers? On 24 oz beers? Show that the budget shares sum to one.
b. Prove that her demand function for S is scale invariant (homogeneous of degree zero).
4. Consider the following Cobb-Douglas utility maximization problem:
max U 
x1 , x2 
8x12x
subject to p1x1 
p2x2 
Y x1 ,x2
a. Are the commodities x1 and x2 goods? Explain your answer.
b. Do the indifference curves exhibit diminishing MRS? Prove your answer.
c. Without transforming U(x1, x2) and without referring to our results for the fully general Cobb-Douglas, use the Lagrangian technique to find the utility maximizing bundle, x1*, x2* as functions ofp1, p2 and Y.
d. Show that these demand functions are scale invariant (homogeneous of degree zero in all prices and income).
e. Find λ* as a function ofp1, p2 and Y. (You probably solved for this in (c).) Use words to interpret.
f. What share of the budget does this consumer spend on x1? On x2? Briefly comment on why these shares don’t depend on any ofthe parameters in the problem.
5. Consider the following constant elasticity of substitution utility maximization problem:
max U 
x1 , x2 
2x12 
2x
subject to p1x1 
p2x2 
Y x1 ,x2
a. Are the commodities x1 and x2 goods? Explain your answer.
b. Do the indifference curves exhibit diminishing MRS? Prove your answer.
c. Use the Lagrangian technique to find the utility maximizing bundle, x1*, x2* as functions ofp1, p2 and Y.
d. Prove that x1 and x2 are normal goods.
e. Are the demand curves downward sloping?
2022-05-11