Question

9. Consider the utility maximization problem max x + y s.t. px + y =m, where the constants p, 9, and m are positive, and the constant a € (0,1). (a) Find the demand functions, x* (p, m) and y* (p, m). (b) Find the partial derivatives of the demand functions w.r.t. p and m, and check their signs. (c) How does the optimal expenditure on the x good vary with p?8 (d) Put a = 1/2. What are the demand functions in this case? Denote the maximal utility as a function of p and m by U* (p, m), the value function, also called the indirect utility function. Verify that aU*/ap = -x*(p, m).

Answer 1

Og Ulny) = x + y Budget line : pray=m. L€(oil). Pq,m> o. (a) The found demand function for both so that both satisfies goods would be the following equation: MRS = 8 by om MRS= a 2 na 30/on dolay -= dna- ana-1 denard of x is independent of income. Putting the demand of is we get P. (2) 2 + y = m. Budget line equation P. (pgati 2 12-1) = m-y paniti (-) 19 ам.

Bernard functions are 4) : * - - - (16)** (b) * = (since de Col1). Sol a leo. Since & € (0,1). Segons would be gn would be negative o as Nh is independent of income OM y* = m-pat Taatio Since & E (0,1) oy>0. 3. 8igin would be the Bigboy would

(C). As prises the Total expenditure for Increase in polices. foon enample: 2= | demand for good a decreases. good n would fall with epal then if p=2 then m = 13 = 14). ? - to When price is equal tol, pal Then Total Expenditure a p. na 1xt at Total Expenditure when p=2 is Total Expenditure = p. uz 2x1 = + <f (d) for x = 1 * 2 (P (t-1 = (apsti) * * = m-pt

putting these values of y* an* in Umay we Utom) = lep+ [in ] et tema (ap) ² + m - upp w* (m) 2 + m up M - O (Pim) = 2 tm. 2 4 TM op up -ve(im). Herce proved
The total expenditure on good x falls as price rises. Also, since the function is quasi linear, demand for good x is independent of income.