Concisely show your working/explain your reasoning.



[Answer the following questions. Concisely show your working/explain your reasoning. Total points = 56] 1. [8 pts] A consumer’s utility function is u(x1, x2) = √ x1 + 2√ x2. (i) [6 pts] Derive expressions for the CV and EV that depend only on the parameters p1, p2 and M. In principle, CV and EV also make sense when both prices change – you’re on some indifference curve before, and on another one after, and the CV/EV just tell you how much money you’d need to move between these two indifference curves. So allow your formulas to capture an arbitrary price change from (p1, p2) to (p 0 1 , p0 2 ). (ii) [2 pts] Suppose prices change from (p1, p2) = (1, 1) to (p 0 1 , p0 2 ) = (2, 3). What is the CV? Interpret the number you get. 2. [2 pts] Adapted from Exercise 5B.2 from the textbook: How would we interpret an upward sloping region of an isoquant? Think of an example where this would make sense. 3. [12 pts] A production function is said to be homothetic if it can be written as an increasing transformation of a function that is homogeneous of degree 1 in the inputs. That is, we can write f(z1, z2) = F(g(z1, z2)) for some F(·) and g(·), where F(·) is strictly increasing, F(0) = 0 and g(·) is homogeneous of degree 1 in (z1, z2). (i) [6 pts] For the following functions say whether it is homothetic and, if so, write down F(·) and g(·): (a) f(z1, z2) = α ln z1 + (1 − α) ln z2 (b) f(z1, z2) = z 2α 1 z 2(1−α) 2 (c) f(z1, z2) = z1 + z2 + 5. [Hint: compare them to some homogeneous of degree 1 functions you know. How would you have to transform those functions you know to get the functions above?] (ii) [3 pts] Show that a homogeneous of degree 1 production function is also homothetic. What are F(·) and g(·)? [Interesting fact: more generally, a homogeneous function of any degree is homothetic – I’ll provide a proof in the solutions – see if you can show it yourself (not graded)] (iii) [3 pts] We saw in class that the MRTS is scale invariant for a homogeneous function. Show that the MRTS is also scale invariant for a homothetic production function. [It is this property, among others, that makes homothetic functions useful in economics – they behave similarly to homogeneous functions but are a little bit more general and include all homogeneous functions as a subset] 2 4. [9 pts] For each of the following production functions, plot the total product, marginal product and average product against z1, holding z2 fixed at z2. (i) [3 pts] f(z1, z2) = z1 + z1z2. (ii) [3 pts] f(z1, z2) = max{z1, z2}. [This could be a case where z1 are two alternative inputs z2 that are mutually exclusive and can’t be combined – e.g., a machine that can run on gasoline or on batteries but not both at the same time] (iii) [3 pts] f(z1, z2) = min  z 1 2 1 z 1 2 2 , 10 . [That is, we have a normal Cobb-Douglas production function except that its output is capped at 10. This could be a machine that physically can’t produce more than 10, or maybe the firm is regulated by the government and not permited to produce more than 10] 5. [25 pts] A firm has the following production function: f(z1, z2) = z 1 4 1 z 1 2 2 . (i) [2 pts] Solve the short-run cost minimization problem where z2 is fixed at z2. (ii) [2 pts] Write down the short-run cost function. Indicate the variable and the fixed costs. (iii) [2 pts] Solve the short-run profit maximization problem. That is, find the short-run supply function. (iv) [1 pts] Write down the short-run profit function. (v) [3 pts] Solve the long-run cost minimization problem. (vi) [3 pts] Draw the “expansion path”. That is, draw the set of input bundles that minimize cost as y varies from 0 to ∞ (i.e., varying the target isoquant). [Your vertical axis should be z2 and horizontal axis should be z1. ] (vii) [2 pts] Write down the long-run cost function. (viii) [3 pts] Verify Shephard’s lemma with respect to p1. (ix) [3 pts] Using the long-run cost function derived above, solve the long-run profit maximization problem. That is, find the long-run supply function. (x) [1 pts] Write down the long-run profit function. (xi) [3 pts] Verify Hotelling’s lemma with respect to the output price p. [To simplify the algebra, if you see a big constant that doesn’t depend on p, just call the whole thing k]

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