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2. Let $y = f(z_1, z_2) = f(z)$ be a production function, the price of y is p, and the factor prices are given by $w_i$ for input $z_i$, for $i = 1, 2$ a. Write down the profit maximization problem. b. Assume $f(z)$ is Cobb-Douglas such that the factor \"weights\" $a_i$ are nonnegative and sum to 1. Show this technology is constant returns to scale. Derive the RTS for this technology. c. Now, assume the factor weight sum to less than 1. Show the technology is decreasing returns to scale. Show if they sum to greater than 1, we have increasing returns to scale. d. That is the first order condition to the profit maximization problem in part (a)? e. Say production is $y = \sum_{i} a_i z_i$ with $a_i > 0$. Show this technology has constant returns to scale. f. Say production is $y = (z_1^a + z_2^a)$ for $a, b > 0$. When does this technology have constant returns to scale? How about decreasing returns to scale? g. Say the technology is constant returns to scale. Show economic profits are zero for the firm in the long-run. 3. For question 2, do the following: (a) write down the expenditure minimization problem. (b) What is the first order conditions for the expenditure minimization problem? (c) Show that in either the profit max problem or the expenditure min problem, for an interior solution for both factors, the slope of an isoquant is the RTS. (d) Now, draw a picture that shows how the profit max problem is \"dual\" to the expenditure min problem. Explain this \"duality\" in detail. 4. Say the production technology is $y = f(z_1, z_2)$, and (a) $z_2$ is fixed at a level strictly below $z_m(w)$ where $y = f(z_m(w), z_2)$ and (b) $f_a(z)$ is strictly

          2. Let $y = f(z_1, z_2) = f(z)$ be a production function, the price of y is p, and the factor prices are given by $w_i$ for input $z_i$, for $i = 1, 2$
a. Write down the profit maximization problem.
b. Assume $f(z)$ is Cobb-Douglas such that the factor \"weights\" $a_i$ are nonnegative and sum to 1. Show this technology is constant returns to scale. Derive the RTS for this technology.
c. Now, assume the factor weight sum to less than 1. Show the technology is decreasing returns to scale. Show if they sum to greater than 1, we have increasing returns to scale.
d. That is the first order condition to the profit maximization problem in part (a)?
e. Say production is $y = \sum_{i} a_i z_i$ with $a_i > 0$. Show this technology has constant returns to scale.
f. Say production is $y = (z_1^a + z_2^a)$ for $a, b > 0$. When does this technology have constant returns to scale? How about decreasing returns to scale?
g. Say the technology is constant returns to scale. Show economic profits are zero for the firm in the long-run.
3. For question 2, do the following:
(a) write down the expenditure minimization problem.
(b) What is the first order conditions for the expenditure minimization problem?
(c) Show that in either the profit max problem or the expenditure min problem, for an interior solution for both factors, the slope of an isoquant is the RTS.
(d) Now, draw a picture that shows how the profit max problem is \"dual\" to the expenditure min problem. Explain this \"duality\" in detail.
4. Say the production technology is $y = f(z_1, z_2)$, and (a) $z_2$ is fixed at a level strictly below $z_m(w)$ where $y = f(z_m(w), z_2)$ and (b) $f_a(z)$ is strictly

2. Let y = f(z1, z2) = f(z) be a production function, the price of y is p, and the factor prices are given by wi for input zi, for i = 1, 2
a. Write down the profit maximization problem.
b. Assume f(z) is Cobb-Douglas such that the factor ẅeights$̈aiare nonnegative and sum to 1. Show this technology is constant returns to scale. Derive the RTS for this technology.
c. Now, assume the factor weight sum to less than 1. Show the technology is decreasing returns to scale. Show if they sum to greater than 1, we have increasing returns to scale.
d. That is the first order condition to the profit maximization problem in part (a)?
e. Say production isy = ∑i ai ziwithai > 0. Show this technology has constant returns to scale.
f. Say production isy = (z1^a + z2^a)fora, b > 0. When does this technology have constant returns to scale? How about decreasing returns to scale?
g. Say the technology is constant returns to scale. Show economic profits are zero for the firm in the long-run.
3. For question 2, do the following:
(a) write down the expenditure minimization problem.
(b) What is the first order conditions for the expenditure minimization problem?
(c) Show that in either the profit max problem or the expenditure min problem, for an interior solution for both factors, the slope of an isoquant is the RTS.
(d) Now, draw a picture that shows how the profit max problem is d̈ualẗo the expenditure min problem. Explain this d̈ualityïn detail.
4. Say the production technology isy = f(z1, z2), and (a)z2is fixed at a level strictly belowzm(w)wherey = f(zm(w), z2)and (b)fa(z)is strictly

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