Homework Assignment 7 Suppose that the output q of a firm depends on the quantities of z1 and z2 that it employs as inputs. Its output level is determined by the production function 1. q = 26z1 + 24z2 − 7z2 1 − 12z1z2 − 6z2 2 2. Write down the firm's profit function when the price of q is $1 and the factor prices are w1 and w2(per unit) respectively. 3. Find the levels of z∗ 1 and z∗ 2 which maximize the firm's profits. Note that these are the firm's demand curves for the two inputs. 4. Verify that your solution to [2] satisfies the second order conditions for a maximum. 5. What will be the effect of an increase in w1 on the firm's use of each input and on its output q? [hint: You do not have to explicitly determine the firm's supply curve of output to determine ∂q/∂w1. Instead write out the total derivative of q and make use of the very simple expressions for ∂q/∂z1 and ∂q/∂z2 at the optimum that can be obtained from the first order conditions.] 6. Is the firm's production function strictly concave? Explain

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1. Write down the firm's profit function when the price of q is $1 and the factor prices are w1 and w2(per unit). P = W1 + W2 ? q 2. Find the levels of z? 1 and z? 2 which maximize the firm's profits. Note that these are the firm's demand curves for the two inputs. z? 1 = 26 ? 1w1 z? 2 = 24 + 1w2 3. Verify that your solution to [2] satisfies the second order conditions for a maximum. Yes, the solution does satisfy the second order conditions for a maximum.

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7 . Consider a perfectly competitive firm that produces output from capital and labor, according to the production function: q f(K,L): The production function has continuous first and second partial derivatives with the following properties: f > 0,fk > 0,fu < 0,fkk <0, fuk = fkl >0. The firm can purchase as much capital (K) and labor (L) as it wants to at their market prices: > 0 (the unit price of capital) and w > 0 (the unit price of labor), which the firm takes as given. It can sell as much output as it likes at the market price for its product: p > 0, which the firm takes as given. Find both first-order partial derivatives of the firm'$ profit function: T(K,L) = p f(K,L) (rK + wL): What two equations describe the first order, necessary conditions for a local max of the profit function? Find the total differentials of the two equations that describe the first order, necessary conditions for a local max of the profit function_ [dK Letting the vector of unknowns, X, be defined as x = write the system of two dL equations from part € as a matrix equation, Ax = b_ Use Cramer's rule to find the partial derivatives dK* /dr and dK* /dw of the capital demand function, K* g (r ,W,p), implicitly defined by the first order, necessary conditions for a local max of the profit function. What are the signs of dK* /dr and dK* /dw?

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Comma l, minus n k, plus w l, find first order derivativeso to solve this question, because we have been given production function sown by pi, equal to p, cube minus w, l plus r, and I equal to p f of k. The parts are divided by de k, small p del f, divided by de k, minus del k, r, and a plus w n. The tail pi, divided by l, is equal to p tal. minus tail f, divided by tall arch, a plus w l equal to p d f, divided by tail n plus w tall divided by tail n part b. We need to find condition for local maximum 4 local and maximum d f equal to 0, so tal f, divide by tall multiply, b, l plus aLF, divided by a multiply dek equal to 0, which means tail f. Since t's equal to p f of k, l minus w l plus r o d pi equal to tail pi divided. If you divide deca by del pi, you get deca plus del pi. Minus r t k plus p de divided by tail l, minus wt, l, p y equal to p al f idyl k, t k, plus delta f, divided. W l plus arca is equal to 0 del f, divided by a l and tall. It is equal to 0 in part b. From the past questions, let x equal to t k, t n p so that the solution is equation number 1 and equation number 2. It is possible to write kisinumer, 1 and 2 in matrix form capital a equal to tal, f, divided by 10 k. The answer is equal to a x and a v.

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