Show that the Cobb-Douglas production function Y = KOL, where 0 < a < 1 and 0 0 and L> 0 by constructing its bordered determinant and inspecting the signs of its two leading principal minors.
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- Question: Orla manages a loom that produces flags (F) using thread (T) and dye (D) as inputs. Herproduction function is given by: Q(T,D) = (T1/2 D1/2)1/2*For this problem, assume F, T, and D are infinitely divisible so you don’t need to worryabout restricting to whole-number answers. a.) Does Orla’s production function exhibit increasing, constant, or decreasing returns toscale? Explain. b.) Set up Orla’s cost-minimization problem to find the lowest-cost combination of inputsrequired to produce a specific level of output (bar Q) given factor prices PT and PD. (Note: You can write this either as a minimization subject to constraints or in Lagrangian form. *You do not need to solve it.)10. Extension of the Cobb-Douglas Production Function-The Cobb-Douglas produc- tion function (Equation 7.16) can be shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form:" Q = y[@K* + (1 – a)L°]W/P where y is an efficiency parameter that shows the output resulting from given quantities of inputs; a is a distribution parameter (0 sas 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the pro- duction process; and v is a scale parameter (v > 0) that indicates the type of re- turns to scale (increasing, constant, or decreasing). Show that when v = 1, this function exhibits constant returns to scale. [Hint: Increase capital K and labor L each by a factor of A, or K* = (A)K and L* = (1)L, and show that output Q also increases by a factor of A, or Q* = (A)(Q).]Monsters, Inc. has two plants for producing electricity, one in Monsterville and another is in New Yeti. The Monsterville plant produces according to eM(x1,x2) = min{x1,2x2} and the New Yeti plant produces according to eNY(x1,X2) = min{2x1,X2}, where x1 and x2 are the inputs. i. Describe in words how the typical graphs of isoquants for Monsterville and New Yeti plants would look like. Identify the main aspects of the isoquants. ii. Suppose that the firm wishes to produce 20 units of electricity in each plant. How much of each input will the firm need in order to produce 20 units of electricity at the Monsterville plant?
- 1. Pekan Nenas has 20 competitive pineapple orchards, all of which sell pineapples at the world price of RM2 per pineapple. The following equations describe the production function and the marginal product of labor in each orchard: where Q is the number of pineapples produced in a day, L is the number of workers, and MPL is the marginal product of labor. (a) (b) (c) Q = 100L - L² MPL = 100 - 2L, (d) What is each orchard's labor demand as a function of the daily wage W? What is the market's labor demand? Pekan Nenas has 200 workers who supply their labor inelastically. Solve for the wage W. How many workers does each orchard hire? How much profit does each orchard owner make? Calculate what happens to the income of workers and orchard owners if the world price doubles to RM4 per pineapple. Now suppose that the price is back at RM2 per pineapple but a flood destroys half the orchards. Calculate how the flood affects the income of each worker and of each remaining orchard owner. What…1. Let y = f(x1, x2)=x11/2+ X1X2 be a firm's production function, where x20, x220. - a. Write down the firm's production possibility set, and its input requirement set.“ b. Is this production function concave, quasi-concave? c. Is this production function homogenous, homothetic? + d. Find its returns to scale when x1=1, and x2=1.e6. Consider the following Cobb - Douglas utility function: U = χαγβην *Note, it should be assumed that a, B.y > 0 Show that this production function can exhibit increasing returns to scale globally while maintaining diminishing returns for each individual input.
- 1. The total cost of producing a quantity q is C(q). The average cost a(q) is given in figure below. The following rule is used by economists to determine the marginal cost C'(q0), for any qo: Construct the tangent line ti to a(q) at qo. Let t, be the line with the same vertical intercept as t, but with twice the slope of t, . Then C'(q0), is the vertical distance shown in figure below. Explain why this rule works. $/unit a(q) t2 t1 C'(qo)Given TC = 120 + 50Q – 10Q2 + Q3Finda. The equations of the TVC, AVC, and MC functions.b. The level of output at which AVC and MC are minimum, and prove that the AVC andMC curves are U-shaped.c. Find the AVC and MC for the level of output at which the AVC curve is minimum.7 which of the following are examples of fixed-proportions production functions?
- QUESTION Given TC = 100 + 60Q – 12Q2 + Q3 find: (a) The equations of the TVC, AVC, and MC functions. (b) The level of output at which AVC and MC are minimum, and prove that the AVC and MC curves are U-shaped. (c) Find the AVC and MC for the level of output at which the AVC curve is minimumY-2x11 y2-3X12 Q 4. ( A) Consider the following production functions: y2 are products and x is the use of input i in the production of productj. The total use of x is a function of y, and y2 given by: x1 =;+ where yi and %3D a) Find the product transformation rate. b) What will be the shape of production possibility curve? c) Are the productive competitive ORIf the production function y = 0.5 = 2x at an input price Of (DR. 300 and a fixed cost of IDR-5000 and an Output price of 10R-500- find : MPP, APP, AVC, AC, MC, AVP. MVP, MFC. describe the maximization of profit using the principle MVP = MFC or MR = MC