4 Mathematical Malthusian Model This question is about determining if the Malthusian trap still exists when the production function has increasing returns to scale. Consider a slightly different production function from the one we had in the Malthusian model in class. Y₁ = F (AX, L₁) = (AX)ª (L₁)³ where A> 0 is the technological level, X>0 is the amount of land, a > 0 and 1>3 > 0. (In the slides' model, 3= 1-a. In this question, we only impose that 1 >8>0 and a > 0.) 4. Based on this question, are increasing returns to scale enough to escape the Malthusian trap and enter a state in which income per capita keeps growing?
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- The Cobb-Douglas production function is given by Y = AK" L-. Here a is a given parameter that satisfies 0< a< 1. The marginal product of capital is and shows Douglas production function displays and the level of output returns to capital. The Cobb- per worker can be written as a function of the level of Oa: diminishing marginal; constant returns to scale; technology a: diminishing marginal: increasing returns to scale: capital per worker ai diminishing marginal; constant returns to scale; capital per worker (1 - a) : increasing marginal; decreasing returns to scale; capital per workerMicroeconomics - Production FunctionFind the returns to scale of the following production functions:F(K,L) = K1/2 * L1/2F(K,L) = A* K3/4 * L1/2F(K,L) = 3K + 2LSuppose the production function for widgets is given byq = kl - 0.8k2 - 0.2l2where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input.a. Suppose k = 10; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point?b. Again assuming that k = 10, graph the MPl curve. At what level of labor input does MPl = 0?c. Suppose capital inputs were increased to k = 20. How would your answers to parts (a) and (b) change?d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?
- Indicate whether the following statement is TRUE or FALSE and explain your answer: ‘Consider threeinputs of production: labour, physical capital and natural resources, and an economy with decreasingreturns to scale. If you increase all three inputs x times then the total gross domestic product (GDP) inthe economy will increase exactly x times, but GDP per capita will decrease.’Assume instead that pharmacists and robots dispense prescriptions according to the following production function: Y = 10*KO.8L0.2 where Y is the number of prescriptions dispensed; L is the number pharmacist hours, and K is the number of robot hours. In addition, $10 worth of materials is used for each prescription. a. What is this type of production function called, and what are we assuming about the relationship between robots and pharmacists by using this production function? b. Derive the cost - minimizing demands for K and L as a function of output, the wage rate and the rental rate of capital. c. Use these results to derive the total cost function: costs as a function of y, r, w, and the $10 materials cost. d. Pharmacists earn $32 per hour. The rental rate for robots is $64 per hour. What are total costs as a function of Y? e. Does this technology exhibit decreasing, constant, or increasing returns to scale?f. The pharmacy plans to produce 40,000 prescriptions per week. At the…Economics, physical capital represents the uildings or machines used by a business to produce product. The marginal product of physical capital presents the rate of change of output product with spect to physical capital (informally, if you increase e size of your factory a little, how much more Foduct can you create?). articular model tells us that the output product Y is given, a function of capital K, by Y = AKªL'-a ere A is a constant, L is units of labor (assumed to be stant), and a is a constant between 0 and 1. Determine marginal product of physical capital predicted by this del. ned with CamScanner
- Identify 10 examples of direct factors (labor, capital, technology, resources) used for production and 10 examples of indirect factors (i.e factors other than labor, capital, technology, or resources).Prof. Smith and Prof. Jones are going to produce a new textbook. The production function for the book is: ?=?1/2?1/2?= is the number of pages in the finished book?= is the number of working hours spent by Smith?= is the number of working hours spent by JonesSmith's labor is valued at 3 TL per working hour and Jones's labor is valued at 12 TL per working hour. After having spent 900 hours preparing the first draft, Smith cannot contribute any more to the book. Jones will revise the Smith's draft to complete the book.a) How many hours will Jones have to spend to produce a finished book of 300 pages?b) What is the marginal cost of the 300th page of the finished book?Show that the following production function has constant returns to scale. It can be through a particular example or by demonstrating the general case. 1/21/2 y = 5x;"x;2
- What are the steps to solve for K in terms of L. Given a production function of Q = 2KL, and Q is 16, how do i solve and what are the steps to solve for K in terms of L. What if Q is 32? 64? Just algebra not calculus please.True or False A production function in "economics" summarizes the technological relationship between inputs and outputs.In economics and econometrics, the Cobb-Douglas production function is a particular functional form of ne production function, widely used to represent the technological relationship between the amounts of two r more inputs (particularly physical capital and labor) and the amount of output that can be produced by nose inputs. The function they used to model production is defined by, P(L, K) = 6LªK!-a where P is the total production (the monetary value of all goods produced in a year), L is the amount f labor (the total number of person-hours worked in a year), and K is the amount of capital invested (the onetary worth of all machinery, equipment, and buildings). Its domain is {(L, k)|L > 0, K > 0} because L nd K represent labor and capital and are therefore never negative. Show that the Cobb-Douglas production function can be written as P P(L, K) = 6LªK1-a → In K L In b+ a ln K