Consider a firm that borrows to invest in production of capital goods. Let's say theproduction function is given by:Y = 2AK/2where, K is physical capital borrowed and A is 'technology'. From what we havelearnt, the 'real' profit function of this firm would be given as II = Y - (r+ 8) K, wherer and ô respectively represents the real interest rate and depreciation rate on capital.Given this:a) Find the firm's optimal demand for capital K".b) Given the optimal demand for capital what is firm's profit II'?c) Given your answer to part a) above, explain intuitively the observed relationshipbetween K and A, r and 8?

Given: The production function is given by:Y = 2AK1/2 K is = Capital A is = Technology The profit…

Consider a firm that borrows to invest in production of capital goods. Let's say the production function is given by: Y = 2AK/2 where, K is physical capital borrowed and A is 'technology'. From what we have learnt, the 'real' profit function of this firm would be given as II = Y-(r+ 8) K, where r and 8 respectively represents the real interest rate and depreciation rate on capital. Given this: a) Find the firm's optimal demand for capital K". b) Given the optimal demand for capital what is firm's profit II'? c) Given your answer to part a) above, explain intuitively the observed relationship between K" and A, r and 8?

Consider a firm that borrows to invest in production of capital goods. Let's say the production function is given by: Y = 2AK/2 where, K is physical capital borrowed and A is 'technology'. From what we have learnt, the 'real' profit function of this firm would be given as II = Y-(r+ 8) K, where r and 8 respectively represents the real interest rate and depreciation rate on capital. Given this: a) Find the firm's optimal demand for capital K". b) Given the optimal demand for capital what is firm's profit II'? c) Given your answer to part a) above, explain intuitively the observed relationship between K" and A, r and 8?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter11: Profit Maximization

Section: Chapter Questions

Problem 11.9P

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Consider a firm that borrows to invest in production of capital goods. Let's say the production function is given by: Y = 2AK!/2 where, K is physical capital borrowed and A is 'technology'. From what we have learnt, the 'real' profit function of this firm would be given as II=Y-(r+ô) K, where r and õ respectively represents the real interest rate and depreciation rate on capital. Given this: a) Find the firm's optimal demand for capital K*. b) Given the optimal demand for capital what is firm's profit II*? c) Given your answer to part a) above, explain intuitively the observed relationship between K* and A, r and ô?
Suppose a firm optimizes the profit function: Y - wN, and the production function is given by Y = 500N-/4. What is the value of dN*/dw? Evaluate at w = 10.
a) Assume that the output of a company is given by y (K, L) = K^0.5 * L^0.5, where the amount of capital on shortterm is given by K = 1 and wage costs are given by w = 1. Calculate and illustrate themaverage variable costs and marginal costs (marginal costs).b) In addition, assume that the price of capital is given by r = 1. Calculate the average totalsthe costs and illustrate using a figure where you also include the marginal cost asyou found in a). Comment on what you find.
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).]
Whoville Marble Works (WMW) produces Bespoke Marble Kits (BMKs) for discerning marble connoisseurs. To produce their BMKS, WMW uses raw materials, labor and energy, and their production function is given by q = 2070-410.250.35, where o q = WMW's monthly output, measured in BMKS, or = WMW's monthly raw material input, measured in tons, • 1 = WMW's monthly labor input, measured in hours of labor, • m = WMW's monthly energy input, measured in megawatt hours. Raw material costs WMW $100/ton, labor costs WMW $30/hour and energy costs WMW $50/megawatt hour. Find the input levels of raw materials, labor and energy that maximize WMW's output, given that their monthly production budget is $30,000. What is the minimum cost?
(04.Suppose we are given (he constant returns to scale (CES) production function q= { ßa + aa }1/a a. Show that the marginal product of Beta is (g/B) and marginal product of Alpha (q/a)** b. Show that RTS = (2/B); I lence or otherwise shov that the ò =-1/(1-a) 1-a
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1. Assume that a firm has the following production function: q = L0.5K0.6 a. Calculate output when L = 64 K = 32. b. What returns to scale does this production function exhibit? c. Calculate output if technology utilized by the firm increases by 25%. d. Determine, both, the marginal product of labour and capital functions (ignore c.). e. Calculate, both, the marginal product of labour and capital when L = 64, K = 32 and interpret your answers. f. Calculate the marginal rate of technical substitution when L = 64, K = 32 and interpret your answer.
According to the Law of Diminishing Returns As more units of a variable factor are used with a fixed factor, beyond some point total production will begin to decline.. Investing more money into the same financial investment will always result in a diminished fiancial return. O If a firm makes another type of product available to its customers, then it will generate less revenue from its new type of product compared to its existing products. As more units of a variable factor are used with a fixed factor, beyond some point it becomes harder and harder for a firm to increase production.
Suppose the representative firm of the economy has a production function of the form F(K, N) = AKO.5N 0.5 The marginal product of labor is then given by MP N = 0.5AK0.5N –0.5 . The current capital 3. stock is K = 40. (a) Holding fixed capital at 40, draw a graph of output as a function of labor. What are some important features of this graph? (b) If A = 4 what is the Labor Demand Curve, ND(w), as a function of the real wage w? (c) Suppose labor is supplied inelastically with NS(w)= 10. What is the equilibrium wage w, employment level N, and full employment output Y ? (d) Suppose that productivity unexpectedly increases to A = 6. What is the new equilibrium wage w, employment level N, and full employment output Y ? (e) Depict this change graphically. Denote the original labor supply and demand curves by NS and ND respectively. Denote the new labor demand curve after the productivity shock by ND' . (Does not need to be to scale).
Consider the following Cobb-Douglas production function: Y = 10L04K04. Suppose that the price of labor is w = 20 and the price of capital ifr = 40. %3D a- Derive the total cost curve for this production function. b- Derive the marginal cost curve for this production function. c- Plot the marginal and total cost curves for q = 1, 2, 3, 4, 5. What does it tell you about economies of scale for the production function? d- Suppose that the wage rate went up from 20 to 30. What would happen to the total wage bill relative to total costs, wL ? What would happen to total costs? wL+rK
Being driven by the high profit from digital asset trading, Doggycoin, Sulfikar has decided to set-up his own digital mining company. Let’s assume that two main essential inputs in digital asset production be the high-quality graphic card (C) and electricity (E). After observing the business for few months, he finds the production function as Cobb-Douglas technology as seen in image: Where q is continuous, strictly increasing, and strictly quasi-concave and q(0) = 0. Answer the following questions: a. If the prices of graphic card and electricity denoted by c and e, respectively, and we assume the cost-minimizing behavior. Solve for C and E in their prices and output level, and what is the total cost function C(e, c, q)? b. What are the conditional demands for input C and E?