Find the optimum level of capital that generates the lowest cost of producing 178 units of output if output is produced according to the Cobb-Douglas Q = 10*(K^0.4)*(L^(1-0.4)), and r = 9 and w = 7.
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- A firm's production function can be written as:Q = 5LKwhere Q represents output per day. The unit costs of inputs are $150 for labor (L) and$1,000 for capital (K). Determine the least cost combination of L and K when output isproduced at the rate of 1,000 tons per day. Determine the required outlay for 1,000 tons perday.Determine the combination of input factors K, L that minimizes the cost of producing Q. Q(K, L) = KL², r = 5, w = 10Consider the following production function:q = (KL)^α, where α > 0. (a) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale.(b) Derive an expression for the cost function of a firm using the production function to produce output of a good.(c) Find the first and second partial derivatives of the cost function with respect to q. Interpret the second partial derivative and relate the sign of the derivative to the returns to scale.
- Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure,where K(t) =2t^4 + 3t + 149 /t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time?The following production function, q (L.) falls stays constant rises (L,K) = — KVALV² + — K increasing returns to scale constant returns to scale decreasing returns to scale will save this response. will have an average cost function that ✓ because it hasx+2 (1) Find the rate of change of the function f(x) (ii) The number of units Q of a particular commodity that will be produced with K with respect to x when x = 1. 1- 8x thousand dollars of capital expenditure is modeled by Q(K) = 500 Kố. Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure, where 2t* + 3t + 149 K(t) t+2 (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production be increasing or decreasing at this time?
- You have been hired as a consultant to advise KatPillar's COO (Chief Operations Officer). Your first task was to estimate the production function for the firm., which is: Q=K0.25L0.75 The marketing department is forecasting that, in the coming year, the price of K is $4 per unit, and the price of L is $6 per unit. You are now asked to calculate the optimal capital/labor ratio.The following production function has been estimated for wheat: W = 4 F0.5 L0.25 where, W = Wheat yield (kg/ha), F = Fertiliser (kg/ha) and L = Labour (hours/ha) a) Derive the expression for the least-cost expansion path given that fertilizers cost R12 per kg and labour cost R6 per hour b) If the wheat producer has a budget of R9 000 to spend on fertiliser and labour, how much of each input should he use and what is the quantity of wheat he will produce? c) Derive expressions for the value of marginal product (VMP) of fertiliser and labour given that wheat sells at R72 per kg.Do the following production functions exhibit increasing, constant, or decreasing returns to scale in K and L (assuming A is fixed)? (a) Y=AK^⅓ L^⅓ (b) Y=AK^⅔ L^
- firm's average cost (AC) per unit of output depends on the number of hours of skilled labor (S) and the number of hours of unskilled labor (U) employed in the production process. The firm's AC function is defined below. Determine the quantities of the two inputs that will result in production at the lowest possible average cost and determine the associated minimum average cost. AC-3S2+2U2-2SU-2S-6U+ 20What is the maximum profit when the the profit-versus-production function is given below? P is profit and x is unit of production P = 200,000 – 1x- (1.1/(X+1)) to the 8th power?A company makes solar panels. The company’s revenue function, in dollars, is R(n)=10n , where n is the number of panels produced. The cost function is C(n)=100(2)π/30 . R and C are shown on the graphbelow. a) Estimate from the graph i) the break-even points ii) the number of panels that should be produced for maximum profit b) Write the equation for the profit function P. c) Graph P. d) Use your graph of P to estimate the number of panels that give maximum profit. e) How would your answers for break-even points and maximum profit change if i) the number of dollars of revenue per panel is increased slightly? ii) the cost function is C(n)=100(2)π/35 changed to ? f) What does the number that was changed in part e) ii) represent?