Question

PLEASE SHOW ME HOW THESE PROBLEMS ARE SOLVED

STEP BY STEP WITH CLEAR EXPLANATION WOULD BE HIGHLY APPRECIATED!
1. For each of the following production functions, decide if (and EXPLAIN why) it describes a constant, increasing or decreasing returns to scale technology (a) f(K,L) = 0.520.8 (b) f(K, L) = 5(VK + 0) 2. Assume that a competitive market is populated by many identical firms producing a homogenous output good. If their common) cost function is c(q) = 36 +249 + qand the current market price of the output is $40 then: a. How much would each firm produce at this price? Write down the firm's profit function and explain how you find the optimal quantity. b. Is this price stable in the long-run? Why or why not? If not, what is the long-run equilibrium price in this market? C. Assume now that the market price is p= $36 and the government decides to tax producers at a rate of $2 per unit of output produced. What will be the short-run consequence of this taxation? What will happen in the long-run if the tax is maintained?

Answer 1

1.
Let K = tK and L = tL where t is a constant and t > 1.
a. f(K,L) = K^0.5L^0.8
f(tK,tL) = (tK)^0.5(tL)^0.8 = t^0.5K^0.5t^0.8L^0.8 = t^0.5+0.8K^0.5L^0.8 = t^1.3f(K,L)
So, power of t is 1.3 which is greater than 1. Thus, there are increasing returns to scale.

b. f(K,L) = 5(K^1/2 + L^1/2)³
f(tK,tL) = 5[(tK)^1/2 + (tL)^1/2]³ = 5(t^1/2K^1/2 + t^1/2L^1/2)³ = 5[t^1/2(K^1/2 + L^1/2)]³ = t^3/25(K^1/2 + L^1/2)³ = t^1.5f(K,L)
So, power of t is 1.5 which is greater than 1. Thus, there are increasing returns to scale.

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