# 1. A production function is given by f(K, L) = L/2+ v K. Given this form,...

1. A production function is given by f(K, L) = L/2+ v K. Given this form, MPL = 1/2 and MPK-2 K (a) Are there constant returns to scale, decreasing returns to scale, or increasing returns to scale? (b) In the short run, capital is fixed at -4 while labor is variable. On the same graph, draw the 2. A production function is f(LK)-(L" + Ka)", where a > 0 and b > 0, For what values of a and b are 3. Firm A has a production function fA(L, K) = min(2L, L + K). Carefully graph the isoquants associated Explain why. short run marginal product and average product curves. there constant returns to scale? Increasing returns to scale? with production levels of Q- 10 and returns to scale? 20. Does this exhibit constant, increasing, or decreasing 4. A firm has the production function f(L, K)-vL+ K. Given this, MPL = 2 T and MPK = 2K. (a) Show that this function does not satisfy the definition of either increasing, constant, or decreasing returns to scale. (Hint: Show that there are some values of L and K for which this will show increasing or decreasing returns to scale.)

(1)

Q = f(K, L) = (L/2) + (K)0.5

(a)

When both inputs are doubled, new production function becomes

Q* = (2L/2) + (2K)0.5 = (2L/2) + (2)0.5 x (K)0.5

The first component of production function (L/2) is exactly doubled, but since (2)0.5 < 2, the second component of production function is less than doubled. Therefore, output is less than doubled. Since doubling both inputs less than doubles the output, there are decreasing returns to scale.

(b)

When K = 4,

Q = (L/2) + (4)0.5 = (L/2) + 2

Marginal product (MPL) = dQ/dL = 1/2 = 0.5

Average product (APL) = Q/L = (1/2) + (2/L) = 0.5 + 2/L

Data table used for graph:

 L MPL APL 1 0.5 2.5 5 0.5 0.9 10 0.5 0.70 15 0.5 0.63 20 0.5 0.6 25 0.5 0.58 30 0.5 0.57 35 0.5 0.56 40 0.5 0.55 45 0.5 0.54 50 0.5 0.54

Graph:

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