Question

A firm uses two inputs x1 and x2 to produce output y. The production function is given by f(x1, x2) = p min{2x1, x2}. The price of input 1 is 1 and the price of input 2 is 2. The price of output is 10.

4. A firm uses two inputs 21 and 22 to produce output y. The production function is given by f(x1, x2) = V min{2x1, x2}. The price of input 1 is 1 and the price of input 2 is 2. The price of output is 10. (a) Calculate Marginal Products for each input. Are they increasing or decreasing? (b) Calculate Marginal Rate of Technical Substitution. Is it increasing or diminishing? (c) Draw two or more isoquants (Your drawing may not be exact, but it needs to display the key features of technology such as convexity/concavity accurately). (d) Does this production function exhibit increasing, decreasing or constant returns to scale? (e) Solve the firm's cost minimization problem. Derive the firm's cost function c(y). (f) Find the profit maximizing choice of inputs x and x and output y*. Calculate the firm's profit from these choices.

Answer 1

f(11,12)= min (2:11,12)

a) Marginal Products

MPA - {oifai 2 ngandza, "aifa <a

MP - - {oifei 5 randza, "Pifo >

Marginal Products are neither increasing or decreasing when the inputs are used in fixed proportions. More of one input does not increase the total production.

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b)

MRTS = MPI,

=

{0if(1 > czandocifci <12}

MRTS is not defined at the point of kink where 2x₁ = x_2,

MRTS is strictly not decreasing because the goods are always consumed in proportion. More of x₁ along x-axis with x₂ kept constant does not actually increases the production. Similarly, More of x₂ along y-axis with x₁ kept constant does not actually increases the production. MRTS changes from $\infty$ to 0.

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c)

2X1 = X2 Y2= 4 II Y1 = 2 2 8

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d) Returns to scale

f(t.11, tr2) = V min(t2.11, tx2)

, where t>1

f(t. ct.c) = /min (2.61, 22) <t/min (2:21,22)

Hence, the production function exhibits decreasing returns to scale.

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