Assume a production economy with a finite supply of capital and labor and two fi

Question

Assume a production economy with a finite supply of capital and labor and two firms that are perfect substitutes. (So that the price of the goods they sell are equal to one.) Let each firm have technology f = K1/5 L4/5. Assume an equilibrium wage of 1 and a rental rate of capital of 2. Compute each firm's equilibrium marginal products of capital and labor. Define production efficiency. Does this economy satisfy production efficiency? Consider a slight modification. Assume that firm one must pay a tax of one dollar on each unit of labor and capital it uses. Repeat the above exercises. Does this economy satisfy production efficiency? Suppose instead the exponent on labor was equal to 3/5 so that the sum of the exponents was less than one. Repeat the above exercises. What would be necessary for there to be production efficiency in this economy. Without a formal analysis, what would production efficiency require if the exponent on L was equal to one? Consider a different modification. Assume firm one pays a tax of one unit on the labor it uses but not on capital. Does the economy with the original production function satisfy production efficiency. Why or why not?

Explanation / Answer

Production process involves the transformation of inputs into output. The inputs could be land, labour, capital, entrepreneurship etc. and the output could be goods or services. In a production process managers take four types of decisions: whether to produce or not, how much output to produce, what input combination to use, and what type of technology to use.

The tangency condition implies [L 1/2 + K 1/2] K-(1/2)/r = [L 1/2 + K 1/2] L-(1/2)/w

1/r^K = 1/w^L where ^ implies underoot

w^L=r^K .. K/L = w square/r square...Given that w=10 and r=1, this implies 100=K/L or 100L=K

Returning to the production function and assuming Q=  121,000 yields

121,000 = [L 1/2 + K 1/2]^2

121,000 = [L 1/2 + (100L)1/2]^2

121,000 =   [L 1/2 + 10L 1/2]^2

121,000 = 11[L 1/2]^2

121,000 = 121L

L= 1000 Since K= 100 L and K = 100 (1000) = The cost minimizing quantities of capital and labour too produce 121,000 X is K = 100,000 and L = 1,000,

Production efficiency is an economic level at which the economy can no longer produce additional amounts of a good without lowering the production level of another product. This will happen when an economy is operating along its production possibility frontier.The ability to produce a good using the fewest resources possible. Efficient production is achieved when a product is created at its lowest average total cost. Yes.

ii) No in this case the economy doesnt satisfy production efficieny.

If a firm has a monopoly in the output market and is a monopsony in the labor market, its profit is where Q(L) is the production function, p(Q)Q is its revenue, and w(L)L—the wage times the number of workers—is its cost of production. The firm maximizes its profit by setting the derivative of profit with respect to labor equal to zero (if the secondorder condition holds): Rearranging terms in the first-order condition, we find that the maximization condition is that the marginal revenue product of labor, equals the marginal expenditure, where is the elasticity of demand in the output market and is the supply elasticity of labor.

No, the economy with original production function does not satisfy production efficiency because the firm in the question is not paying tax on capital.

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