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q3,4

Suppose that a firm employs two inputs, Capital (K) and Labour (L), to produce its final output (Q). This firm pays the price of capital ( $r$ for each unit of capital it employs and pays the price of labour (w) for each unit of labour it employs. It sells its final product at a price of $p$. Assume that we are in the long-run in this question, and so firms choose L \& K. Assume also that this firm is a price-taker. The production function for this firm is given by: $Q=F(K,L)=(LK)_{2}$ 1) Draw the isoquant for this Firm's production technology for $Q=100$ ? Do the same for $Q$ $=200$. Label all $x$-intercepts and $y$-intercepts, and also find and label two interior points for each isoquant. [2 points] 2) Does this production function exhibit diminishing returns for each of its inputs? Explain your answer using calculus. [ 1 point] 3) Does this production function exhibit Increasing Returns to Scale (IRS), Constant Returns to Scale (CRS), or Decreasing Returns to Scale (DRS)? Explain your answer with an example. [2 points] Now assume going forward that the price of labour was equal to $$20$ per unit and the price of capital was equal to $$10$ per unit. 4) Using the LaGrangian, solve for this firm's cost minimizing choices of $K$ and $L$ for a given and general quantity (Q). [4 points] 5) What is the cost minimizing choice of this firm for $Q=100$ ? [ 1 point] 6) Write down a function for the cost of production for the cost-minimizing choices of both $L&K$ as a function of $Q$ (i.e. the optimized cost of production for a given quantity). [3 points]

(Question 3)To test the returns to scale, we take an example as fo

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