Problem 5 Let t€ (0.1) be a labour tax rate paid by consumer, so the labour income is now (1-t)wL. Assume that production function is Cobb-Douglas Y = AKL with A = 1, K = 1 due to proper normalization. Preferences of agent are given by: U(C.L)=aC- 1. Derive labour supply function. 2. Derive labour demand function. 3. Find equilibrium wage rate w* and equilibrium labour L*. 4. Write tax revenue as a function of tax rate. Find optimal tax rate and sketch the graph for revenue.

Suppose that the following production finetion is given: Q= =D4KL For the above production function, find the elasticity of substitution? Find the Retams to Scale. Using the above production funetion: find the labor demand and capital demand as functions of output (Q). price of labor (w) and price of capital (r). Does the Law of Demand hold for each inpur? Are these inputs normal or infenior inputs in the production process? Are inputs complements or substitutes? Why? Find the cost function for the above production finction. Verify the properties of the cost function. Suppose that a fim wants to produce 144 units of output and w-1, r-l. Find long run total cost. Suppose now that wage goes up to 4. Find the new long run total cost. Does fim substitute capital for labor? What is the percentage of cost saving relative to the case where firm is not able to substitute? Suppose that w=1, r-1 and a fimm has fixed amount of capital K 16 in short nun (SR). Find the short run total cost, average…

Suppose we are given a production function: q [al® + (1 – a)kºj a. Please show that MP= (1 – a))P-1 and MP= al b. Please derive the RTS c. Please derive the elasticity of substitution o. d. Does this production function exhibit constant, increasing, or decreasing returns to scale?

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Consider the following production function: x = f0.K) = A e°k where xis the output e is the labour input, kis the capital input, and A, bare positive constants. (a) Set up the cost minimization problem and solve for the first order conditions using the Lagrange Method. Let wbe the wage rate and rthe rental rate of capital. (b) Using your answer in (a), find how much labour and capital would the firm use to produce x outputs, given wage rate wand rental rate of capital ? what are these functions called? (e) Using your answer in (b), find the minimum cost it takes to produce xoutputs, What is this function called?

Suppose the Cobb-Douglas production function is given as: , where b0 = level of technology, b1 = proportion with which labor is used in production, b2 = proportion with which capital is used in production. Then, a. Find marginal product of factors (L and K). b. Compute marginal rate of substitutions (MRSLK and MRSKL). c. Find the elasticity of substitution for this production function. d. Compute elasticity of output with respect to labor and capital.

Answer the Constrained Optimization: Cobb-Douglas Production Function:3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product of Capital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combination of K and L that the firm should use to produce the maximum possible output. Do not solve the problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces

dien There are two factors in a production function y = x 113 x₂¹1³. The market price of each unit ofy is p=3, and the factor prices are w₁=1 and W₂=2 for x₁ and X₂ respectively. variable (a). Calculate your cost function as a function of y if X, and x₂ are both barible factors. (b). Now derive the functions of average and marginal cost and plot them against quantity (C). Solve for your optional output of y. Calculate the ratio of two factors (X₁/X₂) (d). In the short run the fixed factor is set at X₂=1. What is the new optimal output level now?

Q5 solution needed