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.

A firm's production function is given by 1/2 1/2 y = 10x1x2 where x₁ indicates the input labor and x₂ indicates the input capital. (a) Compute the partial derivatives of the production function. Provide an economic interpretation of each derivative. (b) When an amount of capital available is fixed by x2 = 9, compute the par- tial derivative with respect to ₁ and sketch its graph. Determine whether the derivative is monotone increasing or decreasing, and provide an inter- pretation.

Production Function and Input Demand  Consider the production function Q = K + √L. Derive the inputdemand curves for L and K, as a function of the output Q, inputprices w (price of labor services) and r (price of capitalservices). Does the production function exhibit IRS, CRS orDRS?

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…