Problem 4.1: Labor market Equilibrium. (30 points) Consider an economy with the following preferences for consumption and labor: 1 . n cJt :10 o 1711+? ( } s 1 + 7 The consumer's budget constraint is c = (1 1') wh where T is a labor tax levied by the govermnent. Suppose there is one rm that operates with technology Y = H\". (There is no capital in the economy). Suppose that L = 1, so aggregate labor supply is simply h. (a) Write the Lagrangean for the household optimization problem and derive the rst-order con- ditions for consumption and hours worked. (b) What is the Frisch elasticity, given by $3363 holding oonsnmptton constant? Hint: Combine the two FOCs to substitute our the Lagrange multiplier and then write hours worked as a function of the wage (and consumption). (c) Derive the labor supply curve. This should be h. as a function of parameters only. (d) What happens to labor supply when the wage increases? Can you provide some economic intuition in terms of income effect and substitution effect? [Hintz logo is akin to setting (I = 1 in the preferences from the lecture slides.] (e) Set 11p the firm's prot maximization problem and fin the labor demand curve (f) Find the equilibrium wage rate and level of employment