Economics Assignment-

Q1. In the Shapiro-Stiglitz model, what happens to the equilibrium level of unemployment and real wage in each of the following scenarios? Please provide a diagram and 1-2 sentences of explanation for each part.

(a) Workers become less patient, that is, their time discount rate rises

(b) The pace of technological change increases, so that the job destruction rate increases

(c) The labor force grows (at a fixed capital stock)

(d) The economy becomes more productive

(e) Unemployment insurance is introduced

(f) Firms get better at detecting shirking

Q2. Consider the following equation in the Shapiro-Stiglitz model:

ρV_U = a(V_E - V_U)

(a) During a small time interval Δt, what is the first-order approximation of the probability of an unemployed worker becoming employed?

(b) During a small time interval Δt, what is the first-order approximation of the probability of an unemployed worker becoming employed and then becoming unemployed again?

(c) Suppose workers were paid unemployment insurance u^- per unit time in unemployment. How would the above equation change?

(d) Suppose instead that within a small time interval Δt, there is probability xΔt that the worker exits the labor force entirely and receives 0 payment thereafter. How does the above equation change?

Q3. Consider a simplified version of the baseline real business cycle model with no depreciation and no growth. Justify (with both words and equations) each of the following Romer quotes. Each comes from his argument that depreciation causes investment and employment to be more responsive to shocks.

(a) "Investment is zero in the absence of shocks."

(b) "A positive technology shock [raises] marginal product of capital...next period"

(c) "[Rising] marginal product of capital in the next period makes it optimal for households to undertake some investment"

(d) "Temporarily high saving means expected consumption growth is higher than it would be with a constant saving rate"

(e) "Expected consumption growth requires the expected interest rate to be higher"

(f) "Higher interest rate increases current labor supply"

Q4. Please concisely answer each of the following questions.

(a) If we were to assume total depreciation and no government in the baseline RBC model, what would we conclude about the responsiveness of investment relative to consumption during business cycles?

(b) What must be true if the baseline RBC model generates a positive and significant autocorrelation in output growth? (Actual output has an autocorrelation of about 1/3) Why is this considered a limitation of the baseline RBC model?

Q5. Log-linearize each of the following equations:

• y_t = k_t^α
• y_t = c_t + i_t
• k_t+1 = (1 - δ)k_t + i_t
• w_t = (1 - α)k_t^α
• rt = αy_t/k_t - δ
• bc_tl_t = (1 - l_t)w_t
• 1/ct = e^-ρE_t[(1 + r_t+1)/c_t+1]

For the steady state where all of the above variables are constant, derive each of the following as a function of parameters:

• c*/y*
• i*/k*
• αy*/(r*k*)