1 Log Utility

Suppose you have the following utility function of a consumer

where C is consumption, l is leisure and α is some preference parameter that tells you how much you like leisure compared to consumption. The maximum time available for work and leisure is 1.

1. Set l = 0:5; α = 0:3; and plot the utility of consumption over the range C 2 [0; 5]

2. Set C = 4; α = 0:3; and plot the utility of leisure over the range l 2 [0; 1]

3. Derive this function w.r.t. C. Set l = 0:5, α = 0:3; and plot the marginal utility of consumption for C 2 [0; 5]

4. Derive this function w.r.t. l. Set C = 4;  α = 0:3; and plot the marginal utility of leisure for l 2 [0; 1]

2 Utility

Suppose you have the following utility function of a consumer

where C is consumption, l is leisure and α is some preference parameter that tells you how much you like leisure compared to consumption. The maximum time available for work and leisure is 1.

1. Set l = 0.5, α = 0.3, and plot the utility of consumption over the range C ∈ [0,5]

2. Set C = 4, α =0.3, and plot the utility of leisure over the range l ∈ [0,1] 3. Derive this function w.r.t. C. Set l = 0.5, α = 0.3, and plot the marginal utility of consumption for C ∈[0,5]

4. Derive this function w.r.t. l. Set C = 4, α = 0.3, and plot the marginal utility of leisure for l ∈ [0,1]

3 Linear Production Function

Suppose a firm uses the following production function

where Y is output, z is TFP (i.e., technology) and Nd is labor.

1. Set z = 2 and plot the production function over the range Nd ∈ [0,5]

2. Derive this function w.r.t. Nd. Set z = 0.5 and plot the marginal product of labor (MPN) over the range Nd ∈ [0,5]

4 Cobb-Douglas Production Function

Suppose a firm uses the following production function

where Y is output, z is TFP, K is capital, and Nd is labor.

1. Set z = 10, K =5 and plot the production function over the range Nd ∈ [0,5]

2. Derive this function w.r.t. Nd.Set z = 10, K = 5, and plot the marginal product of labor (MPN) over the range Nd ∈ [0,5]

3. What does the MPN measure. Explain in words.