Assume a consumer gains utility over consumption (C) and leisure (R). The utility function is defined as u(C,R)=36R+CR-R^2. the wage the consumer faces is 'w' and the price of consumption is normalized to one. Suppose we are only examining one day, where, if the consumer chooses, he can work the entire day (24 hrs). The amount the consumer works is defined as L=24-R, where 'L' is labour.
a) Derive the consumer's budget constraint and its choice variables (ie. the total amount of hours worked and leisure as a function of wage)
b) Derive the consumer's marginal rate of substitution dC/dR
c) Determine how the optimal amount of hours the consumer works changes over wage dR*/dw. Explain your results.